A Theory of Equivalent Expectation Measures for Expected Prices of Contingent Claims
AA Theory of Equivalent Expectation Measures forExpected Prices of Contingent Claims
Sanjay K. Nawalkha and Xiaoyang Zhuo ∗ June 26, 2020
Abstract
This paper introduces a theory of equivalent expectation measures , such as the R measure and the R T measure, generalizing the martingale pricing theory of Harrisonand Kreps (1979) for deriving analytical solutions of expected prices—both the expectedcurrent price and the expected future price—of contingent claims. We also present new R -transforms which extend the Q -transforms of Bakshi and Madan (2000) and Duffieet al. (2000), for computing the expected prices of a variety of standard and exotic claimsunder a broad range of stochastic processes. Finally, as a generalization of Breedenand Litzenberger (1978), we propose a new concept of the expected future state pricedensity which allows the estimation of the expected future prices of complex Europeancontingent claims as well as the physical density of the underlying asset’s future price,using the current prices and only the first return moment of standard European OTMcall and put options. Keywords:
Equivalent expectation measure theory, R measure, R T measure, Risk-neutral measure, Forward measure, Contingent claims, Expected returns ∗ Sanjay K. Nawalkha (e-mail: [email protected]) is with the Isenberg School of Management,University of Massachusetts, MA, USA. Xiaoyang Zhuo (e-mail: [email protected]) is with thePBC School of Finance, Tsinghua University, Beijing, China. We gratefully acknowledge the comments andfeedback of Fousseni Chabi-Yo, Darrell Duffie, Jens Jackwerth, Nikunj Kapadia, Hossein Kazemi, LongminWang, Yongjin Wang, and Hao Zhou. This version is available at the arXiv, Open Science Framework (OSF),and SSRN. a r X i v : . [ q -f i n . P R ] J u l ontents R ∗ and R ∗ Measures . . . . . . . . . . . . . . . . . . . . . . 132.4 A Classification of the Equivalent Expectation Measures . . . . . . . . . . . 15 R Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1.1 The Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . . . 173.1.2 The Affine Term Structure Models . . . . . . . . . . . . . . . . . . . 193.2 The R T Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.1 The Geman and Jamshidian Models . . . . . . . . . . . . . . . . . . 253.2.2 The Collin-Dufresne and Goldstein Model . . . . . . . . . . . . . . . 26 R -Transforms 28 R -Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 The Extended R -Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 Applications of the R -Transforms . . . . . . . . . . . . . . . . . . . . . . . . 344.3.1 Stochastic Volatility Jump-Based Equity Option Model . . . . . . . . 354.3.2 Expected Option Return Simulation . . . . . . . . . . . . . . . . . . 404.3.3 Other Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 A The Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52A.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 521.2 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 54A.3 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 55A.4 Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 55A.5 Proof of the Black-Scholes Model . . . . . . . . . . . . . . . . . . . . 56A.6 Proof of Proposition 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 58A.7 Proof of Proposition 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 59A.8 Proof of Proposition 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 60A.9 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61B Equivalent Expectation Measures for Multidimensional Stochastic Processes 62B.1 The R Measure for Multidimensional Brownian Motions . . . . . . . 62B.2 The R T Measure for Multidimensional Brownian Motions . . . . . . . 64B.3 The R Measure for L´evy Jump Processes . . . . . . . . . . . . . . . . 65
References 71Figures 79Tables 80A Internet Appendix 1
IA The R ∗ Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1IB Expected Option Prices under the Margrabe Model . . . . . . . . . . . . . . 2IC Expected Bond Prices under the A r (3) Model . . . . . . . . . . . . . . . . . 3ID Expected Bond Prices under the Quadratic Term Structure Model . . . . . . 6IE Expected Prices of Interest Rate Derivatives . . . . . . . . . . . . . . . . . . 9IF Expected Option Prices under the CGMY L´evy Model . . . . . . . . . . . . 15IG Expected Return Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 18IH A Procedure to Extract the Expected FSPD . . . . . . . . . . . . . . . . . . 22 Figures 24Tables 30
Introduction
This paper introduces a theory of equivalent expectation measures generalizing the martingalepricing theory of Harrison and Kreps (1979) for deriving analytical solutions of expected prices—both the expected current price and the expected future price—of contingent claims.The martingale pricing theory (MPT) of Harrison and Kreps (1979) and Harrison and Pliska(1981) has revolutionized the field of financial valuation. This theory allows the applicationof equivalent martingale measures (EMMs)—such as the risk-neutral measure Q , and theforward measure Q T —to different types of contingent claims and alternative stochasticprocesses, extending the option pricing models of Black and Scholes (1973) and Merton(1973) and formalizing the risk-neutral valuation approach pioneered by Cox and Ross (1976).Under the MPT, every EMM is associated with a specific numeraire asset. By the firstfundamental theorem of asset pricing, the absence of arbitrage is equivalent to the existenceof at least one EMM under which the price of any asset discounted by the price of thenumeraire asset is a martingale. The given EMM is also unique if the markets are completeby the second fundamental theorem of asset pricing. The risk-neutral measure Q and the forward measure Q T are the most widely used EMMsin the financial markets. Extending Heston’s (1993) Fourier inversion framework, Bakshi andMadan (2000) and Duffie et al. (2000) develop multivariate transforms based upon the Q measure for pricing a wide variety of contingent claims in incomplete markets with stochasticvolatility and jump-based price processes. The analytical and numerical flexibility providedby these EMMs and the transforms have led to a plethora of valuation models used in theTreasury bond market, the corporate bond market, and the financial derivatives market allover the world. The combined market value of these security markets in which these EMMsare routinely used is significantly greater than the world equity market.Despite the ubiquitous applications of the Q and Q T measures (and the associated trans-forms based upon the Q measure) for the valuation of Treasury bonds, corporate bonds, andfinancial derivatives, an important gap remains in the understanding of the returns of thesefinancial securities. For example, consider the following questions from the perspective ofa buy-side investment firm: What is the expected return of a 10-year Treasury bond overthe next two months under the A (3) affine model of Dai and Singleton (2000)? What isthe expected return of a 5-year, A-rated corporate bond over the next quarter under the See Jamshidian (1989); Geman et al. (1995). The origins of the first fundamental theorem of asset pricing can be traced to Ross (1977, 1978), beforeits formal development by Harrison and Kreps (1979). The second fundamental theorem of asset pricing isgiven by Harrison and Pliska (1981). See also Artzner and Heath (1995), B¨attig et al. (1999), and Battigand Jarrow (1999). expected future prices of these securities—which are necessary inputs for computing expectedreturns—over an arbitrary finite horizon H , do not exist in the finance literature.Estimating expected returns becomes difficult, if not impossible, for most buy-side firmsand other buy-and-hold investors—who do not continuously rebalance and hedge theirportfolios—without the analytical solutions of expected future prices of the financial securi-ties in their portfolios. This problem is faced not just by sophisticated hedge fund managerswho hold complex contingent claims, but also by numerous asset managers at mutual funds,banks, insurance companies, and pension funds, who hold the most basic financial securities,such as the U.S. Treasury bonds and corporate bonds. In an insightful study, Becker andIvashina (2015) illustrate this problem in the context of insurance companies, whose assetmanagers reach for higher (promised) yields, and not necessarily higher expected returns.They find that the “yield-centered” holdings of insurance companies are related to the busi-ness cycle, being most pronounced during economic expansions. More generally, it is commonto use metrics based upon yield, rating, sector/industry, optionality, and default probability,to make portfolio holding decisions by fixed income asset managers. Since expected returnsare not modeled explicitly , these metrics provide at best only rough approximations, and atworst counter-intuitive guidance about the ex-ante expected returns as shown by Becker andIvashina (2015). This is quite unlike the equity markets, where much effort is spent bothby the academics and the portfolio managers for obtaining the conditional estimates of theexpected returns on stocks. Part of this widespread problem in the fixed income markets andthe financial derivatives markets is due to the lack of a simple and parsimonious frameworkthat can guide the conditional estimation of finite-horizon expected returns. This paper fillsthis gap in the finance theory.The main contribution of this paper is to provide a general framework for the derivationof an analytical solution of the expected price (both the current price and the expected futureprice) of a contingent claim under virtually any model which admits an analytical solutionto the claim’s current price using an equivalent martingale measure. To the extent theexpected future price of a contingent claim is a sufficient input for computing its expectedreturn—which is true for many or most contingent claims—the results of this paper can beused for computing the expected return of the claim over any finite horizon less than or2qual to the expiration/maturity date of the claim. When the expected future price is onlya necessary but not a sufficient input for computing the expected return, the results of thispaper would have to be extended to compute the expected return of the claim. We considersome examples of how such extensions can be made in section 6 of this paper. We will usethe expressions “expected future price” and “expected price” interchangeably to mean thetime t expectation of the future price of the contingent claim at time H , where t ≤ H .In order to derive the analytical solution of the expected price of a contingent claim,our theoretical framework requires the construction of a new class of equivalent probabilitymeasures called the “equivalent expectation measures (EEMs)”, as generalizations of theEMMs of Harrison and Kreps (1979). We find that multiple classes of EEMs can be derived,out of which the EEMs in two classes, given as the R ∗ class and the R ∗ class, are the mostuseful. Section 2 develops a formal theory of equivalent expectations measures, focusingspecifically on the EEMs in the R ∗ class and the R ∗ class. We discuss the main propertiesof the EEMs in these two classes and provide a finer classification of the EEMs withineach class based upon some commonly used numeraires. We identify three specific EEMswithin each of these two classes, corresponding to the three numeraires given as the moneymarket account, the T -maturity pure discount bond, and a general traded asset, respectively,resulting in a total of six EEMs. Out of these six EEMs, we find that only three EEMs areuseful for obtaining the analytical solutions of the expected prices of contingent claims, givenas follows:i) the R measure from the R ∗ class, corresponding to the risk-neutral measure Q of Black-Scholes-Merton (Black and Scholes, 1973; Merton, 1973), Cox and Ross (1976), and Harrisonand Kreps (1979);ii) the R T measure from the R ∗ class, corresponding to the forward measure Q T of Geman(1989) and Jamshidian (1989), and Geman et al. (1995);iii) the R S measure from the R ∗ class, corresponding to the generalized Q S measure ofGeman et al. (1995).The three EEMs, R , R T , and R S , can be used to derive analytical solutions of theexpected prices of contingent claims that are priced under the following models: the equityoption model of Black-Scholes-Merton; the corporate debt pricing models of Merton (1974),Black and Cox (1976), Leland and Toft (1996), Jarrow et al. (1997), and Collin-Dufresne andGoldstein (2001); the affine term structure models of Dai and Singleton (2000, 2002) andCollin-Dufresne et al. (2008); the quadratic term structure models of Ahn et al. (2002) andLeippold and Wu (2003); the selected Heath et al. (1992) forward rate-based term structuremodels with closed-form bond option formulas; the credit default swap pricing model ofLongstaff et al. (2005); the VIX futures and the variance swaps models of Dew-Becker et al.32017), Eraker and Wu (2017), Johnson (2017), Cheng (2019), and Hu and Jacobs (2020a); the exchange option pricing model of Margrabe (1978); and the currency option pricingmodels of Grabbe (1983) and Amin and Jarrow (1991), among others. Section 3 and theInternet Appendix (sections IB, IC, and ID) obtain the analytical solutions of the expectedprices of contingent claims using some of the above models.Section 4 applies the EEM theory to obtain the expected prices of a wide variety ofcontingent claims that are priced by the transform-based methods of Bakshi and Madan(2000), Duffie et al. (2000), and Chacko and Das (2002). Unlike the traditional contingentclaim models considered in section 3, which require different numeraires—the selection ofwhich depends upon the type of payoff structure and whether the short rate is constantor stochastic—for obtaining the analytical solutions of the expected prices of the claims,the transform-based methods can obtain the analytical solutions of the expected prices ofthe claims using virtually any numeraire, with enough flexibility built in the form of thetransform itself. Since the money market account is the most natural numeraire, we usethis numeraire and the corresponding EEM R from the R ∗ class to derive the R -transformsas extensions of the “ Q -transforms” of Bakshi and Madan (2000), Duffie et al. (2000), andChacko and Das (2002). The R -transforms nest the corresponding Q -transforms for thespecial case when t = H .The R -transforms can be used to derive analytical solutions of the expected prices ofcontingent claims that are priced under the following models: the affine option pricingmodels of Heston (1993), Bates (1996, 2000), Bakshi et al. (1997), Pan (2002), Bakshi andMadan (2000), Duffie et al. (2000), and Chacko and Das (2002); the L´evy option pricingmodels of Carr et al. (2002) and Carr and Wu (2003); the bond option pricing models inthe maximal affine classes by Dai and Singleton (2000, 2002), Collin-Dufresne et al. (2008),Collin-Dufresne et al. (2009), and in the maximal quadratic class by Ahn et al. (2002), amongothers. Section 4.3 and Internet Appendix (sections IE, IF) obtain the analytical solutionsof the expected prices of equity options and a variety of interest rate derivatives using someof the above models.We wish to underscore an important point regarding the derivations of the expected pricesof different contingent claims in sections 3 and 4.3. The derivation of the analytical solutionof a contingent claim’s expected price at a finite horizon date H , using an EEM, requires Eraker and Wu (2017), Cheng (2019), and Hu and Jacobs (2020a) consider only VIX futures and notvariance swaps. Dew-Becker et al. (2017) and Johnson (2017) consider both VIX futures and variance swaps. We label the transforms of Bakshi and Madan (2000), Duffie et al. (2000), and Chacko and Das (2002)as “ Q -transforms” because all of these transforms are derived to compute the current prices of contingentclaims under Q , and to distinguish these transforms from the “ R -transforms” presented in section 4 of thispaper. H , usingan R -transform requires about the same theoretical effort as required for the derivation of theanalytical solution of the contingent claim’s current price using the associated Q -transform.Moreover, the analytical solution of the expected price is similar to the analytical solutionof the current price, with a few extra parameters related to the market prices of risk andthe time H parameter. Since the EEMs and the R -transforms provide a single analyticalsolution for the expected price (which includes both the current price for H = t , and theexpected future price for all H > t ), and since a given EEM and a given R -transform alwaysnest the associated EMM and the associated Q -transform, respectively, for the special casewhen t = H , all future work on the derivations of the analytical solutions of the expectedprices of contingent claims can use either the EEMs or the R -transforms instead of usingthe associated EMMs or the associated Q -transforms, respectively. Since the expected priceis the most relevant input, and in many cases the only input required for computing theexpected return of a contingent claim, the use of the EEMs or the R -transforms kills twobirds with one stone—it allows the derivation of a single analytical solution that does both valuation and computes the expected return of the claim.Section 5 extends the Breeden and Litzenberger (1978) framework to derive some general“no-arbitrage” results related to the EEM theory that do not depend upon the specificparametric assumptions for the underlying return generating process. First, we demonstratethat the second derivative of the expected future price of a standard European call optionwith respect to the strike price gives the expected future state price density (FSPD). Theorem2 in this section shows that the expected FSPD equals the discounted R T probability density,similar to how the state price density or SPD equals the discounted Q T forward probabilitydensity. Second, we demonstrate that the expected FSPD can be obtained from the expectedreturns of standard European call and put options estimated using historical option pricedata, and then can be used for estimating the expected future prices (and expected returns)of a wider range of complex European contingent claims with terminal payoffs that arearbitrary functions of the underlying asset’s future price. Finally, we show that the expectedterminal FSPD estimated using the current prices and only the first return moment ofstandard European OTM call and put options provides an estimate of the physical densityof the future price of the underlying asset. Though Breeden and Litzenberger (1978) do not explicitly differentiate between the risk-neutral measure Q and the forward measure Q T , the above result holds assuming that the state price density strictly means the Arrow-Debreu price density. Consider a contingent claim denoted by F , with a maturity/expiration date T on which theclaim pays out a single contingent cash flow. The cash flow can depend upon the entirehistory of the underlying state variables in arbitrary ways (as in the papers of Harrison andKreps, 1979; Bakshi and Madan, 2000; Duffie et al., 2000; Chacko and Das, 2002), allowinga variety of regular and exotic contingent claims, such as Asian options, barrier options, andother claims, which make a single payment at a fixed terminal date T .We fix a probability space (Ω , F , P ) and a filtration F s , 0 ≤ s ≤ T ≤ S , satisfying theusual conditions, where S is a fixed terminal date and P is the physical probability measure.We assume that the contingent claim is being valued by a unique EMM Q ∗ defined on (Ω , F ),which uses a traded asset G as the numeraire for valuing the claim. The absence of arbitrageguarantees that the process F s /G s is a martingale under Q ∗ for all 0 ≤ s ≤ T (see Harrisonand Kreps, 1979). According to the martingale valuation results (see, e.g., Geman et al.,1995; Bj¨ork, 2009, Theorem 26.2), the current time t price of the claim is given as F t = G t E Q ∗ t (cid:20) F T G T (cid:12)(cid:12)(cid:12) F t (cid:21) . (1)Due to the equivalence of Q ∗ and P , there exists an almost surely positive random variable L ∗ T such that Q ∗ ( A ) = (cid:90) A L ∗ T ( ω )d P ( ω ) for all A ∈ F T , (2)where L ∗ T is the Radon-Nikod´ym derivative of Q ∗ with respect to P , and we can write it as L ∗ T = d Q ∗ d P (cid:12)(cid:12)(cid:12) F T , F s is right-continuous and F contains all the P -negligible events in F . The numeraire G could be the money market account, zero-coupon bond, or any other asset, for instance,stock, commodity, etc. E [ L ∗ T ] = 1. The Radon-Nikod´ym derivative process is defined as L ∗ s = E [ L ∗ T |F s ] , or L ∗ s = d Q ∗ d P (cid:12)(cid:12)(cid:12) F s , for 0 ≤ s ≤ T. (3)Though equation (1) solves the contingent claims valuation problem, it does not addressthe next most important issue for investors, which is computing the expected return of acontingent claim over a given finite horizon H . The analytical solution of both the currentprice and the expected return of the claim requires the analytical solution of the expectedprice of the claim, E t [ F H ], which can be solved as follows, using equation (1): E t [ F H ] = E P t (cid:20) G H E Q ∗ H (cid:20) F T G T (cid:12)(cid:12)(cid:12) F H (cid:21) (cid:12)(cid:12)(cid:12) F t (cid:21) . (4)When t = H , the above expectation gives the current price of the claim. When t < H , theabove expectation gives the expected future price of the claim. The expected future priceand the current price can be used to obtain the expected return of the claim.Surprisingly, analytical solutions of the above equation for different contingent claimmodels do not exist in the finance literature except for two minor exceptions. The firstexception is Rubinstein’s (1984) formula for the expected future price of a European equityoption, which assumes that the underlying asset price follows a geometric Brownian motion.The second exception is computing the expected future price of a contingent claim overthe horizon equal to its maturity/expiration date T , for which the expectation E t [ F T ] canbe solved directly under those models that allow analytical solutions of the claim’s currentprice. The last observation has been used in recent papers (see, e.g., Broadie et al., 2009;Chaudhuri and Schroder, 2015; Hu and Jacobs, 2020b) to derive analytical solutions of theexpected returns of “held-to-expiration” options under stochastic volatility jump models.However, more generally, when the horizon H is neither equal to t , nor equal to T ,equation (4) cannot easily be used to derive analytical solutions of E t [ F H ], under even thosecontingent claim models that admit an analytical solution to the claim’s current price. Themain reason for this is that the law of iterated expectations cannot be used since the innerand the outer expectations are not under the same probability measure in equation (4).The kernel of the main idea underlying this paper is to ensure that the inner and the outerexpectations in equation (4) are under the same probability measure , thereby allowing the useof the law of iterated expectations. To use this law, we propose a new equivalent probabilitymeasure R ∗ in Theorem 1 (see section 2.2), under which the following two conditions aremet for all 0 ≤ t ≤ H ≤ T : C1.
The conditional P probabilities at time t of events occurring at time H are the same as7he corresponding conditional R ∗ probabilities of those events. C2.
The conditional Q ∗ probabilities at time H of events occurring at time T are the sameas the corresponding conditional R ∗ probabilities of those events.If both the above conditions are satisfied, then equation (4) can be simplified usingiterated expectations under the R ∗ measure as follows: E t [ F H ] = E R ∗ t (cid:20) G H E R ∗ H (cid:20) F T G T (cid:12)(cid:12)(cid:12) F H (cid:21) (cid:12)(cid:12)(cid:12) F t (cid:21) = E R ∗ t (cid:20) E R ∗ H (cid:20) G H F T G T (cid:12)(cid:12)(cid:12) F H (cid:21) (cid:12)(cid:12)(cid:12) F t (cid:21) = E R ∗ t (cid:20) G H F T G T (cid:12)(cid:12)(cid:12) F t (cid:21) . (5)We find that equation (5) provides an analytical solution of the expected price of acontingent claim at a finite horizon date H for most but not all contingent claim modelsthat admit an analytical solution to the time t price of the claim. For those models which donot lead to analytical solutions using equation (5), consider separating the single expectationof the expression G H F T /G T into a product of two expectations, as follows: E t [ F H ] = E R ∗ t (cid:2) G H (cid:12)(cid:12) F t (cid:3) E R ∗ t (cid:20) F T G T (cid:12)(cid:12)(cid:12) F t (cid:21) = E P t (cid:2) G H (cid:12)(cid:12) F t (cid:3) E R ∗ t (cid:20) F T G T (cid:12)(cid:12)(cid:12) F t (cid:21) , (6)where for any F T -measurable variable Z T , R ∗ is defined with respect to R ∗ as follows: E R ∗ t [ Z T ] = E R ∗ t (cid:20) G H E R ∗ t [ G H ] · Z T (cid:12)(cid:12)(cid:12) F t (cid:21) . (7)Of course, equation (6) is not the only way to separate the expectation of the expression G H F T /G T in equation (5) into a product of two or more expectations. Another way maybe to separate this expectation into a product of the expectations of G H /G T and F T usinganother equivalent probability measure obtained as yet another transformation of the R ∗ measure, similar to the transformation shown in equation (7). A final way may be to This can be done as follows: Let E t [ F H ] = E R ∗ t (cid:104) G H G T (cid:12)(cid:12)(cid:12) F t (cid:105) E R ∗ t (cid:2) F T (cid:12)(cid:12) F t (cid:3) , where for any F T -measurablevariable Z T , R ∗ is defined with respect to R ∗ as follows: E R ∗ t [ Z T ] = E R ∗ t (cid:20) G H /G T E R ∗ t [ G H /G T ] · Z T (cid:12)(cid:12)(cid:12) F t (cid:21) . G H , F T , and 1 /G T usingeven more equivalent probability measures. However, we find that the expectations givenunder the two equivalent probability measures R ∗ and R ∗ on the right hand side of the twoequations (5) and (6), respectively, are sufficient for obtaining the analytical solutions ofthe expected prices of contingent claims under all models in finance that admit analyticalsolutions to the current prices of these claims.In this paper we study the properties of the two measures R ∗ and R ∗ , and then, using thesemeasures, we derive the analytical solutions of the expected prices of a variety of contingentclaims under the major classes of contingent claim valuation models used in the equity,interest rate, and credit risk areas of finance. To get more insight into these two measures,recall that i) the R ∗ measure was proposed in equation (5) as a hybrid of the physical measure P and the equivalent martingale measure Q ∗ using a general numeraire G , and ii) the R ∗ measure is obtained as a transformation of the R ∗ measure given in equation (7) using thenumeraire G . Since the R ∗ and R ∗ measures are obtained as hybrid measures with probabilityinformation from both the physical measure P , and the equivalent martingale measure Q ∗ ,they do not belong in the class of equivalent martingale measures (EMMs) of Harrison andKreps (1979). Moreover, since the R ∗ and R ∗ measures are equivalent probability measures(as shown in Theorem 1) and are used for obtaining the expectation E t [ F H ] in equations (5)and (6), respectively, we denote these measures as equivalent expectation measures (EEMs).Before deriving Theorem 1 in section 2.2, the next section provides some intuition on theconstruction of the EEMs using an extension of the binomial tree for the underlying assetprice process (see Cox et al., 1979). We show how to build a binomial tree under the R ∗ measure for the special case when the numeraire is the money market account and the shortrate is constant. For illustrative purpose assume that the EMM Q ∗ = Q for the special case when the nu-meraire G is the money market account, and let R ∗ = R be the equivalent probabilitymeasure that satisfies the two conditions C1 and C2 given earlier, under this special case.Consider a European call option C on an underlying asset price process S , which follows thediscrete-time binomial process as in Cox et al. (1979). The call option matures at time T with strike price K . Substituting F T = C T = max( S T − K,
0) = ( S T − K ) + as the terminalpayoff from the European call option, and using a constant interest rate r , the expectedfuture price of the call option at time H , simplifies using equation (5) as follows: We show in the Internet Appendix IA that if the numeraires are restricted to be either the money marketaccount or the pure discount bond, then the R ∗ measure subsumes the R ∗ measure. t [ C H ] = E R t (cid:2) e − r ( T − H ) ( S T − K ) + (cid:3) , (8)where C H is the price of the call option at time H . A simple and intuitive demonstration ofthe R measure is made using a discrete 3-year binomial tree example in Figure 1, in whichthe current time t = 0, H = 2 years, T = 3 years, and the 1-year short rate is constant. [Insert Figure 1 about here.] To construct the discrete 3-year binomial tree for the stock price process under the R measure, assume that the current price S is $100, the stock’s annualized expected return µ is 0.1, the annualized risk-free rate r is 0.03, the annualized stock return volatility σ is0.15, and the discrete interval ∆ t = 1 year. The stock prices at different nodes are identicalunder all three measures, P , Q , and R , and are given by the binomial model of Cox et al.(1979): the stock price either moves up by the multiplicative factor u = e σ √ ∆ t = 1 . d = e − σ √ ∆ t = 0 . R measure is constructed by satisfying conditions C1 and C2 given earlier. To satisfycondition C1, the next-period up and down R probabilities are equal to the up and down P probabilities, respectively, from time t = 0 until before time H = 2 years. These probabilitiesare calculated as: p u = e µ ∆ t − du − d = e . × − . . − . . , p d = 1 − p u = 0 . . To satisfy condition C2, the next-period up and down R probabilities are equal to the up anddown Q probabilities, respectively, at time H = 2 years. These probabilities are calculatedas: q u = e r ∆ t − du − d = e . × − . . − . . , q d = 1 − q u = 0 . . Now, consider the computation of the expected future price E [ C ] of a 3-year Europeancall option written on this stock with a strike price K = $100. The expected future callprice can be computed using the R measure as shown in Figure 1(b). The option prices C at the terminal nodes of the tree are calculated by the payoff function max ( S − K, C at time H = 2 years are computed using risk-neutral discounting (see Cox10t al., 1979). For example, the option value C at the up-up node is C uu = e − r ∆ t (cid:0) q u × $56 .
83 + q d × $16 . (cid:1) = e − . × (0 . × $56 .
83 + 0 . × $16 .
18) = $37 . . Similar calculations give C ud = $8.85 and C dd = $0 at the up-down node and the down-downnode, respectively. At the future time 1 year, the expected future call price E [ C ] at the upnode is calculated by taking the expectation of call prices C uu and C ud , using the up anddown P probabilities, as follows: E [ C ] u = p u × $37 .
94 + p d × $8 .
85= 0 . × $37 .
94 + 0 . × $8 .
85 = $32 . . A similar calculation gives E [ C ] d = $7.19, at the down node. Finally, at the current time t = 0, the expected future call price E [ C ] is calculated by taking the expectation of E [ C ] u and E [ C ] d , using the up and down P probabilities, as follows: E [ C ] = p u × E [ C ] u + p d × E [ C ] d = 0 . × $32 .
47 + 0 . × $7 .
19 = $27 . . The binomial-tree example in section 2.1 shows how to construct the R ∗ measure under thespecial case with the numeraire as the money market account and a constant short rate.This section shows how to construct the R ∗ measure which satisfies conditions C1 and C2given earlier under very general conditions with any numeraire G without making restrictiveassumptions about the short rate process or other state variable processes. The existence ofthe R ∗ measure also guarantees the existence of the R ∗ measure as defined in equation (7). Wenow present the main theorem of this paper which derives the R ∗ and R ∗ measures formally .The theorem demonstrates that both R ∗ and R ∗ are equivalent probability measures andderives their Radon-Nikod´ym derivatives with respect to the physical probability measure P .After presenting Theorem 1, we study the properties of the R ∗ and R ∗ measures. We showthat the R ∗ measure satisfies conditions C1 and C2, which were assumed to hold in order toderive the expectations in equations (5) and (6). We use E [ · ] and E P [ · ] interchangeably todenote expectation under the physical measure P .11 heorem 1 (i) For a fixed H with ≤ H ≤ T , define a process L ∗ s ( H ) as L ∗ s ( H ) = L ∗ s L ∗ H , if H ≤ s ≤ T , , if ≤ s < H. (9) Let R ∗ ( A ) = (cid:90) A L ∗ T ( H ; ω )d P ( ω ) for all A ∈ F T , (10) then the absence of arbitrage implies that R ∗ is a probability measure equivalent to P ,and L ∗ s ( H ) is the Radon-Nikod´ym derivative process of R ∗ with respect to P .(ii) For a fixed H with ≤ H ≤ T , define a process L ∗ s ( H ) as L ∗ s ( H ) = G H E P [ G H ] · L ∗ s L ∗ H , if H ≤ s ≤ T , E P s [ G H ] E P [ G H ] , if ≤ s < H. (11) Let R ∗ ( A ) (cid:44) (cid:90) A L ∗ T ( H ; ω )d P ( ω ) for all A ∈ F T , (12) then the absence of arbitrage implies that R ∗ is a probability measure equivalent to P ,and L ∗ s ( H ) is the Radon-Nikod´ym derivative process of R ∗ with respect to P .(iii) For all ≤ t ≤ H ≤ T , the expected price of the contingent claim at the horizon date H , E t [ F H ] , can be obtained by equations (5) and (6) , using the equivalent expectationmeasures R ∗ and R ∗ , respectively, as defined above. Proof.
See Appendix A.1.The EEMs R ∗ and R ∗ proposed in Theorem 1 provide a significant generalization ofthe risk-neutral valuation approach of Black-Scholes-Merton and Cox and Ross (1976), themartingale pricing theory of Harrison and Kreps (1979), and the different EMMs given byGeman (1989), Jamshidian (1989), and Geman et al. (1995). Since the expected price inequations (5) and (6) solves both the current price (i.e., for all 0 ≤ t = H ≤ T ) and theexpected future price (i.e., for all 0 ≤ t < H ≤ T ) using a single analytical solution , it We use E t [ · ] and E t [ ·|F t ] interchangeably, as we denote by E t the conditional expectation with respectto the filtration {F t } t ≥ , i.e. E t [ · ] = E [ ·|F t ]. equivalent expectation measures R ∗ and R ∗ instead of using the equivalent martingale measure Q ∗ for the derivations of analyticalsolutions in the future work on contingent claims modeling. R ∗ and R ∗ Measures
This section gives some general properties of the R ∗ and R ∗ measures that provide insightsinto their relationship with the physical measure P and the equivalent martingale measure Q ∗ . We begin with Proposition 1, which gives the relationship of the R ∗ measure with the P and Q ∗ measures. Proposition 1
For all ≤ t ≤ H ≤ s ≤ T , the R ∗ measure has the following properties:(i) R ∗ ( A | F t ) = P ( A | F t ) for all A ∈ F H .(ii) R ∗ ( A | F s ) = Q ∗ ( A | F s ) for all A ∈ F T .(iii) When H = T , R ∗ ( A | F t ) = P ( A | F t ) for all A ∈ F T .(iv) When H = t , R ∗ ( A | F t ) = Q ∗ ( A | F t ) for all A ∈ F T .(v) E R ∗ t [ Z T | F t ] = E P t (cid:104) E Q ∗ H [ Z T | F H ] (cid:12)(cid:12) F t (cid:105) for any random variable Z T at time T . Proof.
See Appendix A.2.According to property (i) of Proposition 1, the conditional R ∗ probabilities at any time t until time H , of the events at time H , are the same as the corresponding P probabilities ofthose events, satisfying condition C1 given earlier. According to property (ii) of Proposition1, the conditional R ∗ probabilities at any time s on or after time H , of the events at time T , are the same as the corresponding conditional Q ∗ probabilities of those events, satisfyingcondition C2 given earlier. Properties (iii) and (iv) of Proposition 1 show that the R ∗ measure nests both the physical measure P and the equivalent martingale measure Q ∗ . When H = t , the R ∗ measure becomes the Q ∗ measure, and the expected value computing problembecomes the traditional claim pricing problem. When H = T , the R ∗ measure becomesthe P measure. For all other values of H that are strictly between the current time t andthe maturity/expiration date T , the R ∗ measure remains a hybrid measure. Property (v)is useful for certain types of problems, such as pricing a pure discount bond, in which it iseasier to obtain the analytical solution of the expected price using the iterated expectationthat involves both the P measure and Q ∗ measure. Proposition 2
The expectation of the price ratio process F s /G s has the following properties: i) For all ≤ H ≤ s ≤ s ≤ T , E R ∗ s [ F s /G s |F s ] = E Q ∗ s [ F s /G s |F s ] = F s /G s .(ii) For all ≤ s ≤ s ≤ H ≤ T , E R ∗ s [ F s /G s |F s ] = E P s [ F s /G s |F s ] (cid:54) = E Q ∗ s [ F s /G s |F s ] = F s /G s . Proof.
See Appendix A.3.According to property (i) of Proposition 2, the price ratio process F s /G s is a martingaleunder R ∗ from time H to time T . However, according to property (ii) of Proposition 2,unless P ( A ) = Q ∗ ( A ) for all A ∈ F H , the price ratio process F s /G s is not a martingaleunder R ∗ from time 0 to time H . This shows the essential difference between the equivalent martingale measure Q ∗ , under which the price ratio F s /G s is a martingale from time 0 untiltime T , and the corresponding equivalent expectation measure R ∗ under which F s /G s is amartingale only from time H until time T .We now turn our attention to the properties of the R ∗ measure. Recall that the R ∗ measure can be obtained as a transformation of the R ∗ measure in equation (7), or deriveddirectly from the physical P measure in equation (12). The properties of R ∗ are given by thefollowing proposition: Proposition 3
For all ≤ t ≤ H ≤ s ≤ T , the R ∗ measure has the following properties:(i) R ∗ ( A | F t ) = E P t (cid:104) G H E P t [ G H ] A (cid:12)(cid:12) F t (cid:105) for all A ∈ F H .(ii) R ∗ ( A | F s ) = R ∗ ( A | F s ) = Q ∗ ( A | F s ) for all A ∈ F T .(iii) When H = T , R ∗ ( A | F t ) = E P t (cid:104) G T E P t [ G T ] A (cid:12)(cid:12) F t (cid:105) for all A ∈ F T .(iv) When H = t , R ∗ ( A | F t ) = R ∗ ( A | F t ) = Q ∗ ( A | F t ) for all A ∈ F T .(v) E R ∗ t [ Z T | F t ] = E R ∗ t (cid:104) G H E R ∗ t [ G H ] · Z T (cid:12)(cid:12) F t (cid:105) = E P t (cid:104) G H E P t [ G H ] E Q ∗ H [ Z T | F H ] (cid:12)(cid:12) F t (cid:105) for any randomvariable Z T at time T . Proof.
See Appendix A.4.Properties (ii) and (iv) of Proposition 3 for the R ∗ measure are identical to the corre-sponding properties (ii) and (iv) of Proposition 1 for the R ∗ measure, respectively. Accordingto property (ii) of Proposition 3, the conditional R ∗ probabilities at any time s on or aftertime H , of the events at time T , are the same as the corresponding conditional R ∗ probabili-ties, as well as the conditional Q ∗ probabilities of those events. According to property (iv) ofProposition 3, when H = t , the R ∗ measure becomes the same as the R ∗ measure and the Q ∗ measure, and the expected value computing problem becomes the traditional claim pricingproblem. Properties (i) and (iii) of Proposition 3 for the R ∗ measure are different from the14orresponding properties (i) and (iii) of Proposition 1 for the R ∗ measure, respectively. Ac-cording to property (i) of Proposition 3, the conditional R ∗ probabilities at any time t untiltime H , of any events at time H , are not the same as the corresponding P probabilities ofthose events, due to an adjustment term equal to G H / E P t [ G H ], related to the numeraire G .According to property (iii) of Proposition 3, when H = T , the R ∗ measure does not becomethe P measure, due to an adjustment term equal to G T / E P t [ G T ], related to the numeraire G .However, there is one exception to the last argument. When the numeraire G is given as the T -maturity pure discount bond, P ( · , T ), then G T = P ( T, T ) = 1. Therefore, for the specialcase of the numeraire G = P ( · , T ), property (iii) of Proposition 3 implies that when H = T ,the R ∗ measure becomes the P measure, since G T / E P t [ G T ] = P ( T, T ) / E P t [ P ( T, T )] = 1.Property (v) shows that it is easier to obtain the analytical solution of a contingent claimusing the R ∗ measure directly (whenever this measure leads to an analytical solution), sincethe term G H / E P t [ G H ] complicates the solution of the iterated expectation that involves boththe P measure and Q ∗ measure.Finally, though not explicitly stated here in the form of a proposition, it can be shownthat similar to the result in Proposition 2 for the R ∗ measure, the price ratio process F s /G s is a martingale under the R ∗ measure from time H to time T , but not from time 0 to time H . Until now we have used a general specification of the EMM Q ∗ based upon a general nu-meraire G . According to the results in sections 2.2 and 2.3, for every EMM based on a specificnumeraire, there exist at least two specific EEMs corresponding to that numeraire, which canbe obtained as special cases of R ∗ and R ∗ , respectively. Thus, R ∗ and R ∗ measures repre-sent two distinct classes of equivalent expectation measures (EEMs), and each class consistsof multiple EEMs corresponding to the multiple numeraires in that class, respectively. Wenow consider three specific numeraires which are used commonly in the contingent claimsliterature, given as the money market account B , the T -maturity pure discount bond P ( · , T ),and an arbitrary traded asset S (see, e.g., the exchange option model of Margrabe, 1978).For simplicity of exposition, and also to be consistent with existing contingent claims liter-ature, we denote the Q ∗ measures corresponding to these three numeraires (i.e., B , P ( · , T ),and S ), as Q , Q T , and Q S , respectively. Using similar notation, we denotei) the EEMs in the R ∗ class corresponding to the above three numeraires as R , R T , R S Additional EEMs may exist, such as the R ∗ measure as shown in footnote 8. However, we have foundthat the EEMs R ∗ and R ∗ are sufficient for all contingent claims applications in finance. R ∗ class corresponding to the above three numeraires as R , R T , and R S respectively.We summarize these six EEMs in Table 1. By investigating the major classes of contingentclaims models in the equity, interest rate, and credit risk areas of finance, we have found R , R T , and R S as the three EEMs that are useful for computing expected future pricesof contingent claims. The next section considers these three EEMs in the context of somecommonly used contingent claim models in finance. [Insert Table 1 about here.] This section presents a few examples of the applications of the R measure, the R T measure,and the R S measure for obtaining the analytical solutions of the expected prices of contingentclaims using the three numeraires: B , P ( · , T ), and S , respectively. Section 3.1 presents theapplications of the R measure using the Black and Scholes (1973) option pricing model andthe affine term structure models of Dai and Singleton (2000) and Collin-Dufresne et al.(2008). Section 3.2 presents the applications of the R T measure using the bond optionmodels of Geman (1989) and Jamshidian (1989), and the structural debt pricing model ofCollin-Dufresne and Goldstein (2001). The application of the R S measure is demonstratedin the Internet Appendix IB using the Margrabe (1978) exchange option pricing model. Webegin with the following corollary which presents three expectation results that hold underthe three EEMs: R , R T , and R S . Corollary 1
The expected price of the contingent claim F , can be represented under the R measure, the R T measure, and the R S measure as follows: • When the numeraire asset is the money market account, i.e., G = B = e (cid:82) · r u d u , R ∗ = R , and equation (5) becomes E t [ F H ] = E R t (cid:104) e − (cid:82) TH r u d u F T (cid:12)(cid:12) F t (cid:105) . (13) • When the numeraire asset is a pure discount bond, i.e., G = P ( · , T ) , R ∗ = R T , andequation (6) becomes E t [ F H ] = E P t (cid:2) P ( H, T ) (cid:12)(cid:12) F t (cid:3) E R T t (cid:2) F T (cid:12)(cid:12) F t (cid:3) . (14)16 When the numeraire asset is any arbitrary traded asset, i.e., G = S , R ∗ = R S , andequation (6) becomes E t [ F H ] = E P t (cid:2) S H (cid:12)(cid:12) F t (cid:3) E R S t (cid:20) F T S T (cid:12)(cid:12)(cid:12) F t (cid:21) . (15) Proof.
Equation (13) is a direct application of equation (5), while equations (14) and (15)are direct applications of equation (6).Notably, we can also write equation (13) as E t [ F H ] = E Q t (cid:104) L t L H e − (cid:82) TH r u d u F T | F t (cid:105) , or E t [ F H ] = E P t (cid:104) L T L H e − (cid:82) TH r u d u F T | F t (cid:105) = E P t (cid:104) M T M H F T | F t (cid:105) , where M s = L s /B s is the pricing kernel and M T /M s is the stochastic discount factor (SDF) for all 0 ≤ s ≤ T . R Measure
The geometric Brownian motion for the asset price process under the Black and Scholes(1973) model is given as follows: d S s S s = µ d s + σ d W P s , (16)where µ is the drift, σ is the volatility, and W P is the Brownian motion under the physicalmeasure P . The Girsanov theorem allows the transformation of Brownian motions undertwo equivalent probability measures. Defining W Q s = W P s + (cid:82) s γ d u , where γ = ( µ − r ) /σ isthe market price of risk (MPR), the asset price process under the risk neutral measure Q isd S s S s = r d s + σ d W Q s . (17)Using part (i) of Theorem 1, the Brownian motion under the R measure is derived in Ap-pendix A.5 as follows: W R s = W P s + (cid:90) s { u ≥ H } γ d u. (18)Substituting W P s from the above equation into equation (16) gives the asset price processunder the R measure as follows:d S s S s = (cid:0) r + σγ { s 0) =( S T − K ) + as the terminal payoff from the European call option, equation (13) in Corollary1 gives E t [ C H ] = E R t (cid:2) e − r ( T − H ) ( S T − K ) + (cid:3) , (20)where C H is the price of the call option at time H . Since equations (19) and (19 a ) implythat S T is lognormally distributed under the R measure, the expected call price can easilybe solved (see Appendix A.5) and is given as E t [ C H ] = S t e µ ( H − t ) N (cid:16) ˆ d (cid:17) − K e − r ( T − H ) N (cid:16) ˆ d (cid:17) , (21)18here N ( · ) is the standard normal cumulative distribution function, andˆ d = ln( S t /K ) + µ ( H − t ) + r ( T − H ) + σ ( T − t ) σ √ T − t , ˆ d = ln( S t /K ) + µ ( H − t ) + r ( T − H ) − σ ( T − t ) σ √ T − t . Rubinstein (1984) first derived this expected price formula by using the property of doubleintegrals of normally distributed variables, while our derivation is much simpler.When t = H , the R measure becomes the risk-neutral measure Q (see property (iv) ofProposition 1), and equation (20) reduces to the familiar risk-neutral valuation equation, C t = E Q t (cid:2) e − r ( T − t ) ( S T − K ) + (cid:3) . (22) Surprisingly, the analytical solution of the expected price of a T -maturity pure discountbond at time H (for all t < H < T ), has not been derived under any of the dynamic termstructure models, such as the affine models of Dai and Singleton (2000) and Collin-Dufresneet al. (2008), the quadratic model of Ahn et al. (2002), and the forward-rate models of Heathet al. (1992). Using the maximal affine term structure models of Dai and Singleton (2000)as an expositional example, this section shows how such an expectation can be solved underthe dynamic term structure models using the R measure. Substituting F T = P ( T, T ) = 1 inequation (13) of Corollary 1, the time t expectation of the future price of the pure discountbond at time H can be given under the R measure as follows: E t [ P ( H, T )] = E R t (cid:20) exp (cid:18) − (cid:90) TH r u d u (cid:19)(cid:21) . (23)The above expectation can be generally solved under all dynamic term structure models thatadmit an analytical solution to the bond’s current price, and is mathematically equivalentto the iterated expectation in equation (4), with the money market account as the numeraire, Rubinstein (1984) uses the following property of normal distribution: (cid:90) ∞−∞ N ( A + Bz ) 1 √ π e − z / d z = N (cid:18) A √ B (cid:19) . This double integral property is hard to extend to other distributions. The only condition required is that the functional forms of the market prices of risks should allowmaintaining the similar admissible affine or quadratic forms under both the P and Q measures, so that theadmissible affine or quadratic form will also apply under the R measure (which is a hybrid of P and Q measures as shown by Proposition 1). E t [ P ( H, T )] = E P t (cid:20) E Q H (cid:20) exp (cid:18) − (cid:90) TH r u d u (cid:19)(cid:21)(cid:21) . (24)However, instead of using the above equation to obtain expected (simple) return, empiricalresearchers in finance have generally used expected log returns based upon the followingequation: E t [ln P ( H, T )] = E P t (cid:20) ln (cid:18) E Q H (cid:20) exp (cid:18) − (cid:90) TH r u d u (cid:19)(cid:21)(cid:19)(cid:21) . (25)Analytical solutions of the exponentially affine and exponentially quadratic form exist for theinside expectation term E Q H (cid:104) exp (cid:16) − (cid:82) TH r u d u (cid:17)(cid:105) in the above equation for the affine and thequadratic term structure models, respectively. Taking the log of the exponential-form solu-tion of this inside expectation term simplifies equation (25) considerably since log(exp(x))= x. Moreover, for different econometric reasons, researchers have used expected log re-turns based upon equation (25), instead of using expected simple returns based upon equa-tion (24) for studying default-free bond returns. For example, Dai and Singleton (2002),Bansal and Zhou (2002), Cochrane and Piazzesi (2005), and Eraker et al. (2015), all use E P t [ln ( P ( H, T ) /P ( t, T ))] for the models studied in their papers.We would like to make two important observations in the context of the above discussion.First, an analytical solution of E t [ P ( H, T )] given by equation (23), can be generally derivedusing the R measure for any dynamic term structure model which admits an analyticalsolution for the current price of the zero-coupon bond using the Q measure. Second, itis generally not possible to know the analytical solution of E t [ P ( H, T )] by knowing theanalytical solution of E t [ln ( P ( H, T ))], or vice versa, for most term structure models exceptover the infinitesimal horizon H = t + dt . Thus, even though the estimates of the expectedlog returns over a finite horizon have been obtained by many researchers (usually usingone month returns), such estimates cannot be used by fixed income investors to computethe expected returns of bonds and bond portfolios over a finite horizon, without makingadditional return distribution assumptions.In order to derive the expected returns of default-free bonds, the following derives theexpected price of a pure discount bond at any finite horizon H (for all t ≤ H ≤ T ) usingthe affine term structure models (ATSMs) of Dai and Singleton (2000) and Collin-Dufresne Expected log return is defined as E t [ln ( P ( H, T ) /P ( t, T ))], and expected simple return is defined as E t [ P ( H, T ) − P ( t, T )] /P ( t, T ), where P ( t, T ) is the current price. Ito’s lemma can be used to obtain E t [ P ( H, T )] from E t [ln ( P ( H, T ))], or vice-versa, when H = t + dt ,where dt is an infinitesimally small time interval. 20t al. (2008). Collin-Dufresne et al. (2008) present a new method to obtain maximal ATSMsthat uses infinitesimal maturity yields and their quadratic covariations as the globally iden-tifiable state variables. The new method allows model-independent estimates for the statevector that can be estimated directly from yield curve data. When using three or lessstate variables—which is typical in empirical research on term structure models—the newapproach of Collin-Dufresne et al. (2008) is consistent with the general framework of Daiand Singleton (2000) given below, but with some clear advantages in the estimation andinterpretation of the economic state variables. This framework is also consistent with theunspanned stochastic volatility-based affine term structure models of Collin-Dufresne et al.(2009) with appropriate parameter restriction on the stochastic processes.Assume that the instantaneous short rate r is an affine function of a vector of N statevariables Y = ( Y , Y , ..., Y N ) (which may be observed as in Collin-Dufresne et al. (2008) orlatent as in Dai and Singleton (2000)), given as r s = δ + N (cid:88) i =1 δ i Y is (cid:44) δ + δ y (cid:48) Y s , (26)where Y follows an affine diffusion under the physical measure P , as follows:d Y s = K (Θ − Y s ) d s + Σ (cid:112) V s d W P s . (27) W P is a vector of N independent standard Brownian motions under P , Θ is N × K and Σ are N × N matrices, which may be nondiagonal and asymmetric, and V is a diagonalmatrix with the i th diagonal element given by (cid:104) V s (cid:105) ii = α i + β i (cid:48) Y s . (28)Assume that the market prices of risks, γ s , are given by γ s = (cid:112) V s γ, (29) When using more than three state variables, the maximal affine models of Collin-Dufresne et al. (2008)can have more identifiable parameters than the “maximal” model of Dai and Singleton (2000). For thesecases, the framework presented in this paper can be generalized so that the generalized A m ( N ) class ofmaximal affine models can have m square-root processes and N − m Gaussian state variables, but with M number of Brownian motions, such that N ≤ M . Extending the Dai and Singleton (2000) model so thatthe number of Brownian motions may exceed the number of state variables ensures that the extended modelnests the more general maximal affine models of Collin-Dufresne et al. (2008) when the number of statevariables is more than three. For example, Collin-Dufresne et al. present a maximal A (4) model whichrequires 5 Brownian motions in order to be consistent with the Dai and Singleton (2000) framework. Also,see the discussion in Collin-Dufresne et al. (2008, pp. 764-765, and footnote 19). γ is an N × Y is given as,d Y s = [ K (Θ − Y s ) − Σ V s γ ] d s + Σ (cid:112) V s d W Q s . (30)Using Theorem 1 and Appendix B.1, the stochastic process for Y under the R measureis given as follows: d Y s = (cid:2) K (Θ − Y s ) − { s ≥ H } Σ V s γ (cid:3) d s + Σ (cid:112) V s d W R s . (31)Recall from the previous section that any stochastic process under the R measure isthe physical stochastic process until before the horizon H , and it becomes the risk-neutralprocess on or after the horizon H . Thus, the state variable process under the R measure inequation (31) can also be obtained by a simple inspection of the state variable process underthe P measure given by equation (27), and under the Q measure given by equation (30).As shown by Dai and Singleton (2000), all N -factor ATSMs can be uniquely classifiedinto N + 1 non-nested subfamilies A m ( N ), where m = 0, 1,..., N , indexes the degree ofdependence of the conditional variances on the number of state variables Y . The A m ( N )model must satisfy several parametric restrictions for admissibility as outlined in Dai andSingleton (2000). Under these restrictions, the expected price of T -maturity pure discountbond is given by the following proposition: Proposition 4 The expected price of a T -maturity pure discount bond at time H is givenunder the A m ( N ) model as follows: For all t ≤ H ≤ T , E t [ P ( H, T )] = E R t (cid:20) exp (cid:18) − (cid:90) TH r u d u (cid:19)(cid:21) = E R t (cid:20) exp (cid:18) − (cid:90) TH ( δ + δ y (cid:48) Y u ) d u (cid:19)(cid:21) = exp (cid:16) − δ ( T − H ) − A (0 ,δ y ) K ∗ , Θ ∗ ( T − H ) − A ( b , K , Θ ( H − t ) − B ( b , K , Θ ( H − t ) (cid:48) Y t (cid:17) , (32) where b = B (0 ,δ y ) K ∗ , Θ ∗ ( T − H ) , K ∗ = K + ΣΦ , Θ ∗ = K ∗ − ( K Θ − ΣΨ) , where the i th row of Φ is given by γ i β i (cid:48) , and Ψ is an N × vector whose i th element is givenby γ i α i . For any well-behaved N × vectors b , c , Θ and N × N matrix K , A K , Θ ( τ ) (cid:44) A ( b,c ) K , Θ ( τ )22 nd B K , Θ ( τ ) (cid:44) B ( b,c ) K , Θ ( τ ) are defined by the solution of the following expectation, E P t (cid:20) exp (cid:18) − b (cid:48) Y t + τ − (cid:90) t + τt c (cid:48) Y u d u (cid:19)(cid:21) = exp (cid:0) − A K , Θ ( τ ) − B K , Θ ( τ ) (cid:48) Y t (cid:1) , and are obtained by solving the following ordinary differential equations (ODEs): d A K , Θ ( τ )d τ = Θ (cid:48) K (cid:48) B K , Θ ( τ ) − N (cid:88) i =1 (cid:104) Σ (cid:48) B K , Θ ( τ ) (cid:105) i α i , d B K , Θ ( τ )d τ = −K (cid:48) B K , Θ ( τ ) − N (cid:88) i =1 (cid:104) Σ (cid:48) B K , Θ ( τ ) (cid:105) i β i + c, (33) with the terminal conditions A K , Θ (0) = 0 , B K , Θ (0) = b . A K , Θ ( τ ) and B K , Θ ( τ ) can be solvedthrough numerical procedures such as Runge-Kutta. A K ∗ , Θ ∗ ( τ ) (cid:44) A ( b,c ) K ∗ , Θ ∗ ( τ ) and B K ∗ , Θ ∗ ( τ ) (cid:44) B ( b,c ) K ∗ , Θ ∗ ( τ ) are defined by the solution of the following expectation, E Q H (cid:20) exp (cid:18) − b (cid:48) Y H + τ − (cid:90) H + τH c (cid:48) Y u d u (cid:19)(cid:21) = exp (cid:0) − A K ∗ , Θ ∗ ( τ ) − B K ∗ , Θ ∗ ( τ ) (cid:48) Y H (cid:1) , and are obtained similarly as in equation (33) with the coefficients ( K , Θ) replaced withcoefficients ( K ∗ , Θ ∗ ) . Proof. See Appendix A.6.Proposition 4 can be used to obtain the expected price of a pure discount bond under anyspecific ATSMs of Dai and Singleton (2000) and Collin-Dufresne et al. (2008). For example,using a tedious derivation, we obtain the expected price of the pure discount bond under the A r (3) model of Dai and Singleton (2000) in Internet Appendix IC. For the ease of exposition,the following illustrates the solutions of the expected future price of a pure discount bondunder the classical one-factor models of Vasicek (1977) and Cox-Ingersoll-Ross (CIR, Coxet al., 1985). The Vasicek Model. The Ornstein-Uhlenbeck process for the short rate under the Vasicekmodel is given as follows: d r s = α r ( m r − r s ) d s + σ r d W P s , (34)where α r is the speed of mean reversion, m r is the long-term mean of the short rate, and σ r is the volatility of the changes in the short rate. Assuming W Q s = W P s + (cid:82) s γ r d u , the shortrate process under the R measure can be derived using Theorem 1 and Appendix B.1, as23ollows: d r s = [ α r ( m r − r s ) − { s ≥ H } σ r γ r ]d s + σ r d W R s . (35)The expected price of a T -maturity pure discount bond at time H is given under the Vasicekmodel as follows: For all t ≤ H ≤ T , E t [ P ( H, T )] = E R t (cid:20) exp (cid:18) − (cid:90) TH r u d u (cid:19)(cid:21) = exp (cid:16) − A (0 , α ∗ r ,m ∗ r ( T − H ) − A ( b , α r ,m r ( H − t ) − B ( b , α r ,m r ( H − t ) · r t (cid:17) , (36)where b = B (0 , α ∗ r ,m ∗ r ( T − H ), α ∗ r = α r , m ∗ r = m r − σ r γ r /α r , and for any well-behaved b , c , α ,and m , A ( b,c ) α,m ( τ ) and B ( b,c ) α,m ( τ ) are given by A ( b,c ) α,m ( τ ) = αm (cid:18) bB α ( τ ) + c τ − B α ( τ ) α (cid:19) − σ r (cid:32) b − e − ατ α + bcB α ( τ ) + c τ − B α ( τ ) − αB α ( τ ) α (cid:33) ,B ( b,c ) α,m ( τ ) = b e − ατ + cB α ( τ ) , with B α ( τ ) = (1 − e − ατ ) /α . When H = t , equation (36) reduces to the Vasicek bond pricesolution. The CIR Model. The square-root process for the short rate under the CIR model is givenas follows: d r s = α r ( m r − r s ) d s + σ r √ r s d W P s . (37)Assume that the market price of risk γ s satisfies γ s = γ r √ r s . Thus, W Q s = W P s + (cid:82) s γ r √ r u d u , and the short rate process under the R measure can bederived using Theorem 1 and Appendix B.1, as follows:d r s = (cid:2) α r ( m r − r s ) − { s ≥ H } γ r σ r r s (cid:3) d s + σ r √ r s d W R s . (38)The expected price of a T -maturity pure discount bond at time H is given under the CIR24odel as follows: For all t ≤ H ≤ T , E t [ P ( H, T )] = E R t (cid:20) exp (cid:18) − (cid:90) TH r u d u (cid:19)(cid:21) = exp (cid:16) − A (0 , α ∗ r ,m ∗ r ( T − H ) − A ( b , α r ,m r ( H − t ) − B ( b , α r ,m r ( H − t ) · r t (cid:17) , (39)where b = B (0 , α ∗ r ,m ∗ r ( T − H ), α ∗ r = α r + γ r σ r , m ∗ r = α r m r / ( α r + γ r σ r ), and for any well-behaved b , c , α , and m , A ( b,c ) α,m ( τ ) and B ( b,c ) α,m ( τ ) are given as A ( b,c ) α,m ( τ ) = − αmσ r ln (cid:32) β e ( β + α ) τ bσ r (e βτ − 1) + β − α + e βτ ( β + α ) (cid:33) ,B ( b,c ) α,m ( τ ) = b (cid:0) β + α + e βτ ( β − α ) (cid:1) + 2 c (cid:0) e βτ − (cid:1) bσ r (e βτ − 1) + β − α + e βτ ( β + α ) , with β = (cid:112) α + 2 σ r c . When H = t , equation (39) reduces to the CIR bond price solution.To our knowledge, equations (36) and (39) provide the first analytical solutions of theexpected price of a pure discount bond at a future time H , under the Vasicek and CIRmodels in the finance literature.The Internet Appendix ID extends the results of this section, and derives the expectedprice of a pure discount bond under the quadratic term structure models (QTSMs) of Ahnet al. (2002) and under the specific QTSM3 model. R T Measure Geman (1989) and Jamshidian (1989) derived the forward measure Q T , for valuing contin-gent claims with stochastic interest rates. As shown by equation (14) of Corollary 1, theappropriate EEM for obtaining the expected price of a European option under stochasticinterest rates is the R T measure (this measure converges to the forward measure Q T , when t = H , as shown by property (iv) of Proposition 3, with the T -maturity pure discount bondas the numeraire). This section shows how to use the EEM R T to get the expected price ofa call option on a pure discount bond under the Vasicek (1977) model. Consider a Europeancall option C written on an S -maturity pure discount bond with a strike price of K , andan option expiration date equal to T , such that T ≤ S . The expected price of this optionis given by equation (14), with F T = C T = ( P ( T, S ) − K ) + as the terminal payoff from the Also, see Geman et al. (1995). E t [ C H ] = E P t [ P ( H, T )] E R T t (cid:2) ( P ( T, S ) − K ) + (cid:3) . (40)The second expectation term E R T t [( P ( T, S ) − K ) + ] can be solved by expressing the shortrate process under the R T measure using Theorem 1 and Appendix B.2, as follows:d r s = (cid:2) α r ( m r − r s ) − { s ≥ H } σ r γ r − σ r B α r ( T − s ) + σ r B α r ( H − s ) { s The expected price of the risky discount bond at time H is E t [ D ( H, T )] = E P t [ P ( H, T )] (cid:16) − ω E R T t (cid:104) { τ [ t,T ] See Appendix A.7 for proof, and for the definitions of expressions M , ˜ M , S and ˜ S . R -Transforms In two path-breaking papers, Bakshi and Madan (2000) and Duffie et al. (2000) derivenew transforms that extend the Fourier transform-based method of Heston (1993) for the28aluation of a wide variety of contingent claims. Though Chacko and Das (2002) also in-dependently develop a generalized Fourier transform-based method for the valuation of avariety of interest rate derivatives, their approach can be obtained as a special case of themore general transform-based method of Duffie et al. (2000). Since the specific transforms inthese papers use the risk-neutral measure Q to obtain current prices or to do valuation, werefer to these transforms as Q -transforms. As extensions of these Q -transforms, this sectionderives new R -transforms using the equivalent expectation measure R , to obtain the expected price of a contingent claim at any future date H until the claim’s maturity or expirationdate.Unlike the traditional methods for contingent claims analysis that select a specific nu-meraire for valuation, depending upon the assumptions about the discount rate (i.e., constantor stochastic) or the form of the claim’s terminal payoff, the transform methods of Bakshiand Madan (2000), Duffie et al. (2000), and Chacko and Das (2002)—which allow multiplestate variables in the definition of their Q -transforms—can value different types of contingentclaims using only the money market account as the numeraire. However, there is nothingspecial about using the money market account as the numeraire, and general transform-based methods can be derived using even other numeraires. For example, it is possible toobtain another transform using the bond price P ( t, T ) as the numeraire, which can also priceall of the contingent claims that are priced using the Q -transform with money market as thenumeraire. In other words, the choice of numeraire is not important for valuation when usingthe transform-based methods with multiple state variables. The money market numeraireis chosen for convenience, and also because empirical estimation methods already exist forobtaining the risk-neutral parameters in the finance field.This section presents new R -transforms based upon the R measure, which can be usedfor the derivation of the analytical solution of the expected price of a contingent claim ata future time H . The R -transforms derived in this section nest the Q -transforms of Bakshiand Madan (2000), Duffie et al. (2000), and Chacko and Das (2002), when t = H . R -Transform As in Duffie et al. (2000), we define an affine jump diffusion state process on a probabilityspace (Ω , F , P ), and assume that Y is a Markov process in some state space D ∈ R N , solvingthe stochastic differential equationd Y s = µ ( Y s )d s + σ ( Y s )d W P s + d J s , (49) A simple change of variable can be used to show the equivalence between the Q -transforms of Bakshiand Madan (2000) and Duffie et al. (2000). W P is a standard Brownian motion in R N ; µ : D → R N , σ : D → R N × N , and J is a pure jump process whose jumps have a fixed probability distribution ν on R N andarrive with intensity { λ ( Y s ) : s ≥ } , for some λ : D → [0 , ∞ ). The transition semi-group ofthe process Y has an infinitesimal generator D of the L´evy type, defined at a bounded C function f : D → R , with bounded first and second derivatives, by D f ( y ) = f y ( y ) µ ( y ) + 12 tr [ f yy ( y ) σ ( y ) σ ( y ) (cid:48) ] + λ ( y ) (cid:90) R N [ f ( y + j ) − f ( y )] d ν ( j ) . (50)As in Duffie et al. (2000), Duffie and Kan (1996), and Dai and Singleton (2000), weassume that ( D, µ, σ, λ, ν ) satisfies the joint restrictions, such that Y is well defined, and theaffine dependence of µ , σσ (cid:48) , λ , and r are determined by coefficients ( k, h, l, ρ ) defined by • µ ( y ) = k + k y , for k = ( k , k ) ∈ R N × R N × N . • ( σ ( y ) σ ( y ) (cid:48) ) ij = ( h ) ij + ( h ) ij · y , for h = ( h , h ) ∈ R N × N × R N × N × N . • λ ( y ) = l + l (cid:48) y , for l = ( l , l ) ∈ R × R N . • r ( y ) = ρ + ρ (cid:48) y , for ρ = ( ρ , ρ ) ∈ R × R N .For x ∈ C N , the set of N -tuples of complex numbers, we define θ ( x ) = (cid:82) R N exp ( x (cid:48) j ) d ν ( j )whenever the integral is well defined. Hence, the “coefficients” ( k, h, l, θ ) capture the distri-bution of Y , and a “characteristic” χ = ( k, h, l, θ, ρ ) captures both the distribution of Y aswell as the effects of any discounting. Furthermore, we assume the state vector Y to be anaffine jump diffusion with coefficients ( k ∗ , h ∗ , l ∗ , θ ∗ ) under the risk-neutral measure Q , andthe relevant characteristic for risk-neutral pricing is then χ ∗ = ( k ∗ , h ∗ , l ∗ , θ ∗ , ρ ∗ ).Duffie et al. (2000) define a transform φ : C N × R + × R + → C of Y T conditional on F t ,when well defined for all t ≤ T , as φ ( z ; t, T ) (cid:44) E Q t (cid:20) exp (cid:18) − (cid:90) Tt r ( Y u )d u (cid:19) exp ( z (cid:48) Y T ) (cid:21) . (51)The current prices of various types of contingent claims can be computed conveniently usingthe above transform. Since this transform is derived to value the current price of contingentclaims, we refer to it as the Q -transform. To extend the Q -transform and compute theexpected prices of contingent claims, we define an R -transform φ R : C N ×R + ×R + ×R + → C Notably, when z = iu (where i = √− φ becomes the Q -transform of Bakshi and Madan (2000)(see equation (6)), defined as the characteristic function of the state-price density. Y T conditional on F t , when well defined for all t ≤ H ≤ T , as φ R ( z ; t, T, H ) (cid:44) E R t (cid:20) exp (cid:18) − (cid:90) TH r ( Y u )d u (cid:19) exp ( z (cid:48) Y T ) (cid:21) (52)= E P t (cid:20) E Q t (cid:20) exp (cid:18) − (cid:90) TH r ( Y u )d u (cid:19) exp ( z (cid:48) Y T ) (cid:21)(cid:21) (52 a )= E Q t (cid:20) L t L H exp (cid:18) − (cid:90) TH r ( Y u )d u (cid:19) exp ( z (cid:48) Y T ) (cid:21) (52 b )= E P t (cid:20) L T L H exp (cid:18) − (cid:90) TH r ( Y u )d u (cid:19) exp ( z (cid:48) Y T ) (cid:21) , (52 c )where L s is the Radon-Nikod´ym derivative process of Q with respect to P .Equations (52 a ), (52 b ), and (52 c ) can be obtained from equation (52) using Proposition1(v) and a change of measure. While all of the above equations lead to an identical solu-tion for the R -transform φ R , the exponentially-affine “solution-form” and the correspondingODEs are much simpler and easier to derive using equations (52) and (52 a ), and morecomplicated to derive using equations (52 b ) and (52 c ), due to the presence of the Radon-Nikod´ym derivative terms in the latter two equations. Derivation of a simpler solution-formis also the reason why the traditional Q -transforms in Bakshi and Madan (2000), Duffieet al. (2000), and Chacko and Das (2002) are derived under the Q measure, and not underthe physical measure P with additional Radon-Nikod´ym terms. The following propositionobtains the solution of the R -transform φ R . Proposition 6 Under technical regularity conditions given in Duffie et al. (2000), the R -transform φ R of Y defined by equation (52) is given by φ R ( z ; t, T, H ) = exp (cid:16) − A ( ˆ b R , ) χ ( H − t ) − B ( ˆ b R , ) χ ( H − t ) (cid:48) Y t (cid:17) , (53) where ˆ b R = (cid:16) A ( b R , ρ R ) χ ∗ ( T − H ) , B ( b R , ρ R ) χ ∗ ( T − H ) (cid:17) , b R = (0 , − z ) , and ρ R = ( ρ , ρ ) .For some well-defined b (cid:44) ( b , b ) ∈ C N +1 , c (cid:44) ( c , c ) ∈ R N +1 and a coefficient charac-teristic χ = ( k, h, l, θ ) , A χ ( τ ) (cid:44) A ( b , c ) χ ( τ ) and B χ ( τ ) (cid:44) B ( b , c ) χ ( τ ) are defined by the solutionof the following expectation, E P t (cid:20) exp (cid:18) − b − b (cid:48) Y t + τ − (cid:90) t + τt ( c + c (cid:48) Y u ) d u (cid:19)(cid:21) = exp (cid:0) − A χ ( τ ) − B χ ( τ ) (cid:48) Y t (cid:1) , nd are obtained by solving the following complex-valued ODEs: d A χ ( τ )d τ = k (cid:48) B χ ( τ ) − B χ ( τ ) (cid:48) h B χ ( τ ) − l ( θ ( − B χ ( τ )) − 1) + c , d B χ ( τ )d τ = k (cid:48) B χ ( τ ) − B χ ( τ ) (cid:48) h B χ ( τ ) − l ( θ ( − B χ ( τ )) − 1) + c , (54) with boundary conditions A χ (0) = b , and B χ (0) = b . The solutions can be found numer-ically, for example by Runge-Kutta. A χ ∗ ( τ ) (cid:44) A ( b , c ) χ ∗ ( τ ) and B χ ∗ ( τ ) (cid:44) B ( b , c ) χ ∗ ( τ ) are definedby the solution of the following expectation, E Q H (cid:20) exp (cid:18) − b − b (cid:48) Y H + τ − (cid:90) H + τH ( c + c (cid:48) Y u ) d u (cid:19)(cid:21) = exp (cid:0) − A χ ∗ ( τ ) − B χ ∗ ( τ ) (cid:48) Y H (cid:1) , and are obtained similarly as in equation (54) with the coefficient characteristic χ = ( k, h, l, θ ) replaced with a coefficient characteristic χ ∗ = ( k ∗ , h ∗ , l ∗ , θ ∗ ) . Proof. See Appendix A.8.The expositional examples given in section 4.3.1 and Internet Appendix IE demonstratethe application of the R -transform for the derivation of the analytical solutions of the ex-pected prices of equity options and interest rate derivatives, respectively, at a future time H . For the special case of t = H , the R -transform and its solution given by equations (52)and (53), reduce to the Q -transform and its solution given by equations (2.3) and (2.4),respectively, in Duffie et al. (2000). R -Transform While the Q -transform given in equation (51) is sufficient for the valuation of many contin-gent claims—e.g., standard equity options, discount bond options, caps and floors, exchangerate options, chooser options, digital options, etc.—certain pricing problems like the valu-ation of Asian options or estimating default-time distributions require the extended trans-form given by equation (2.13) in Duffie et al. (2000). The following proposes an extended R -transform corresponding to their extended transform, which we refer to as the extended Q -transform. We define the extended R -transform ϕ R : R N × C N × R + × R + × R + → C of Y T conditional on F t , when well defined for t ≤ H ≤ T , as ϕ R ( v, z ; t, T, H ) (cid:44) E R t (cid:20) exp (cid:18) − (cid:90) TH r ( Y u )d u (cid:19) ( v (cid:48) Y T ) exp ( z (cid:48) Y T ) (cid:21) (55)= E P t (cid:20) E Q t (cid:20) exp (cid:18) − (cid:90) TH r ( Y u )d u (cid:19) ( v (cid:48) Y T ) exp ( z (cid:48) Y T ) (cid:21)(cid:21) (55 a )32 E Q t (cid:20) L t L H exp (cid:18) − (cid:90) TH r ( Y u )d u (cid:19) ( v (cid:48) Y T ) exp ( z (cid:48) Y T ) (cid:21) (55 b )= E P t (cid:20) L T L H exp (cid:18) − (cid:90) TH r ( Y u )d u (cid:19) ( v (cid:48) Y T ) exp ( z (cid:48) Y T ) (cid:21) , (55 c )where L s is the Radon-Nikod´ym derivative process of Q with respect to P .As in the case of the R -transform given earlier, the exponentially-affine “solution-form”and the corresponding ODEs of the extended R -transform ϕ R are much simpler and easierto derive using equations (55) and (55 a ), and more complicated to derive using equations(55 b ) and (55 c ), due to the presence of the Radon-Nikod´ym derivative terms. The followingproposition gives the solution to the extended R -transform ϕ R . Proposition 7 Under technical regularity conditions given in Duffie et al. (2000), the ex-tended R -transform ϕ R of Y defined by equation (55) is given as ϕ R ( v, z ; t, T, H ) = φ R ( z ; t, T, H ) (cid:16) D ( ˆ b R , , ˆ v R ) χ ( H − t ) + E ( ˆ b R , , ˆ v R ) χ ( H − t ) (cid:48) Y t (cid:17) , (56) where ˆ b R = (cid:16) A ( b R , ρ R ) χ ∗ ( T − H ) , B ( b R , ρ R ) χ ∗ ( T − H ) (cid:17) , ˆ v R = (cid:16) D ( b R , ρ R , v R ) χ ∗ ( T − H ) , E ( b R , ρ R , v R ) χ ∗ ( T − H ) (cid:17) ,with b R = (0 , − z ) , ρ R = ( ρ , ρ ) , and v R = (0 , v ) .For some well-defined b (cid:44) ( b , b ) ∈ C N +1 , c (cid:44) ( c , c ) ∈ R N +1 , d (cid:44) ( d , d ) ∈ R N +1 , anda coefficient characteristic χ = ( k, h, l, θ ) , A χ ( τ ) (cid:44) A ( b , c ) χ ( τ ) , B χ ( τ ) (cid:44) B ( b , c ) χ ( τ ) , D χ ( τ ) (cid:44) D ( b , c , d ) χ ( τ ) and E χ ( τ ) (cid:44) E ( b , c , d ) χ ( τ ) are defined by the solution of the following expectation, E P t (cid:20) exp (cid:18) − b − b (cid:48) Y t + τ − (cid:90) t + τt ( c + c (cid:48) Y u ) d u (cid:19) ( d + d (cid:48) Y t + τ ) (cid:21) = exp (cid:0) − A χ ( τ ) − B χ ( τ ) (cid:48) Y t (cid:1) · (cid:0) D χ ( τ ) + E χ ( τ ) (cid:48) Y t (cid:1) , where A χ ( τ ) and B χ ( τ ) are obtained as in Proposition 6, and D χ ( τ ) and E χ ( τ ) satisfy thecomplex-valued ODEs d D χ ( τ )d τ = k (cid:48) E χ ( τ ) − B χ ( τ ) (cid:48) h E χ ( τ ) + l ∇ θ ( − B χ ( τ )) E χ ( τ ) , d E χ ( τ )d τ = k (cid:48) E χ ( τ ) − B χ ( τ ) (cid:48) h E χ ( τ ) + l ∇ θ ( − B χ ( τ )) E χ ( τ ) , (57) with boundary conditions D χ (0) = d , and E χ (0) = d , and ∇ θ ( x ) is gradient of θ ( x ) with respect to x ∈ C N . A χ ∗ ( τ ) (cid:44) A ( b , c ) χ ∗ ( τ ) , B χ ∗ ( τ ) (cid:44) B ( b , c ) χ ∗ ( τ ) , D χ ∗ ( τ ) (cid:44) D ( b , c , d ) χ ∗ ( τ ) and χ ∗ ( τ ) (cid:44) E ( b , c , d ) χ ∗ ( τ ) are defined by the solution of the following expectation, E Q H (cid:20) exp (cid:18) − b − b (cid:48) Y H + τ − (cid:90) H + τH ( c + c (cid:48) Y u ) d u (cid:19) ( d + d (cid:48) Y H + τ ) (cid:21) = exp (cid:0) − A χ ∗ ( τ ) − B χ ∗ ( τ ) (cid:48) Y H (cid:1) · (cid:0) D χ ∗ ( τ ) + E χ ∗ ( τ ) (cid:48) Y H (cid:1) , where A χ ∗ ( τ ) and B χ ∗ ( τ ) are obtained as in Proposition 6, and D χ ∗ ( τ ) and E χ ∗ ( τ ) are ob-tained similarly as in equation (57) with the coefficient characteristic χ = ( k, h, l, θ ) replacedwith a coefficient characteristic χ ∗ = ( k ∗ , h ∗ , l ∗ , θ ∗ ) . Proof. The proof follows similarly from the proof of Proposition 6, we omit the specificsteps.As an expositional example, we demonstrate the application of the extended R -transformfor the derivation of the analytical solution of the expected price of an Asian option on interestrates at a future time H , in the Internet Appendix IE (see equations (IA32) to (IA36)). Forthe special case of t = H , the extended R -transform given by equations (55) and (56) reducesto the extended Q -transform given by equations (2.13) and (2.14) in Duffie et al. (2000). R -Transforms This section obtains the analytical solutions of the expected prices of a few equity-basedand interest rate-based contingent claims using the R -transform and extended R -transform.For the equity-based claims, we first consider the derivation of the analytical solutions ofthe expected price of equity options under the general affine jump-diffusion (AJD) modelproposed by Duffie et al. (2000). We then consider two general models nested in the AJDmodel that nest virtually all other equity-based option valuation models in the finance liter-ature. As the first general model, this section considers the specific example given in Duffieet al. (2000), i.e., the stochastic volatility model with jumps in price and volatility, referredto as the SVJJ model in the options literature. This model nests the stochastic volatility(SV) model (see, e.g., Heston, 1993), the stochastic volatility with price jumps (SVJ) model(see, e.g., Bakshi et al., 1997; Bates, 2000; Pan, 2002), and the stochastic volatility withindependent jumps in price and volatility (SVIJ) model (see, e.g., Eraker et al., 2003). Asthe second general model for equity options, the Internet Appendix IF considers the CGMYmodel of Carr et al. (2002), which nests other infinite-activity L´evy jump models, such asthe variance-gamma model of Madan and Seneta (1990) and Madan et al. (1998).Given the affine jump-diffusion model setting as in section 4.1, we assume that the un-derlying asset is e d (cid:48) Y , for some d ∈ R N . Recall that the market value at time t of anyoption written on the underlying asset e d (cid:48) Y that pays an F T -measurable random variable34 e d (cid:48) Y T − K (cid:1) + at time T is given by C t = E Q t (cid:20) exp (cid:18) − (cid:90) Tt r ( Y s )d s (cid:19) (cid:16) e d (cid:48) Y T − K (cid:17) + (cid:21) . Therefore, using equation (13) in Corollary 1, the current time t expectation of the futurecall price at a horizon date H and strike at K with exercise date T with t ≤ H ≤ T underthe general AJD model is given by E t [ C H ] = E R t (cid:20) exp (cid:18) − (cid:90) TH r ( Y s )d s (cid:19) (cid:16) e d (cid:48) Y T − K (cid:17) + (cid:21) = E R t (cid:20) exp (cid:18) − (cid:90) TH r ( Y s )d s (cid:19) e d (cid:48) Y T { d (cid:48) Y T > ln K } (cid:21) − K E R t (cid:20) exp (cid:18) − (cid:90) TH r ( Y s )d s (cid:19) { d (cid:48) Y T > ln K } (cid:21) = G d,d (ln K ; Y t , t, T, H ) − KG ,d (ln K ; Y t , t, T, H ) . (58)Given some ( x, t, T, H, d , d ) ∈ D × [0 , ∞ ) × [0 , ∞ ) × [0 , ∞ ) × R N × R N , a transform G d ,d ( · ; x, t, T, H ) : R → R + is given by G d ,d ( y ; Y t , t, T, H ) (cid:44) E R t (cid:20) exp (cid:18) − (cid:90) TH r ( Y s )d s (cid:19) exp ( d (cid:48) Y T ) { d (cid:48) Y T >y } (cid:21) = φ R ( d ; t, T, H )2 + 1 π (cid:90) ∞ Re (cid:20) e − iuy φ R ( d + iud ; t, T, H ) iu (cid:21) d u, (59)where Re( c ) is the real part of c ∈ C , and the R -transform φ R is defined in equation (52). The SVJJ model assumes that the log asset price process, Y = ln S , and the volatility factorfollow the following processes under the physical measure P :d Y s = (cid:16) r − q + γ S v s + γ J − v s (cid:17) d s + √ v s − (cid:16) ρ d W P s + (cid:112) − ρ d W P s (cid:17) + d (cid:32) N s (cid:88) i =1 J S,i (cid:33) − λ ¯ µ d s, d v s = α v ( m v − v s ) d s + σ v √ v s − d W P s + d (cid:32) N s (cid:88) i =1 J v,i (cid:33) , (60)35here S = exp( Y ) is the ex-dividend asset price that pays constant dividends at rate q , r isthe constant interest rate, and v is the volatility process that is elastically pulled towards itslong-term mean m v with the speed of mean-reversion α v . W P and W P are two independentBrownian motions. The correlation term ρ allows negative correlation between the assetreturn and changes in volatility. The risk premiums γ S v and γ J are for diffusive return riskand the price-jump risk, respectively. The Poisson process N under P with constant intensity λ leads to jumps in both the asset price and the volatility, where the jump sizes J S,i and J v,i are correlated. The volatility jump size J v,i is exponentially distributed with P -mean µ v , i.e., J v,i ∼ exp ( µ v ). In addition, J S,i | J v,i ∼ N ( µ S + ρ J J v,i , σ S ) with normally distributed P -mean µ S + ρ J J v,i and P -variance σ S . The Poisson process is independent of the Brownian motions W P and W P , and the Brownian motions of inter-jump times. When a jump event arrives, theunconditional mean of the price jump size is ¯ µ = E P (cid:2) e J S,i (cid:3) − µ S + σ S / (1 − ρ J µ v ) − Q , the volatility jump size J v,i is exponentiallydistributed with Q -mean µ ∗ v , i.e., J v,i ∼ exp ( µ ∗ v ). In addition, J S,i | J v,i ∼ N ( µ ∗ S + ρ J J v,i , σ ∗ S )with normally distributed Q -mean µ ∗ S + ρ J J v,i and Q -variance σ ∗ S . Then under Q , theunconditional mean of the price jump size is ¯ µ ∗ = E Q (cid:2) e J S,i (cid:3) − µ ∗ S + σ ∗ S / (1 − ρ J µ ∗ v ) − Q -intensity rate λ ∗ . Finally, the market price ofprice jump risk is given as γ J = λ ¯ µ − λ ∗ ¯ µ ∗ , and that of volatility jump risk is defined as γ vJ = λµ v − λ ∗ µ ∗ v . Similar to Pan (2002), we assume that the market price of risks (MPRs)of diffusions, are given as: γ s = γ v σ v √ v s ,γ s = 1 (cid:112) − ρ (cid:18) − ρ γ v σ v + γ S (cid:19) √ v s . (61)Therefore, the risk-neutral processes corresponding to the physical processes in equation(60) are given as follows:d Y s = (cid:16) r − q − v s (cid:17) d s + √ v s − (cid:16) ρ d W Q s + (cid:112) − ρ d W Q s (cid:17) + d (cid:32) N s (cid:88) i =1 J S,i (cid:33) − λ ∗ ¯ µ ∗ d s, d v s = [ α v ( m v − v s ) − γ v v s ] d s + σ v √ v s − d W Q s + d (cid:32) N s (cid:88) i =1 J v,i (cid:33) . (62)This specification nests many popular equity option pricing models. In the absence ofany jumps, λ = 0, this model reduces to the Heston (1993) stochastic volatility model, orthe SV model. The SVJ models in Pan (2002) and Bates (2000) have normally distributedjumps in returns, i.e., J S,i ∼ N ( µ S , σ S ), but no jumps in volatility, i.e., µ v = 0, ρ J = 0. The36VIJ model of Eraker et al. (2003) is obtained as a special case of the SVJJ model underthe assumption that the size of price jump is independent of the size of volatility jump, or ρ J = 0.An application of Theorem 1, Appendix B.1 and B.3 to a multidimensional process withBrownian motions and jumps risks allows us to express the asset price process and thevolatility factor process given in equation (60) under the R measure, as follows:d Y s = (cid:16) r − q − v s { s 0) = ( S T − K ) + asthe terminal payoff from the European call option, E t [ C H ] = E R t (cid:104) e − (cid:82) TH r d u ( S T − K ) + (cid:105) = E R t (cid:104) e − (cid:82) TH r d u S T { ln S T > ln K } (cid:105) − K E R t (cid:104) e − (cid:82) TH r d u { ln S T > ln K } (cid:105) . (65)Using the Fourier inversion method outlined in Bakshi and Madan (2000) and Duffieet al. (2000), the expected call option price can be solved as E t [ C H ] = E t [e − (cid:82) TH q d u S H ]Π − K E t [ P ( H, T )]Π , (66)where the expected prices of the underlying asset and the T -maturity pure discount bondare given as: E t [e − (cid:82) TH q d u S H ] (cid:44) E R t (cid:104) e − (cid:82) TH r d u S T (cid:105) = φ R (1) , E t [ P ( H, T )] (cid:44) E R t (cid:104) e − (cid:82) TH r d u (cid:105) = φ R (0) , and for j = 1 , 2, the two probabilities Π j are calculated by using the Fourier inversionformula: Π j (cid:44) E R j t (cid:2) { ln S T > ln K } (cid:3) = 12 + 1 π (cid:90) ∞ Re (cid:34) e − iu ln K Φ R j ( u ) iu (cid:35) d u, with the corresponding characteristic functions Φ E ( u ) and Φ E ( u ), given asΦ E ( u ) = φ R ( iu + 1) φ R (1) and Φ E ( u ) = φ R ( iu ) φ R (0) , where i is the imaginary number.Alternatively, for corresponding European put option P , we have E t [ P H ] = K E t [ P ( H, T )] (1 − Π ) − E t [e − (cid:82) TH q d u S H ] (1 − Π ) . (67)The current prices of a call and a put can be obtained as special cases of equations (66)and (67), respectively, with H = t . 39 .3.2 Expected Option Return Simulation This section provides summary results of the simulation of the expected returns on Europeanequity index options using the models nested in the SVJJ model for different combinations ofthe holding period H , and the option expiration date T . We consider the following models:the constant volatility model of Black and Scholes (1973) (BS), the stochastic volatilitymodel of Heston (1993) (SV), the stochastic volatility jump models of Bakshi et al. (1997),Bates (1996, 2000), and Pan (2002) (SVJ), and the stochastic volatility double-jump modelof Duffie et al. (2000) (SVJJ). Due to space constraints we report only a short summaryof the main results here and the full details of the simulation results are reported in theInternet Appendix IG. The parameters for the simulations are borrowed from the empiricalstudies of Broadie et al. (2007) and Eraker et al. (2003).Overall expected returns of both calls and puts display more extreme patterns whenconsidering shorter option maturity, shorter holding period, and out-of-the-money (OTM)options (due to the leverage effect). The expected returns of calls are always positive andexceed the risk-free rate, while those of puts are generally negative and less than the risk-free rate under the BS and SV models. Between the BS model and the SV model, theexpected returns are slightly lower for both calls and puts under the SV model due to thenegative effect of the market price of volatility risk on the expected returns of options. Theexpected returns of both the calls and the puts are very highly negative (more negativefor puts than calls) under the SVJ model due to the negative effect of the market prices ofvolatility risk and volatility of price jump risk; and even more negative under the SVJJ modelwith the additional negative effect of the market price of volatility jump risk. The highlynegative expected option returns under the SVJ and SVJJ models are consistent with variousempirical studies that have reported highly negative put option returns and highly negativestraddle returns (see, e.g., Coval and Shumway, 2001; Broadie et al., 2009; Chambers et al.,2014). Interestingly, the expected returns of the holding-to-maturity call options are notonly less than the risk-free rate, but also highly negative, suggesting significant risk-loving behavior on the upside (see Coval and Shumway, 2001) under the SVJ and the SVJJ models.The last result is consistent with the well-known “pricing kernel puzzle” (see Jackwerth,2000; A¨ıt-Sahalia and Lo, 2000; Rosenberg and Engle, 2002) as well as the application of thecumulative prospect theory by Baele et al. (2019) for explaining the risk-loving behavior onthe upside. More details of these simulations are available in the Internet Appendix IG.40 .3.3 Other Models The theoretical results in sections 3 and 4 can be used for deriving the “finite-horizon” ex-pected prices of contingent claims in different securities markets, including European equityoptions, variance swaps and VIX futures, Treasury bonds, corporate bonds, interest ratederivatives, credit derivatives, and others, under different models. As few more applications,the Internet Appendix IF derives the analytical solutions of the expected prices of Europeanequity options under the CGMY model of Carr et al. (2002)—which nests other infinite-activity L´evy jump models, such as the variance-gamma model of Madan and Seneta (1990)and Madan et al. (1998)—using the R -transform; and the Internet Appendix IE derives theanalytical solutions of the expected prices of three broad types of interest rate derivativesunder a general affine jump-diffusion model and a special case of this model given as thetwo-jump model (see Chacko and Das (2002)), using the R -transform and the extended R -transform. The literature on option pricing contains many well-known no-arbitrage results—such as theput-call parity (see Stoll (1969)); the convexity of the call price function (see Merton (1973));the existence of at least one equivalent martingale measure under which discounted pricesare martingales (see Harrison and Kreps (1979)); and the existence of a positive state pricedensity which equals the second derivative of the call price function with respect to the strikeprice (see Breeden and Litzenberger (1978))—which apply to the current prices under alloption pricing models, whether parametric, non-parametric, arbitrage-based or equilibrium-based. These results have led to significant“model-free” results developed in the papers ofA¨ıt-Sahalia and Lo (1998), Britten-Jones and Neuberger (2000), Bakshi et al. (2003), Jiangand Tian (2005), Ross (2015), and Martin (2017), among others.No-arbitrage results also apply to the expected future prices of European options, but mostof them follow trivially from the no-arbitrage results mentioned above for the current prices. In this section we outline a few non-trivial no-arbitrage results related to the EEM theory.To motivate the need for such no-arbitrage results, consider a complex European contingent For example, a couple of trivial no-arbitrage relations related to the expected future prices of Europeanoptions are given as follows:i) the put-call expectations parity can be given as: E t [ P H ] = E t [ C H ] + K E t [ P ( H, T )] − E t [ S H ] , ii) the absence of arbitrage guarantees the existence of at least one EEM via Theorem 1, which can be usedto obtain the expected future price of a European contingent claim using either equation (5) or (6). We show how to obtain the expected future price ofthis complex European contingent claim using market-observed current prices and expectedreturns of the standard European OTM calls and puts written on the underlying asset.In other words, we show how the current prices and expected returns from a very liquidoption market of standard calls and puts, can be used to estimate the expected future priceof a complex claim in a non-liquid market. As another no-arbitrage result, we show howthe market-observed current prices and the “holding-to-maturity” expected returns of thestandard European OTM calls and puts, are related to the physical density of the underlyingasset’s future price.Breeden and Litzenberger (1978) show that the second derivative of the call option pricefunction with respect to the strike price gives the continuum of Arrow-Debreu prices or thestate price density (SPD). This result has been used by researchers to get non-parametricmodel-free estimates of the SPD, which are then used for valuing a wider range of complexEuropean derivative securities and to study the fundamental asset-pricing dynamics. Thissection demonstrates that analogous to the Breeden and Litzenberger result, the secondderivative of the expected future call price function with respect to the strike price givesthe expected future state price density (FSPD). The expected FSPD equals the discounted Many empirical papers on option expected returns do not make strong assumptions about the returngenerating process, which are typically made under the parametric option pricing models that allow analyticalsolutions (e.g., see Coval and Shumway, 2001; Bondarenko, 2003, 2014; Jones, 2006; Driessen and Maenhout,2007; Santa-Clara and Saretto, 2009; Bakshi et al., 2010; Constantinides et al., 2013; Israelov and Kelly,2017; Baele et al., 2019; B¨uchner and Kelly, 2019). For example, see A¨ıt-Sahalia and Lo (1998). Unlike the conventional definition of the SPD as Arrow-Debreu price density, A¨ıt-Sahalia and Lo (1998) normalize the SPD so that it equals the risk-neutral densityfor the case of constant interest rates, and the forward measure density for the case of stochastic interestrates. We use the conventional definition of the SPD as the Arrow-Debreu price density in this paper, sincewe do not normalize it. For example, see A¨ıt-Sahalia and Lo (1998), Jackwerth (2000), Rosenberg and Engle (2002), Bakshiet al. (2003), Chabi-Yo et al. (2008), Xing et al. (2010), Bollerslev and Todorov (2011), Conrad et al. (2013),Kozhan et al. (2013), Christoffersen et al. (2013), An et al. (2014), Chaudhuri and Schroder (2015), Ross(2015), Martin (2017), Martin and Wagner (2019), Chabi-Yo and Loudis (2020), and Jensen et al. (2019). T probability density (as shown below in Theorem 2), similar to how the SPD equals thediscounted forward density Q T . If the expected FSPD can be econometrically estimatedusing the expected future prices of standard European call and put options, then it can beused for obtaining the expected future prices (and expected returns) of a wider range ofcomplex European contingent claims with terminal payoffs that are arbitrary functions ofthe underlying asset’s future price. Though a full-blown econometric investigation of theexpected FSPD is beyond the scope of this paper, we present three results related to theexpected FSPD in the following three subsections, which reveal its importance for futureempirical research on the option return models. Assume that the SPD of an asset at current time t is given by the function f t ( S T ), where S T is the future price of this asset at time T . Also, assume that the future state price density(FSPD) of this asset at a future time H is given by the function f H ( S T | Y H ), where Y H is avector of state variables (such as future asset price S H , future volatility, etc.) at time H , forall t ≤ H ≤ T . Let the physical expectation g t ( H, S T ) (cid:44) E t [ f H ( S T | Y H )] define the expectedFSPD. Theorem 2 presents an extension of the Breeden and Litzenberger (1978) result.Consider a European call option with a payoff equal to C T = ( S T − K ) + and a Europeanput option with a payoff equal to P T = ( K − S T ) + at time T . Let C H and P H be the futureprices of the call option and the put option, respectively, at a future time H ≤ T . Theorem 2 The expected FSPD, g t ( H, S T ) (cid:44) E t [ f H ( S T | Y H )] , is given by the second deriva-tive of either the expected call price E t [ C H ] or the expected put price E t [ P H ] with respect tothe strike price K , and it equals the product of the expected bond price E P t [ P ( H, T )] and theprobability density p R T t ( S T ) . Formally, g t ( H, S T ) = (cid:20) ∂ E t [ C H ] ∂K (cid:21) K = S T = (cid:20) ∂ E t [ P H ] ∂K (cid:21) K = S T (68)= E P t [ P ( H, T )] p R T t ( S T ) , for all t ≤ H ≤ T which under the special case of constant or deterministic short rate becomes, g t ( H, S T ) = (cid:104) e − (cid:82) TH r u d u (cid:105) p R t ( S T ) , for all t ≤ H ≤ T, (69) where p R t ( S T ) and p R T t ( S T ) are time t equivalent probability densities of S T under R and R T easures, respectively. Proof. See Appendix A.9.Equation (68) expresses the expected FSPD, g t ( H, S T ) (cid:44) E t [ f H ( S T | Y H )], as a functionof S T and H . According to equation (68), the expected FSPD equals the discounted R T density, analogous to how the SPD equals the discounted (forward) Q T density in Breedenand Litzenberger (1978), as follows: f t ( S T ) = (cid:2) ∂ C t /∂K (cid:3) K = S T = P ( t, T ) p Q T t ( S T ) , for all t = H ≤ T. (70)According to equation (69), the expected FSPD equals the discounted R density for thespecial case when the short rate is either constant or deterministic, analogous to how theSPD equals the discounted (risk-neutral) Q density in Breeden and Litzenberger (1978)under this case, as follows: f t ( S T ) = (cid:2) ∂ C t /∂K (cid:3) K = S T = (cid:104) e − (cid:82) Tt r u d u (cid:105) p Q t ( S T ) , for all t = H ≤ T, (71)When H = t , the expected FSPD becomes the current SPD, and equations (70) and (71)obtain as special cases of equations (68) and (69), respectively. This section demonstrates how the expected FSPD can be obtained using the market-observed current prices and the historical estimates of the expected returns of standardEuropean OTM calls and OTM puts for a finite range of different strikes. We also demon-strate how the expected FSPD can be used for obtaining the expected future prices of awider range of complex European contingent claims (with terminal payoffs that are arbi-trary functions of the underlying asset’s future price), similar to how the non-parametricestimate of the SPD obtained from the current prices of standard European OTM calls andputs can be used for obtaining the current prices of a wider range of complex Europeancontingent claims (see A¨ıt-Sahalia and Lo (1998)). Before deriving these results, Corollary 2given below presents a no-arbitrage relationship between the SPD and the expected FSPD. Equation (70) follows as a special case of equation (68), since by property (iv) of Proposition 3, R T converges to Q T , when t = H . Similarly, equation (71) follows as a special case of equation (69), since byproperty (iv) of Proposition 1, R converges to Q , when t = H . orollary 2 By the absence of arbitrage, R T and Q T are equivalent probability measures,which implies: g t ( H, S T ) > if and only if f t ( S T ) > , (72) and g t ( H, S T ) = 0 if and only if f t ( S T ) = 0 . (72 a ) Proof. Since the absence of arbitrage implies that the P measure is equivalent to boththe R T measure (by Theorem 1(ii)) and the Q T measure (by the first fundamental theoremof asset pricing), it follows that R T and Q T are also equivalent probability measures. Inaddition, since both E P t [ P ( H, T )] and P ( t, T ) are always positive, equations (68) and (70)imply equations (72) and (72 a ), respectively.Since the expected FSPD or g t ( H, S T ) is given as the curvature of E t ( C H ) with respectto the strike price in equation (68), and the SPD or f t ( S T ) is given as the correspondingcurvature of C t in equation (70), the absence of arbitrage implies that whenever the curvatureof the current call price C t with respect to the strike price is greater than (equal to) zero, thecorresponding curvature of the expected future call price E t [ C H ] is also greater than (equalto) zero.We now demonstrate how the expected FSPD can be obtained using the current pricesand the expected returns of standard European OTM calls and OTM puts. Let the time t expectation of the future price of a European option (either call or put) at time H be givenas follows: E t [ C H ] = C t × (1 + R ct ( k t , h, τ )) , (73) (cid:122) (cid:125)(cid:124) (cid:123) Forward-looking (cid:122) (cid:125)(cid:124) (cid:123) Backward-looking E t [ P H ] = P t × (1 + R pt ( k t , h, τ )) , (73 a ) (cid:122) (cid:125)(cid:124) (cid:123) Forward-looking (cid:122) (cid:125)(cid:124) (cid:123) Backward-lookingwhere C t and P t are the time t prices of the call and the put, respectively; k t = K/S t measuresthe moneyness of the option; h = H − t is the holding period, which is less than or equal to τ = T − t , the time remaining until option expiration date; and R ct ( k t , h, τ ) = E t [ C H ] /C t - 1and R pt ( k t , h, τ ) = E t [ P H ] /P t - 1, are the expected simple returns of the call option and theput option, respectively. 45ccording to equations (73) and (73 a ), the historical data-based backward-looking ex-pected option returns R ct ( k t , h, τ ) and R pt ( k t , h, τ ), combined with the market-observed forward-looking prices of these options C t and P t , provide the estimates of the expected future pricesof these options E t [ C H ] and E t [ P H ], respectively. By extending the non-parametric methods,such as Jackwerth (2004) and Figlewski (2009), twice-differentiable functions (with respectto the strike price) can be fitted to the expected future prices E t [ C H ] and E t [ P H ], usinga finite number of out-of-the-money (OTM) calls and OTM puts, and then the expectedFSPD or g t ( H, S T ) is obtained by numerically computing the second derivative of the ex-pected future price functions of the OTM calls and the OTM puts using Theorem 2. As anillustrative example, the technical details of such a non-parametric method is demonstratedwith an extension of the fast and stable method of Jackwerth (2004) in Internet AppendixIH. Obtaining the estimate of g t ( H, S T ) in this manner is economically intuitive. It is wellknown that the backward-looking estimates of expected returns R ct ( k t , h, τ ) and R pt ( k t , h, τ ),depend more on preferences which can be assumed to be time-stationary. For example, ina recent paper, Baele et al. (2019) find that R ct ( k t , h, τ ) and R pt ( k t , h, τ ) are much more sen-sitive to the preference parameters (related to the cumulative prospect theory-based utilityfunction) than the assumed return distribution of the underlying asset. On the other hand,the forward-looking current option prices C t and P t reflect new information, which allows E t [ C H ] and E t [ P H ] to change dynamically in equations (73) and (73 a ). Thus, new infor-mation revealed by the current option prices is incorporated in the dynamically changingestimate of g t ( H, S T ) using Theorem 2.The option expected returns R ct ( k t , h, τ ) and R pt ( k t , h, τ ) can be estimated using time-series data on call and put options of various strikes either unconditionally or conditionally.This approach does not require making strong parametric assumptions about the returngenerating process (as in Bates, 1996; Bakshi et al., 1997; Duffie et al., 2000; Bates, 2000;Andersen et al., 2002; Pan, 2002; Eraker et al., 2003; Eraker, 2004; Broadie et al., 2007, 2009;Chambers et al., 2014; Chaudhuri and Schroder, 2015) since it does not use a parametricoption pricing model to get the expected FSPD, but obtains the expected FSPD from theexpected returns of standard European calls and puts using historical option price data. Inother words, it requires knowing only the first moment of the option returns , and not theentire physical probability distributions assumed under the parametric option pricing models.It can use historical time-series data-based option return models that do not make strongassumptions about the underlying probability distribution (e.g., see Coval and Shumway,2001; Bondarenko, 2003, 2014; Jones, 2006; Driessen and Maenhout, 2007; Santa-Clara andSaretto, 2009; Bakshi et al., 2010; Constantinides et al., 2013; Israelov and Kelly, 2017; Baele46t al., 2019; B¨uchner and Kelly, 2019).The next corollary shows how the empirically-estimated expected FSPD can be used toget the expected future prices of complex European contingent claims. Corollary 3 The expected future price of a contingent claim F with an arbitrary terminalpayoff function F T = h ( S T ) , can be numerically computed using the expected FSPD, g t ( H, S T ) as follows: E t [ F H ] = (cid:90) ∞ g t ( H, S ) h ( S ) dS. (74) Proof. The proof follows by substituting g t ( H, S T ) from equation (68) into equation (14),and simplifying.The expected FSPD, g t ( H, S T ), is the only input needed in the above equation which canbe solved numerically for any non-linear function h ( S T ). The recovery theorems of Ross (2015) and others (see, e.g. Carr and Yu, 2012; Qin and Linet-sky, 2016; Walden, 2017; Martin and Ross, 2019) assume a risk-averse representative investor,a transition independent pricing kernel, and time-homogeneous transition probabilities. Asshown by Ross (2015), under these assumptions a unique physical density exists using thePerron-Frobenius theorem. This theoretical setup places extremely stringent assumptions onthe return distribution process. For example, the time-homogeneity of the transition prob-abilities is inconsistent with the return distributions under virtually all of the known optionpricing and asset pricing models used in the equity returns literature. Moreover, as shownby Boroviˇcka et al. (2016), the time-homogeneity of transition probabilities implies that themartingale component of the stochastic discount factor is a constant equal to unity, whichis untrue given the lower bounds on the martingale component’s variance derived in Alvarezand Jermann (2005) and Bakshi and Chabi-Yo (2012) for investments with a mix of bothequity and fixed income securities. This assumption is also seemingly untrue for the subsetof purely fixed income investments, as shown by i) the volatile behavior of the long bond,which violates this assumption as shown in Qin et al. (2018), and ii) the violations of therestrictions that link the risk-neutral and physical distributions implied by this assumption,as shown by Bakshi et al. (2017) with the data on the options on the 30-year bond futures.Thus, recovering the forward-looking physical density does not seem feasible using the Ross472015) recovery theorem. While the recovery theorems of Ross (2015) and others have focused on how to obtain thephysical density from the SPD, surprisingly a much simpler and direct relation between theunderlying asset’s physical probability density and its expected FSPD has gone unnoticed.To see this define the expected terminal FSPD as the expected FSPD in equation (68)with the horizon date H equal to the option expiration date T . The following corollary toTheorem 2 demonstrates this result: Corollary 4 The expected terminal FSPD (i.e., for the special case H = T ) given by equa-tion (68) of Theorem 2 is equal to the physical density of the underlying asset’s future price.Formally, E t [ f T ( S T )] = (cid:20) ∂ E t [ C T ] ∂K (cid:21) K = S T = p P t ( S T ) , for all t ≤ H = T, (75) where p P t ( S T ) is the time t physical density of S T under the P measure. Proof. This result can be obtained as a special case of Theorem 2 for H = T . While Corollary 4 is not directly related to the debate on the recovery theorems—sincethe relationship of the physical density is not demonstrated with the SPD, but with theexpected terminal FSPD—it still provides a new perspective on the recovery theorems ofRoss (2015) and others. It shows that the relationship between the physical probabilitydensity and option prices can be obtained as an arbitrage-based, tautologically exact resultonly when the relationship is made using the expected terminal FSPD via the expected Relaxing the time-homogeneity assumption, Jensen et al. (2019) extend the Ross (2015) recovery frame-work by recovering the physical density using more equations corresponding to “several” future time periods.Jackwerth and Menner (2020) consider variants of Ross (2015) recovery theorem with additional economicrestrictions, including the generalized recovery theorem of Jensen et al. (2019), which they derive inde-pendently. The density tests and the mean and variance prediction tests of Jackwerth and Menner (2020)further underscore not only the failure of the recovery theorem of Ross (2015) and its variants with additionaleconomic restrictions, but also the failure of the generalized recovery theorem of Jensen et al. (2019)—allof which reject the null hypothesis that future realized one-month S&P 500 returns are drawn from theone-month physical distribution implied by these theorems. In contrast, Jackwerth and Menner (2020)find that two simple benchmark methods (power utility with the risk aversion coefficient equal to 3, andhistorical return distribution method) perform much better as they tend not to reject the null hypothesis intheir various tests. The first equality in equation (75) follows directly by applying Theorem 2 for H = T . The secondequality in equation (75) can be obtained by noting that when H = T , E P t [ P ( T, T )] p R T t ( S T ) = p P t ( S T )or R T = P , in equation (68) by using Proposition 3(iii), and using the T -maturity pure discount bond, P ( · , T ) as the numeraire. With this numeraire, G T = P ( T, T ) = 1 in Proposition 3(iii). Therefore, for thespecial case, Proposition 3(iii) implies that when H = T , the R ∗ measure becomes the P measure, since G T / E P t [ G T ] = P ( T, T ) / E P t [ P ( T, T )] = 1. The second equality in equation (75) also follows by substituting C T = ( S T − K ) + , and obtaining the second derivative with respect to K directly. only the firstmoment of the terminal value of the call option with respect to the strike price. Thus, otherhigher order moments of the terminal values of call options or the higher order momentsof the future price of the asset are not required for approximating the physical probabilitydensity.Interestingly, Corollary 4 makes all of the results derived in sections 5.1 and 5.2 forobtaining the expected FSPD from current prices and expected returns of standard EuropeanOTM calls and puts immediately applicable for obtaining the physical density using thespecial case of H = T . Of course, the robustness of the physical density obtained in thisway depends on the nature of assumptions made in the empirical papers on option returnmodels, and can be investigated by future research. Our more immediate and limited goalin this subsection is similar to that of the previous subsection: to demonstrate that theexpected terminal FSPD can used for obtaining the expected future prices of a wider rangeof complex European contingent claims with terminal payoffs that are arbitrary functions ofthe underlying asset’s future price. Of course, since the expected terminal FSPD equals thephysical density, Corollary 4 can also be used for obtaining the higher order moments of thecomplex claim’s future price or returns. This section considers some extensions and applications of the EEM theory and then con-cludes the paper. Until now the paper has focused only on those European contingent claimmodels that admit an analytical solution to the claim’s current price. As the first extension,consider a general claim (e.g., European, Barrier, Bermudan, quasi-American, etc.) thatremains alive until time H , but the analytical solution of its current price does not exist.Numerical methods can be used to compute the expected price of such a claim at time H using the EEMs, such as the R measure. For example, consider a quasi-American put op-tion which is exercisable on or after time T e ≥ H , until the option expiration date T (i.e., The estimation of expected option returns using historical data can be done without making strongparametric assumptions about the physical density of the underlying asset price process (e.g., see Covaland Shumway, 2001; Bondarenko, 2003, 2014; Jones, 2006; Driessen and Maenhout, 2007; Santa-Clara andSaretto, 2009; Bakshi et al., 2010; Constantinides et al., 2013; Israelov and Kelly, 2017; Baele et al., 2019;B¨uchner and Kelly, 2019). e ≤ T ). The expected price of this option at time H is given as E t (cid:2) P AH (cid:3) = sup τ ∈T Te,T E R t (cid:104) e − (cid:82) τH r u d u P Aτ (cid:12)(cid:12)(cid:12) F t (cid:105) for 0 ≤ t ≤ H ≤ T e ≤ T, (76)where P As is the price of the quasi-American put option at time s , for all H ≤ s ≤ T , P AT = P T = max( K − S T , T is the set of all stopping times with respect to the filtration F , and T T e ,T (cid:44) { τ ∈ T | P ( τ ∈ [ T e , T ]) = 1 } is the subset of T . The above equation followsby applying properties (i) and (ii) of Proposition 1, and valuing the American option as anoptimal stopping problem (see Jacka, 1991) from time T e until time T . The above exampleshows how the EEM theory can be applied for obtaining the expected price of any type ofclaim at time H , assuming the claim remains alive until time H . Numerical methods suchas trees, finite difference methods, and Monte Carlo methods can be used for pricing suchclaims under the R measure.Second, consider those claims that may not be alive until time H in some states of nature.For example, consider a standard American option that can be exercised at any time untilthe expiration date of the option. While the expected future price of an American optioncannot be computed, since the option is not alive at time H in some states of nature, theexpected future price of a reinvestment strategy that invests the value of the American optionupon exercise in another security like a Treasury bond or a money market account, untiltime H , can be computed numerically using the R measure.Third, though this paper has focused on the derivation of the analytical solution of onlythe first moment of the future price of a contingent claim, future research can use the EEMtheory to derive analytical solutions of the variance and other higher order moments ofthe claim’s future price. Also, various numerical methods, such as trees, finite differencemethods, and Monte Carlo methods can be used to obtain the higher order moments of thefuture price of a portfolio of claims under the R measure. Furthermore, various finite-horizontail risk measures (such as Value-at-Risk, conditional VaR, expected shortfall, and others)of contingent claim portfolios can be obtained with Monte Carlo simulations under the R measure. Thus, EEMs such as the R measure can significantly advance the risk and returnanalysis of fixed income and financial derivative portfolios.Finally, perhaps the most important empirical application of the analytical solutionsof the expected future prices of contingent claims obtained using the EEM theory is thatvarious simulation methods, such as the Markov Chain Monte Carlo, generalized method ofmoments, and others become feasible for studying the cross-section of returns of a variety ofcontingent claims, such as Treasury bonds, corporate bonds, interest rate derivatives, creditderivatives, equity derivatives, and others. It is our hope that the EEM theory can shift the50ocus of research from mostly valuation to both valuation and expected returns, for a varietyof finite-maturity securities and claims used in the fixed income markets and the derivativesmarkets, respectively.We conclude this paper by summarizing its main results. First and foremost, this paperhas developed the EEM theory for deriving the analytical solutions of the expected prices(both current prices and expected future prices) of European contingent claims, generalizingthe risk-neutral valuation approach of Black-Scholes-Merton and Cox and Ross (1976), as wellas the martingale pricing theory of Harrison and Kreps (1979). We present three equivalentexpectation measures—the R measure, the R T measure, and the R S measure—as generaliza-tions of the three equivalent martingale measures—the Q measure, the Q T measure, and the Q S measure, respectively (see Harrison and Kreps, 1979; Geman et al., 1995)—for derivinganalytical solutions of the expected prices of a variety of contingent claims including Trea-sury bonds, corporate bonds, interest rate derivatives, credit derivatives, equity derivatives,and others. Second, this paper derives the R -transform and the extended R -transform asextensions of the Q -transforms of Bakshi and Madan (2000) and Duffie et al. (2000) forderiving analytical solutions of the expected prices of both standard and exotic claims undermore complex stochastic processes.Finally, as a new direction of empirical research, we extend the work of Breeden andLitzenberger (1978) and show that the second derivative of the expected future price of astandard European call option with respect to the strike price equals the expected futurestate price density (FSPD). Using the expected FSPD we provide no-arbitrage results whichcan be used to obtain the expected future prices of complex European contingent claims aswell as the physical density of the underlying asset’s future price, using the current prices and only the first return moment of standard European OTM call and put options.51 ppendix A The Proofs A.1 Proof of Theorem 1Proof of Theorem 1 (i) . To prove that R ∗ and P are equivalent probability measures,and L ∗ s ( H ) is the corresponding Radon-Nikod´ym derivative process, we show that L ∗ s ( H ) isa martingale; L ∗ T ( H ) is almost surely positive; and E [ L ∗ T ( H )] = 1 (see, e.g., Shreve, 2004,Theorem 1.6.1 and Definition 1.6.3).Consider the following three cases to show that L ∗ s ( H ) is a martingale (see, e.g., Borodinand Salminen, 2002, Chapter I), or E (cid:2) L ∗ s ( H ) |F s (cid:3) = L ∗ s ( H ), for all 0 ≤ s ≤ s ≤ T .First, for all 0 ≤ s ≤ s ≤ H ≤ T , L ∗ s ( H ) = L ∗ s ( H ) = 1, hence E (cid:2) L ∗ s ( H ) |F s (cid:3) = L ∗ s ( H ).Second, for all 0 ≤ H ≤ s ≤ s ≤ T , E (cid:2) L ∗ s ( H ) |F s (cid:3) = E (cid:20) L ∗ s L ∗ H (cid:12)(cid:12)(cid:12) F s (cid:21) = E (cid:20) E [ L ∗ T |F s ] L ∗ H (cid:12)(cid:12)(cid:12) F s (cid:21) = E [ E [ L ∗ T |F s ] |F s ] L ∗ H = E [ L ∗ T |F s ] L ∗ H = L ∗ s L ∗ H = L ∗ s ( H ) . Third, for all 0 ≤ s ≤ H ≤ s ≤ T , E (cid:2) L ∗ s ( H ) |F s (cid:3) = E (cid:20) L ∗ s L ∗ H (cid:12)(cid:12)(cid:12) F s (cid:21) = E (cid:20) E (cid:20) L ∗ s L ∗ H (cid:12)(cid:12)(cid:12) F H (cid:21) F s (cid:21) = E (cid:20) E (cid:20) E [ L ∗ T |F s ] L ∗ H (cid:12)(cid:12)(cid:12) F H (cid:21) (cid:12)(cid:12)(cid:12) F s (cid:21) = E (cid:20) E [ E [ L ∗ T |F s ] |F H ] L ∗ H (cid:12)(cid:12)(cid:12) F s (cid:21) = E (cid:20) E [ L ∗ T |F H ] L ∗ H (cid:12)(cid:12)(cid:12) F s (cid:21) = E (cid:20) L ∗ H L ∗ H (cid:12)(cid:12)(cid:12) F s (cid:21) = L ∗ s ( H ) . Therefore, L ∗ s ( H ) is a martingale.Next, we show that L ∗ T ( H ) is almost surely positive and E [ L ∗ T ( H )] = 1. First, notethat L ∗ T is almost surely positive since P and Q ∗ are equivalent probability measures. Since L ∗ H = E [ L ∗ T |F H ], then L ∗ H is also almost surely positive. Finally, by definition (see equation(9)), L ∗ T ( H ) = L ∗ T /L ∗ H is also almost surely positive. Additionally, E [ L ∗ T ( H )] = E [ L ∗ T /L ∗ H ] = E [ E [ L ∗ T |F H ] /L ∗ H ] = E [ L ∗ H /L ∗ H ] = 1. Proof of Theorem 1 (ii) . We show that R ∗ and P are equivalent probability measures,and L ∗ s ( H ) is the corresponding Radon-Nikod´ym derivative process similar to the aboveprocess. First, we show that L ∗ s ( H ) is a martingale using the following three cases. For0 ≤ s ≤ s ≤ H ≤ T , E (cid:2) L ∗ s ( H ) |F s (cid:3) = E (cid:20) E s [ G H ] E [ G H ] (cid:12)(cid:12)(cid:12) F s (cid:21) = E s [ E s [ G H ]] E [ G H ] = E s [ G H ] E [ G H ] = L ∗ s ( H ) . ≤ H ≤ s ≤ s ≤ T , E (cid:2) L ∗ s ( H ) |F s (cid:3) = E (cid:20) G H E [ G H ] · L ∗ s L ∗ H (cid:12)(cid:12)(cid:12) F s (cid:21) = G H E [ G H ] · E s (cid:2) L ∗ s (cid:3) L ∗ H = G H E [ G H ] · L ∗ s L ∗ H = L ∗ s ( H ) . For 0 ≤ s ≤ H ≤ s ≤ T , E (cid:2) L ∗ s ( H ) |F s (cid:3) = E (cid:20) G H E [ G H ] · L ∗ s L ∗ H (cid:12)(cid:12)(cid:12) F s (cid:21) = E s (cid:20) E H (cid:20) G H E [ G H ] · L ∗ s L ∗ H (cid:21)(cid:21) = E s (cid:34) G H E [ G H ] · E H (cid:2) L ∗ s (cid:3) L ∗ H (cid:35) = E s (cid:20) G H E [ G H ] · L ∗ H L ∗ H (cid:21) = E s [ G H ] E [ G H ] = L ∗ s ( H ) . Next, we show that L ∗ T ( H ) is almost surely positive and E [ L ∗ T ( H )] = 1. Since thenumeraire asset G is positive, using similar logic as in the proof of Theorem 1(i), L ∗ T ( H ) isalmost surely positive, and E [ L ∗ T ( H )] = E (cid:20) G H E [ G H ] · L ∗ T L ∗ H (cid:21) = E (cid:20) E H (cid:20) G H E [ G H ] · L ∗ T L ∗ H (cid:21)(cid:21) = E (cid:20) G H E [ G H ] · E H [ L ∗ T ] L ∗ H (cid:21) = E (cid:20) G H E [ G H ] · L ∗ H L ∗ H (cid:21) = 1 . Proof of Theorem 1 (iii) . Equation (5) follows from the definition of the R ∗ measure, as E t [ F H ] = E P t (cid:20) E Q ∗ H (cid:20) G H F T G T (cid:12)(cid:12)(cid:12) F H (cid:21) (cid:12)(cid:12)(cid:12) F t (cid:21) = E P t (cid:20) E P H (cid:20) L ∗ T L ∗ H · G H F T G T (cid:12)(cid:12)(cid:12) F H (cid:21) (cid:12)(cid:12)(cid:12) F t (cid:21) = E P t (cid:20) L ∗ T L ∗ H · G H F T G T (cid:12)(cid:12)(cid:12) F t (cid:21) = E P t (cid:20) L ∗ T /L ∗ H · G H F T G T (cid:12)(cid:12)(cid:12) F t (cid:21) = E P t (cid:20) L ∗ T ( H ) L ∗ t ( H ) · G H F T G T (cid:12)(cid:12)(cid:12) F t (cid:21) = E R ∗ t (cid:20) G H F T G T (cid:12)(cid:12)(cid:12) F t (cid:21) . (A1)53o obtain equation (6), we start with the third equality of equation (A1), which gives E t [ F H ] = E P t (cid:20) L ∗ T L ∗ H · G H F T G T (cid:12)(cid:12)(cid:12) F t (cid:21) = E P t [ G H ] E P t (cid:20) G H E P t [ G H ] · L ∗ T L ∗ H · F T G T (cid:12)(cid:12)(cid:12) F t (cid:21) = E P t [ G H ] E P t (cid:20) G H / E [ G H ] · L ∗ T /L ∗ H E P t [ G H ] / E [ G H ] F T G T (cid:12)(cid:12)(cid:12) F t (cid:21) = E P t [ G H ] E P t (cid:20) L ∗ T ( H ) L ∗ t ( H ) · F T G T (cid:12)(cid:12)(cid:12) F t (cid:21) = E P t [ G H ] E R ∗ t (cid:20) F T G T (cid:12)(cid:12)(cid:12) F t (cid:21) . (A2)The relationship defined between R ∗ and R ∗ in equation (7) can also be reconfirmed asfollows. For any F T -measurable variable Z T , we have E R ∗ t [ Z T ] = E P t (cid:20) L ∗ T ( H ) L ∗ t ( H ) Z T (cid:21) = E P t (cid:20) G H / E [ G H ] · L ∗ T /L ∗ H E P t [ G H ] / E [ G H ] Z T (cid:21) = E P t (cid:20) L ∗ T L ∗ H · G H E P t [ G H ] Z T (cid:21) = E P t (cid:20) L ∗ T ( H ) L ∗ t ( H ) · G H E P t [ G H ] Z T (cid:21) = E R ∗ t (cid:20) G H E P t [ G H ] Z T (cid:21) = E R ∗ t (cid:20) G H E R ∗ t [ G H ] Z T (cid:21) , (A3)where the last equality follows since E R ∗ t [ G H ] = E P t [ L ∗ H ( H ) / L ∗ t ( H ) · G H ] = E P t [ G H ]. A.2 Proof of Proposition 1Proof. For the first property, for all 0 ≤ t ≤ H , given A ∈ F H , we have R ∗ ( A |F t ) = E R ∗ t [ A ] = E P t [ L ∗ H ( H ) / L ∗ t ( H ) · A ] = E P t [( L ∗ H /L ∗ H ) / · A ] = E P t [ A ] = P ( A |F t ).For the second property, for all H ≤ s ≤ T , given A ∈ F T , we have R ∗ ( A | F s ) = E R ∗ s [ A | F s ] = E P s (cid:20) L ∗ T ( H ) L ∗ s ( H ) A (cid:12)(cid:12)(cid:12) F s (cid:21) = E P s (cid:20) L ∗ T /L ∗ H L ∗ s /L ∗ H A (cid:12)(cid:12)(cid:12) F s (cid:21) = E P s (cid:20) L ∗ T L ∗ s A (cid:12)(cid:12)(cid:12) F s (cid:21) = E Q ∗ s [ A | F s ] = Q ∗ ( A | F s ) . (A4)The third property is a special case of the first property when H = T , while the fourthproperty is a special case of the second property when H = t .54or the last property, we have E R ∗ t [ Z T | F t ] = E R ∗ t (cid:2) E R ∗ H [ Z T | F H ] (cid:12)(cid:12) F t (cid:3) = E R ∗ t (cid:104) E Q ∗ H [ Z T | F H ] (cid:12)(cid:12) F t (cid:105) = E P t (cid:104) E Q ∗ H [ Z T | F H ] (cid:12)(cid:12) F t (cid:105) , where the second equality is an application of the second property, the last equality isan application of the first property, since E Q ∗ H [ Z T | F H ] is a random variable that is F H measurable. A.3 Proof of Proposition 2Proof. For the first property, for all 0 ≤ H ≤ s ≤ s ≤ T , E R ∗ s (cid:20) F s G s |F s (cid:21) = E P s (cid:20) L ∗ s ( H ) L ∗ s ( H ) · F s G s (cid:12)(cid:12)(cid:12) F s (cid:21) = E P s (cid:20) L ∗ s /L ∗ H L ∗ s /L ∗ H · F s G s (cid:12)(cid:12)(cid:12) F s (cid:21) = E P s (cid:20) L ∗ s L ∗ s · F s G s (cid:12)(cid:12)(cid:12) F s (cid:21) = E Q ∗ s (cid:20) F s G s (cid:12)(cid:12)(cid:12) F s (cid:21) = F s G s . For the second property, for all 0 ≤ s ≤ s ≤ H ≤ T , E R ∗ s (cid:20) F s G s (cid:12)(cid:12)(cid:12) F s (cid:21) = E P s (cid:20) L ∗ s ( H ) L ∗ s ( H ) · F s G s (cid:12)(cid:12)(cid:12) F s (cid:21) = E P s (cid:20) · F s G s (cid:12)(cid:12)(cid:12) F s (cid:21) = E P s (cid:20) F s G s (cid:12)(cid:12)(cid:12) F s (cid:21) , which is generally not equal to E Q ∗ s [ F s /G s | F s ], except when P = Q ∗ . A.4 Proof of Proposition 3Proof. For the first property, for all 0 ≤ t ≤ H , given A ∈ F H , we have R ∗ ( A |F t ) = E R ∗ t [ A ] = E P t (cid:20) L ∗ H ( H ) L ∗ t ( H ) · A (cid:21) = E P t (cid:20) G H / E P [ G H ] · L ∗ H /L ∗ H E P t [ G H ] / E P [ G H ] · A (cid:21) = E P t (cid:20) G H E P t [ G H ] A (cid:12)(cid:12)(cid:12) F t (cid:21) . H ≤ s ≤ T , given A ∈ F T , we have R ∗ ( A | F s ) = E R ∗ s [ A | F s ] = E P s (cid:20) L ∗ T ( H ) L ∗ s ( H ) A (cid:12)(cid:12)(cid:12) F s (cid:21) = E P s (cid:20) G H / E P [ G H ] · L ∗ T /L ∗ H G H / E P [ G H ] · L ∗ s /L ∗ H A (cid:12)(cid:12)(cid:12) F s (cid:21) = E P s (cid:20) L ∗ T L ∗ s A (cid:12)(cid:12)(cid:12) F s (cid:21) = E Q ∗ s [ A | F s ] = Q ∗ ( A | F s ) . (A5)Furthermore, according to property (ii) of Proposition 1, R ∗ ( A | F s ) = R ∗ ( A | F s ) alsoholds for all A ∈ F T .The third property is a special case of the first property when H = T , while the fourthproperty is a special case of the second property when H = t .The first part of the last property is a direct result of equation (7), which was proven inequation (A3). For the second part of the last property, we have E R ∗ t [ Z T | F t ] = E R ∗ t (cid:104) E R ∗ H [ Z T | F H ] (cid:12)(cid:12) F t (cid:105) = E R ∗ t (cid:104) E Q ∗ H [ Z T | F H ] (cid:12)(cid:12) F t (cid:105) = E P t (cid:20) G H E P t [ G H ] E Q ∗ H [ Z T | F H ] (cid:12)(cid:12) F t (cid:21) , where the second equality is an application of the second property, the last equality isan application of the first property, since E Q ∗ H [ Z T | F H ] is a random variable that is F H measurable. A.5 Proof of the Black-Scholes ModelProof of Brownian motion under R . Consider the Radon-Nikod´ym derivative processes L ∗ s ( H ) and L ∗ s in part (i) of Theorem 1 and denote them as L s ( H ) and L s , respectively, forthe special case when the numeraire is the money market account, or G = B (cid:44) e (cid:82) · r u d u . Forthis special case, R ∗ = R , as defined in section 2.4, and we know that L s under the Blackand Scholes (1973) model is given as L s (cid:44) d Q d P (cid:12)(cid:12)(cid:12) F s = exp (cid:18) − (cid:90) s γ d W P u − (cid:90) s γ d u (cid:19) , (A6)where γ = ( µ − r ) /σ is the market price of risk (MPR) and r is the constant risk-free rate.56sing equations (9) and (A6), L s ( H ) can be given as follows: L s ( H ) (cid:44) d R d P (cid:12)(cid:12)(cid:12) F s = L s L H = exp (cid:18) − (cid:90) sH γ d W P u − (cid:90) sH γ d u (cid:19) , if s ≥ H, , if s < H, = exp (cid:18) − (cid:90) s γ u ( H )d W P u − (cid:90) s ( γ u ( H )) d u (cid:19) . (A7)where γ s ( H ) (cid:44) { s ≥ H } γ = (cid:40) γ, if s ≥ H, , if s < H. The Girsanov theorem allows the transformation of Brownian motions under two equiv-alent probability measures. Therefore, given Q and P , we know that W Q s = W P s + (cid:82) s γ d u .Similarly, the relationship of the Brownian motions under the R and P measures is given as W R s = W P s + (cid:90) s γ u ( H )d u = W P s + (cid:90) s { u ≥ H } γ d u. (A8) Proof of Black-Scholes expected price solution. Under the R measure, the expectedvalue and variance in ln S T are E R t [ln S T ] = ln S t + ( r − σ )( T − t ) + σγ ( H − t )= ln S t + µ ( H − t ) + r ( T − H ) − σ ( T − t ) , Var R t [ln S T ] = σ ( T − t ) . (A9)Therefore, we have E R t [ S T ] = E R t [e ln S T ] = exp (cid:18) E R t [ln S T ] + 12 Var R t [ln S T ] (cid:19) = S t e µ ( H − t )+ r ( T − H ) . (A10)Since the numeraire is the money market account with a constant risk-free rate r , ac-cording to equation (20), the expected option price holding until future time H with strikeprice K is E t [ C H ] = E R t (cid:2) e − r ( T − H ) ( S T − K ) + (cid:3) = E R t (cid:2) e − r ( T − H ) ( S T − K ) { ln S T > ln K } (cid:3) = e − r ( T − H ) E R t (cid:2) S T { ln S T > ln K } (cid:3) − K e − r ( T − H ) E R t (cid:2) { ln S T > ln K } (cid:3) E R t [ S T ]e − r ( T − H ) E ˆ R t (cid:2) { ln S T > ln K } (cid:3) − K e − r ( T − H ) E R t (cid:2) { ln S T > ln K } (cid:3) , (A11)where ˆ R is an adjusted probability measure defined by E ˆ R t [ Z T ] = E R t (cid:104) S T E R t [ S T ] · Z T (cid:12)(cid:12) F t (cid:105) for agiven F T -measurable variable Z T . Then, it is easy to obtain the desired equation (21) fromequations (A9) and (A10). A.6 Proof of Proposition 4Proof. We denote Ψ( Y t ; τ ) = E P t (cid:104) exp (cid:16) − b (cid:48) Y t + τ − (cid:82) t + τt c (cid:48) Y u d u (cid:17)(cid:105) . Under the parametricrestrictions for admissibility, according to the Feynman-Kac theorem, the expected valueΨ( Y t ; τ ) should fulfill the following PDE: − ∂ Ψ ∂τ + ∂ Ψ ∂Y K (Θ − Y ) + 12 tr (cid:20) ∂ Ψ ∂Y Σ V Σ (cid:48) (cid:21) = c (cid:48) Y Ψ , with a boundary condition Ψ( Y t ; 0) = exp ( − b (cid:48) Y t ). Then, we can easily show that the solutionto this PDE is Ψ( Y t ; τ ) = exp (cid:0) − A K , Θ ( τ ) − B K , Θ ( τ ) (cid:48) Y t (cid:1) , where A K , Θ ( τ ) and B K , Θ ( τ ) aregiven in equation (33). Similarly, we can also show that the solution of the expectation E Q H (cid:104) exp (cid:16) − b (cid:48) Y H + τ − (cid:82) H + τH c (cid:48) Y u d u (cid:17)(cid:105) is exp (cid:0) − A K ∗ , Θ ∗ ( τ ) − B K ∗ , Θ ∗ ( τ ) (cid:48) Y H (cid:1) .Note that under the R measure, the drift coefficients of Y are K and Θ from t to H andare K ∗ and Θ ∗ from H to T . Therefore, for the expected price of a T -maturity pure discountbond at time H , we have E t [ P ( H, T )] = E R t (cid:20) exp (cid:18) − (cid:90) TH r u d u (cid:19)(cid:21) = E R t (cid:20) exp (cid:18) − (cid:90) TH ( δ + δ y (cid:48) Y u ) d u (cid:19)(cid:21) = exp ( − δ ( T − H )) · E R t (cid:20) E R H (cid:20) exp (cid:18) − (cid:90) TH δ y (cid:48) Y u d u (cid:19)(cid:21)(cid:21) = exp ( − δ ( T − H )) · E P t (cid:20) E Q H (cid:20) exp (cid:18) − (cid:90) TH δ y (cid:48) Y u d u (cid:19)(cid:21)(cid:21) = exp ( − δ ( T − H )) · E P t (cid:104) exp (cid:16) − A (0 ,δ y ) K ∗ , Θ ∗ ( T − H ) − B (0 ,δ y ) K ∗ , Θ ∗ ( T − H ) (cid:48) Y H (cid:17)(cid:105) = exp (cid:16) − δ ( T − H ) − A (0 ,δ y ) K ∗ , Θ ∗ ( T − H ) − A ( b , K , Θ ( H − t ) − B ( b , K , Θ ( H − t ) (cid:48) Y t (cid:17) . .7 Proof of Proposition 5Proof. Under the payoff settings in Collin-Dufresne and Goldstein (2001), recall that thecurrent price of a risky discount bond D ( t, T ) maturing at time T is given by D ( t, T ) = P ( t, T ) (cid:16) − ω E Q T t (cid:104) { τ [ t,T ] Under the integration conditions in the definition of Duffie et al. (2000, page 1351),we can prove that the solution of the expectation E P t (cid:104) exp (cid:16) − b − b (cid:48) Y t + τ − (cid:82) t + τt ( c + c (cid:48) Y u ) d u (cid:17)(cid:105) is exp (cid:0) − A χ ( τ ) − B χ ( τ ) (cid:48) Y t (cid:1) , and the solution of the expectation E Q H (cid:104) exp (cid:16) − b − b (cid:48) Y H + τ − (cid:82) H + τH ( c + c (cid:48) Y u ) d u (cid:17)(cid:105) is exp (cid:0) − A χ ∗ ( τ ) − B χ ∗ ( τ ) (cid:48) Y H (cid:1) , by following the proof of Proposi-tion 1 in Duffie et al. (2000). Note that under the R measure, the coefficient characteristicis χ from t to H and χ ∗ from H to T . Therefore, for the R -transform φ R ( z ; t, T, H ) (cid:44) E R t (cid:20) exp (cid:18) − (cid:90) TH r ( Y u )d u (cid:19) exp ( z (cid:48) Y T ) (cid:21) = E R t (cid:20) E R H (cid:20) exp (cid:18) z (cid:48) Y T − (cid:90) TH ( ρ + ρ (cid:48) Y u ) d u (cid:19)(cid:21)(cid:21) E P t (cid:20) E Q H (cid:20) exp (cid:18) z (cid:48) Y T − (cid:90) TH ( ρ + ρ (cid:48) Y u ) d u (cid:19)(cid:21)(cid:21) = E P t (cid:104) exp (cid:16) − A ( b R , ρ R ) χ ∗ ( T − H ) − B ( b R , ρ R ) χ ∗ ( T − H ) (cid:48) Y H (cid:17)(cid:105) = exp (cid:16) − A ( ˆ b R , ) χ ( H − t ) − B ( ˆ b R , ) χ ( H − t ) (cid:48) Y t (cid:17) . A.9 Proof of Theorem 2Proof. Using the future state price density f H ( S T | Y H ) at time H , where Y H represents thestate variable vector at time H . The future price of a European call option expiring at time T , can be obtained as C H = (cid:90) ∞ K ( S − K ) f H ( S | Y H )d S. (A14)Let p P t ( Y H ) be the probability density of the state variable vector Y H conditioned on time t information under P . For any F H -measurable variable Z H = Z H ( Y H ), which depends onthe state variable vector Y H , its expectation under P is E t [ Z H ] = (cid:90) Z H p P t ( Y H )d Y H . (A15)The expected future price of the call option can be simplified as E t [ C H ] = (cid:90) C H p P t ( Y H )d Y H , = (cid:90) (cid:18)(cid:90) ∞ K ( S − K ) f H ( S | Y H )d S (cid:19) p P t ( Y H )d Y H = (cid:90) ∞ K ( S − K ) (cid:18)(cid:90) f H ( S | Y H ) p P t ( Y H )d Y H (cid:19) d S = (cid:90) ∞ K ( S − K ) E t [ f H ( S | Y H )] d S, (A16)where the first equality uses equation (A15), the second equality uses equation (A14), thethird equality changes the integral order, and the last equality uses equation (A15) again.Taking the second derivative of E t ( C H ) with respect to K gives E t [ f H ( K | Y H )] = ∂ E t [ C H ] ∂K or, E t [ f H ( S T | Y H )] = (cid:20) ∂ E t [ C H ] ∂K (cid:21) K = S T , (A17)61hich proves the first part of equation (68) in Theorem 2. The same logic applies to thecase of the put option, so we omit the proof.Recalling equation (14) in Corollary 1, we can also write the expected price of the calloption as E t [ C H ] = E P t [ P ( H, T )] E R T t (cid:2) ( S T − K ) + (cid:3) = E P t [ P ( H, T )] (cid:90) ∞ K ( S − K ) p R T t ( S )d S, (A18)where p R T t ( S T ) is the probability density of S T under R T .Due to the equivalence of equations (A16) and (A18), we have g t ( H, S T ) (cid:44) E t [ f H ( S T | Y H )] = E P t [ P ( H, T )] p R T t ( S T ) , (A19)which proves the second part of equation (68) in Theorem 2. Equation (69) is a special caseof equation (68), and we omit the proof. B Equivalent Expectation Measures for Multidimensional Stochas-tic Processes B.1 The R Measure for Multidimensional Brownian Motions Here, we consider the R measure construction when there are multidimensional Brownianmotions as the source of uncertainty. Assuming the existence of a risk-neutral measure Q ,we let the Radon-Nikod´ym derivative process L s of Q with respect to P be L s (cid:44) d Q d P (cid:12)(cid:12)(cid:12) F s = exp (cid:18) − (cid:90) s γ u d W P u − (cid:90) s || γ u || d u (cid:19) , (A20)where γ s = ( γ s , ..., γ Ns ), 0 ≤ s ≤ T , is an N -dimensional measurable, adapted processsatisfying (cid:82) T γ iu d u < ∞ almost surely for 1 ≤ i ≤ N , and the Novikov condition, i.e., E (cid:104) exp (cid:16) (cid:82) T || γ u || d u (cid:17)(cid:105) < ∞ . In addition, || γ s || = (cid:16)(cid:80) di =1 γ is (cid:17) denotes the Euclideannorm.Then, similarly to the case of one-dimensional Brownian motion in Appendix A.5, wehave the following proposition: Proposition A1 In the economy coupled with multidimensional Brownian motion as the ource of uncertainty, for a fixed H with ≤ H ≤ T , define a process L s ( H ) as L s ( H ) = exp (cid:18) − (cid:90) s γ u ( H )d W P u − (cid:90) s || γ u ( H ) || d u (cid:19) , where γ s ( H ) (cid:44) { s ≥ H } γ s = (cid:40) γ s , if s ≥ H, , if s < H. Let R ( A ) (cid:44) (cid:90) A L T ( H ; ω )d P ( ω ) for all A ∈ F T . (A21) Then, R is a probability measure equivalent to P , and L s ( H ) is the Radon-Nikod´ym derivativeprocess of R with respect to P .Moreover, define W R s (cid:44) W P s + (cid:90) s γ u ( H )d u = W P s + (cid:90) s { u ≥ H } γ u d u. (A22) The process W R is an N -dimensional Brownian motion under the R measure. Proof. We skip the steps that are consistent with the proof of Theorem 1(i). For multidi-mensional Brownian motions, clearly, we have E (cid:20) exp (cid:18) (cid:90) T || γ u ( H ) || d u (cid:19)(cid:21) = E (cid:20) exp (cid:18) (cid:90) TH || γ u || d u (cid:19)(cid:21) ≤ E (cid:20) exp (cid:18) (cid:90) T || γ u || d u (cid:19)(cid:21) < ∞ . In other words, γ s ( H ) also satisfies the Novikov condition, then the defined L s ( H ) is amartingale and E [ L T ( H )] = 1. Since L T ( H ) is defined as an exponential, it is almost surelystrictly positive; hence, the probability measure R is well-defined and equivalent to P .This Brownian motion transformation between R and P in equation (A22) is an ap-plication of the multidimensional Girsanov theorem. The Girsanov theorem can apply toany N -dimensional process γ s ( H ) that has the properties i. γ s ( H ) is F s -adapted for each s ∈ [0 , T ]; ii. (cid:82) T ( γ iu ( H )) d u < ∞ almost surely for 1 ≤ i ≤ N ; iii. γ s ( H ) satisfies theNovikov condition, which was proven.For a given 0 ≤ H ≤ T , by definition, when s < H , γ s ( H ) = 0 is a constant. Sinceconstant functions are measurable with respect to any σ -algebra, in particular with respect63o F s , s < H , this proves that γ s ( H ) is F s -adapted for each s < H . When s ≥ H , γ s ( H ) = γ s is also F s -adapted by the definition of γ s . Therefore, γ s ( H ) is F s -adapted for each s ∈ [0 , T ],and condition i is proven. Condition ii is also satisfied due to that, for 1 ≤ i ≤ N , (cid:90) T ( γ iu ( H )) d u = (cid:90) T { u ≥ H } γ iu d u = (cid:90) TH γ iu d u ≤ (cid:90) T γ iu d u < ∞ almost surely , which ends the proof. B.2 The R T Measure for Multidimensional Brownian Motions Consider the Radon-Nikod´ym derivative processes L ∗ s ( H ) and L ∗ s in part (ii) of Theorem 1and denote them as L P s ( H ) and L Ps , respectively, for the special case when the numeraire isthe pure discount bond, or G = P ( · , T ). For this special case, R ∗ = R T , and R ∗ = R T , asdefined in section 2.4. We know that L Ps is given as L Ps (cid:44) d Q T d P (cid:12)(cid:12)(cid:12) F s = d Q T d Q · d Q d P (cid:12)(cid:12)(cid:12) F s = P ( s, T ) B P (0 , T ) B s L s , where B is the money market account, L s is defined in equation (A6) for one-dimensionalBrownian motions, or equation (A20) for multidimensional Brownian motions.Assume that the numeraire bond P ( · , T )’s volatility is v ( · , T ) , then the process of L Ps satisfies d L Ps L Ps = ( − γ u − v ( s, T )) d W P s . Then, we have the following proposition: Proposition A2 In the economy coupled with stochastic interest rates, for a fixed H with ≤ H ≤ T , a process L P s ( H ) is defined as L P s ( H ) = P ( H, T ) E P [ P ( H, T )] · P ( s, T ) B H P ( H, T ) B s L s L H = B H P ( s, T ) B s E P [ P ( H, T )] · L s L H , if H ≤ s ≤ T , E P s [ P ( H, T )] E P [ P ( H, T )] , if ≤ s < H. For example, under Q , the pure discount bond followsd P ( s, T ) P ( s, T ) = r s d s + v ( s, T )d W Q s . et R T ( A ) (cid:44) (cid:90) A L P T ( H ; ω )d P ( ω ) for all A ∈ F T , (A23) then R T is a probability measure equivalent to P , and L P s ( H ) is the Radon-Nikod´ym derivativeprocess of R T with respect to P .Moreover, assume that d L P s ( H ) L P s ( H ) = ( − γ u ( H ) − v ( s, T, H )) d W P s , then define W R T s (cid:44) W P s + (cid:90) s γ u ( H )d u + (cid:90) s v ( u, T, H )d u = W P s + (cid:90) s { u ≥ H } γ u d u + (cid:90) s v ( u, T, H )d u. (A24) The process W R T s is a Brownian motion under the R T measure. Proof. The first part is an application of Theorem 1(ii), and the second part follows fromProposition 26.4 in Bj¨ork (2009) with Q = R T and Q = Q .Notably, when s ≥ H , v ( s, T, H ) is equal to v ( s, T ); when s < H , v ( s, T, H ) is generallya term that is adjusted from v ( s, T ). For example, under the Vasicek model defined inequation (34), we have v ( s, T, H ) = σ r − e − α r ( T − s ) α r , if H ≤ s ≤ T ,σ r e − α r ( H − s ) − e − α r ( T − s ) α r , if 0 ≤ s < H. B.3 The R Measure for L´evy Jump Processes Let X be a L´evy process with triplet ( b, σ , ν ) under the physical measure P . Then, accordingto the L´evy-Itˆo decomposition (see, e.g., Cont and Tankov, 2003, Proposition 3.7), X hasthe canonical form X s = bs + σW P s + (cid:90) s (cid:90) R x ( µ X − {| x |≤ } ν X )(d u, d x ) , (A25)65here W is a Brownian motion; µ X is the random measure of the jumps of the process X defined by µ X ( ω ; s, A ) = (cid:80) u ≤ s A (∆ X u ( ω )), which counts the jumps of the process X of thesize in A up to time s ; ν ( A ) is the L´evy measure of the process defined by ν ( A ) = E [ µ X (1 , A )],which describes the expected number of jumps of a certain size A in a time interval of length1. Moreover, ν X , the compensator of the jump’s random measure, is a product measure ofthe L´evy measure with the Lebesgue measure, which we write as ν X (d s, d x ) = ν (d x )d s (see,e.g., Papapantoleon, 2008).Given the current time t , without loss of generality, we assume X t = 0. By the L´evy-Khintchine formula, for a well-defined z ∈ C , the Laplace transform of the variable X T ( t ≤ T ) is φ P ( z ) (cid:44) E P t [exp( zX T )]= exp (cid:18) ( T − t ) (cid:18) zb + 12 z σ + (cid:90) R (cid:0) e zx − − {| x |≤ } zx (cid:1) ν (d x ) (cid:19)(cid:19) . (A26)We further assume that there exists a process Y ( s, x ) such that L s is a positive martingaleof the form L s = exp (cid:18) − (cid:90) s γ u d W P u − (cid:90) s γ u d u + (cid:90) s (cid:90) R ( Y ( u, x ) − (cid:0) µ X − ν X (cid:1) (d u, d x ) − (cid:90) s (cid:90) R ( Y ( u, x ) − − ln ( Y ( u, x ))) µ X (d u, d x ) (cid:19) . (A27)Then, using L s as the Radon-Nikod´ym derivative process, it defines an equivalent martingalemeasure Q under which X has the following canonical decomposition: X s = b ∗ s + σW Q s + (cid:90) s (cid:90) R x ( µ X − {| x |≤ } ν ∗ X )(d u, d x ) , (A28)where ν ∗ X (d s, d x ) = Y ( s, x ) ν X (d s, d x ) is the compensator of the jump’s random measureunder Q , and W Q s = W P s + (cid:82) s γ u d u is a Brownian motion under Q . The drift b ∗ satisfies b ∗ s = bs − (cid:90) s σγ u d u + (cid:90) s (cid:90) − x (cid:0) ν ∗ X − ν X (cid:1) (d u, d x ) . In the remaining part of this section, we require ( γ, Y ) to be deterministic and indepen-dent of time, so that X also remains L´evy process under Q , and its triplet is ( b ∗ , σ , Y · ν )(see, e.g., Papapantoleon, 2008, Remark 12.4 (G1)).66 roposition A3 In the economy coupled with the L´evy process as the source of uncertainty,for a fixed H with ≤ H ≤ T , define a process L s ( H ) as L s ( H ) = exp (cid:18) − (cid:90) s γ u ( H )d W P u − (cid:90) s ( γ u ( H )) d u + (cid:90) s (cid:90) R ( Y ( H ; u, x ) − (cid:0) µ X − ν X (cid:1) (d u, d x ) − (cid:90) s (cid:90) R ( Y ( H ; u, x ) − − ln ( Y ( H ; u, x ))) µ X (d u, d x ) (cid:19) . where γ s ( H ) (cid:44) { s ≥ H } γ s = (cid:40) γ s , if s ≥ H, , if s < H,Y ( H ; s, x ) (cid:44) { s Theorem 12.1 and Remark 12.4 (G2) in Papapantoleon (2008), with ( γ ( H ) , Y ( H ))which are deterministic but dependent on time, yield this result.67herefore, given t ≤ H ≤ T , the (nondiscounted) R -transform of X T under the R measureis φ R ( z ) (cid:44) E R t [exp( zX T )]= exp (cid:18) ( T − t ) (cid:18) zb H + 12 z σ (cid:19) + (cid:18)(cid:90) Tt (cid:90) R (cid:0) e zx − − {| x |≤ } zx (cid:1) ν R (d u, d x ) (cid:19)(cid:19) = exp (cid:18) ( H − t ) (cid:18) zb + 12 z σ + (cid:90) R (cid:0) e zx − − {| x |≤ } zx (cid:1) ν (d x ) (cid:19)(cid:19) × exp (cid:18) ( T − H ) (cid:18) zb ∗ + 12 z σ + (cid:90) R (cid:0) e zx − − {| x |≤ } zx (cid:1) ν ∗ (d x ) (cid:19)(cid:19) . (A31) Example: Compound Poisson Processes. Here, we assume that the L´evy process X hasjumps of finite variation, which means (cid:82) | x |≤ | x | ν (d x ) < ∞ . Moreover, let ν ( R ) = (cid:82) R ν (d x ) < ∞ , so the jumps of X correspond to a compound Poisson process.For a compound Poisson process, the L´evy measure can be expressed in the form of ν (d x ) = λf J ( x )d x , where λ (cid:44) ν ( R ) is the expected number of jumps and f J ( x ) (cid:44) ν (d x ) / ( λ d x )is a probability measure of jump size x .Define a compound Poisson process as X s = (cid:90) s (cid:90) R xµ X (d u, d x ) = N s (cid:88) i =1 J i , (A32)where its L´evy measure is given by ν ( x ) = λf J ( x )d x and ν ∗ ( x ) = λ ∗ f ∗ J ( x )d x under P and Q , respectively. According to Proposition A3, we let ν XR (d s, d x ) = { s 68 exp (cid:0) λ ( H − t ) (cid:0) E P [e zJ ] − (cid:1) + λ ∗ ( T − H ) (cid:0) E Q [e zJ ] − (cid:1)(cid:1) , (A34)where E P [e zJ ] = (cid:82) R e zx f ( x )d x and E Q [e zJ ] = (cid:82) R e zx f ∗ ( x )d x . Example: Compensated Compound Poisson Process. We first assume a process under P as X s = γs + (cid:90) s (cid:90) R x ( µ X − ν X )(d u, d x )= γs + N s (cid:88) i =1 J i − λs E P [ J ] , (A35)where E P [ J ] = (cid:82) R xf J ( x )d x and γ = λ E P [ J ] − λ ∗ E Q [ J ]. Then, the corresponding processunder Q is a compensated compound Poisson process, which satisfies X s = (cid:90) s (cid:90) R x ( µ X − ν ∗ X )(d u, d x )= N s (cid:88) i =1 J i − λ ∗ s E Q [ J ] , (A36)where E Q [ J ] = (cid:82) R xf ∗ J ( x )d x . Hence, according to Proposition A3, we let ν XR (d s, d x ) = { s The Review of Financial Studies 15 (1), 243–288.A¨ıt-Sahalia, Y. and A. W. Lo (1998). 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Journal of Financial and Quantitative Analysis 45 (3),641–662. 78 igures Figure 1: The Binomial Tree under the R Measure (a) Stock Price (b) Expected Call Price Notes : This figure shows a 3-year binomial tree structure for the stock price and the expectedcall price under the R measure. The R measure follows physical P distribution until before H = 2 years, and risk-neutral Q distribution beginning at time H = 2 years. The followingparameters are used: S t = 100, K = 100, µ = 0 . r = 0 . σ = 0 . 15, ∆ t = 1. The upand down node values of the future stock price evolution are computed using the binomial-treemethod given by Cox et al. (1979). The up and down conditional physical probabilities overthe next period are 0.812 and 0.188, respectively, which are used at time 0 and time 1; and theup and down conditional risk-neutral probabilities over the next period are 0.564 and 0.436,respectively, which are used at time 2. ables Table 1: The Definitions of Six EEMsNumeraire (G) R ∗ R ∗ Money Market Account ( B ) R R Pure Discount Bond ( P ( · , T )) R T R T Exchange Asset ( S ) R S R S Notes : This table gives six specific EEM definitions in the R ∗ and R ∗ classes when using B , P ( · , T ), and S as the numeraire. nternet Appendix for “A Theory of Equivalent ExpectationMeasures for Expected Prices of Contingent Claims” Sanjay K. Nawalkha and Xiaoyang Zhuo ∗ June 26, 2020 This Internet Appendix provides the following supplemental sections to the main article: • Section IA. The R ∗ Measure • Section IB. Expected Option Prices under the Margrabe Model • Section IC. Expected Bond Prices under the A r (3) Model • Section ID. Expected Bond Prices under the Quadratic Term Structure Model • Section IE. Expected Prices of Interest Rate Derivatives • Section IF. Expected Option Prices under the CGMY L´evy Model • Section IG. Expected Return Simulation • Section IH. A Procedure to Extract the Expected FSPD ∗ Sanjay K. Nawalkha (e-mail: [email protected]) is with the Isenberg School of Manage-ment, University of Massachusetts, MA, USA. Xiaoyang Zhuo (e-mail: [email protected]) iswith the PBC School of Finance, Tsinghua University, Beijing, China. This version is available at the arXiv,Open Science Framework (OSF), and SSRN. Internet Appendix IA The R ∗ Measure We first give the definition of the R ∗ measure in the following proposition: Proposition IA1 For a fixed H with ≤ H ≤ T , define a process L ∗ s ( H ) as L ∗ s ( H ) = G H E R ∗ s [ G s /G T ] G s E R ∗ [ G H /G T ] · L ∗ s L ∗ H , if H ≤ s ≤ T , E R ∗ s [ G H /G T ] E R ∗ [ G H /G T ] , if ≤ s < H. (IA1) Let R ∗ ( A ) (cid:44) (cid:90) A L ∗ T ( H ; ω )d P ( ω ) for all A ∈ F T , (IA2) then R ∗ is a probability measure equivalent to P , and L ∗ s ( H ) is the Radon-Nikod´ym derivativeprocess of R ∗ with respect to P . We omit the proof of Proposition IA1 because it follows similarly from Appendix A.1.For the relation between R ∗ and R ∗ , we have E R ∗ t [ Z T ] = E P t (cid:20) L ∗ T ( H ) L ∗ t ( H ) Z T (cid:21) = E P t (cid:20) L ∗ T L ∗ H · G H /G T E R ∗ t [ G H /G T ] Z T (cid:21) = E P t (cid:20) L ∗ T ( H ) L ∗ t ( H ) · G H /G T E R ∗ t [ G H /G T ] Z T (cid:21) = E R ∗ t (cid:20) G H /G T E R ∗ t [ G H /G T ] Z T (cid:21) , (IA3)for any F T -measurable variable Z T .Now, we prove that the R ∗ measure subsumes the R ∗ measure when the numeraires arerestricted to be either the money market account or the pure discount bond.According to the classification of EEMs in section 2.4, when G = B = e (cid:82) · r u d u ; in otherwords, the numeraire is the money market account, we have R ∗ = R , R ∗ = R , and L ∗ s = L s .Then, for R ’s Radon-Nikod´ym derivative process L s ( H ) = B H E R s [ B s /B T ] B s E R [ B H /B T ] · L s L H = B H P ( s, T ) B s E P [ P ( H, T )] · L s L H , if H ≤ s ≤ T , E R s [ B H /B T ] E R [ B H /B T ] = E P s [ P ( H, T )] E P [ P ( H, T )] , if 0 ≤ s < H, (IA4)1here E R s [ P ( H, T )] = E P s [ P ( H, T )] for 0 ≤ s ≤ H due to Proposition 1.When G = P ( · , T ), in other words, the numeraire is the pure discount bond, we have R ∗ = R T , R ∗ = R T , and L ∗ s = L Ps = P ( s,T ) B P (0 ,T ) B s L s . Then, for R T ’s Radon-Nikod´ym derivativeprocess L P s ( H ) = P ( H, T ) E R T s [ P ( s, T )] P ( s, T ) E R T [ P ( H, T )] · P ( s, T ) B H P ( H, T ) B s L s L H = B H P ( s, T ) B s E P [ P ( H, T )] · L s L H , if H ≤ s ≤ T , E R T s [ P ( H, T )] E R T [ P ( H, T )] = E P s [ P ( H, T )] E P [ P ( H, T )] , if 0 ≤ s < H, (IA5)where E R T s [ P ( H, T )] = E P s [ P ( H, T )] for 0 ≤ s ≤ H due to Proposition 1.When G = P ( · , T ), we have R ∗ = R T , R ∗ = R T , and L ∗ s = L Ps = P ( s,T ) B P (0 ,T ) B s L s . Then, for R T ’s Radon-Nikod´ym derivative process (also see Appendix B.2) L P s ( H ) = P ( H, T ) E P [ P ( H, T )] · P ( s, T ) B H P ( H, T ) B s L s L H = B H P ( s, T ) B s E P [ P ( H, T )] · L s L H , if H ≤ s ≤ T , E P s [ P ( H, T )] E P [ P ( H, T )] , if 0 ≤ s < H. (IA6)As can be seen, the Radon-Nikod´ym derivative processes satisfy L P s ( H ) = L P s ( H ) = L s ( H ), which means R T = R T = R . Hence, R T subsumes both R T and R measures whenthe numeraires are either the pure discount bond or the money market account. IB Expected Option Prices under the Margrabe Model For simplicity of exposition, we use the Margrabe (1978) model to demonstrate the use ofEEM R S for obtaining the expected price of a call option that gives the option holder theright to exchange an asset S for another asset S . Assume that both assets S and S followgeometric Brownian motions under the physical measure P , as follows:d S s S s = µ d s + σ d W P s , d S s S s = µ d s + σ d W P s , (IA7)where W P and W P are correlated Brownian motions with a correlation coefficient equal to ρ . It is convenient to price the exchange option using asset S as the numeraire asset.2he expected price of the exchange option is given by equation (15) in Corollary 1, with F T = C T = ( S T − S T ) + as the terminal payoff, and the numeraire values given as S T = S T , and S H = S H , as follows: E t [ C H ] = E P t [ S H ] E R S t (cid:20) ( S T − S T ) + S T (cid:21) = E P t [ S H ] E R S t (cid:2) ( V T − + (cid:3) , (IA8)where V = S /S is the asset price ratio. Therefore, it suffices for us to obtain the processof V under the R S measure, which can be derived using Theorem 1 and Appendix B.1 asfollows: d V s V s = { s Here, we employ the commonly used A r (3) form of three-factor ATSMs, which has beendeveloped in various models, including Balduzzi et al. (1996), Dai and Singleton (2000).Specifically, we use the A r (3) MAX model to compute the expected bond price. Under thephysical measure P , the state variable processes under the A r (3) MAX model are defined as3ollows:d v s = α v ( m v − v s ) d s + η √ v s d W P s , d θ s = α θ ( m θ − θ s ) d s + σ θv η √ v s d W P s + (cid:112) ζ + β θ v s d W P s + σ θr (cid:112) δ r + v s d W P s , d r s = α rv ( m v − v s ) d s + α r ( θ s − r s ) d s + σ rv η √ v s d W P s + σ rθ (cid:112) ζ + β θ v s d W P s + (cid:112) δ r + v s d W P s , (IA11)where W P , W P and W P are independent Brownian motions under the physical measure P .We assume the market price of risks (MPRs), γ s = ( γ s , γ s , γ s ), are given by γ s = γ √ v s ,γ s = γ (cid:112) ζ + β θ v s ,γ s = γ (cid:112) δ r + v s . (IA12)Therefore, using Theorem 1 and Appendix B.1, the state variable processes under the A r (3) MAX model under the R measure are given byd v s = (cid:2) α v ( m v − v s ) − { s ≥ H } γ ηv s (cid:3) d s + η √ v s d W R s , d θ s = (cid:2) α θ ( m θ − θ s ) − { s ≥ H } γ σ θv ηv s − { s ≥ H } γ (cid:0) ζ + β θ v s (cid:1) − { s ≥ H } γ σ θr ( δ r + v s ) (cid:21) d s + σ θv η √ v s d W R s + (cid:112) ζ + β θ v s d W R s + σ θr (cid:112) δ r + v s d W R s , d r s = (cid:2) α rv ( m v − v s ) + α r ( θ s − r s ) − { s ≥ H } γ σ rv ηv s − { s ≥ H } γ σ rθ (cid:0) ζ + β θ v s (cid:1) − { s ≥ H } γ ( δ r + v s ) (cid:21) d s + σ rv η √ v s d W R s + σ rθ (cid:112) ζ + β θ v s d W R s + (cid:112) δ r + v s d W R s . (IA13)Hence, using equation (13) in Corollary 1, the time t expectation of the future price of a T -maturity default-free zero-coupon bond at time H is given under the A r (3) MAX model asfollows. For all t ≤ H ≤ T , E t [ P ( H, T )] = E R t (cid:20) exp (cid:18) − (cid:90) TH r u d u (cid:19)(cid:21) = exp − A (cid:32) , , , (cid:33) χ ∗ ( T − H ) − A (cid:32) b, c, d, (cid:33) χ ( H − t ) − B (cid:32) b, c, d, (cid:33) χ ( H − t ) v t − C (cid:32) b, c, d, (cid:33) χ ( H − t ) θ t − D (cid:32) b, c, d, (cid:33) χ ( H − t ) r t , (IA14)4here b = B (cid:32) , , , (cid:33) χ ∗ ( T − H ) , c = C (cid:32) , , , (cid:33) χ ∗ ( T − H ) , d = D (cid:32) , , , (cid:33) χ ∗ ( T − H ) , and χ = ( α v , α θ , α r , α rv , m v , m θ , σ θv , σ θr , σ rv , σ rθ , η, ζ, β θ , δ r , , ,χ ∗ = ( α ∗ v , α ∗ θ , α ∗ r , α ∗ rv , m ∗ v , m ∗ θ , σ θv , σ θr , σ rv , σ rθ , η, ζ, β θ , δ r , α ∗ θv , m ∗ r ) , with α ∗ v = α v + γ η, m ∗ v = α v m v α v + γ η , α ∗ θ = α θ , m ∗ θ = α θ m θ − γ ζ − γ σ θr δ r α θ ,α ∗ r = α r , α ∗ rv = α rv + γ σ rv η + γ σ rθ β θ + γ , α ∗ θv = − γ σ θv η − γ β θ − γ σ θr ,m ∗ r = α rv m v − α ∗ rv m ∗ v − γ σ rθ ζ − γ δ r . For any well-defined λ i , µ i ( i = 1 , , 3) and the characteristic χ , A χ ( τ ) (cid:44) A (cid:32) λ ,µ λ ,µ λ ,µ (cid:33) χ ( τ ), B χ ( τ ) (cid:44) B (cid:32) λ ,µ λ ,µ λ ,µ (cid:33) χ ( τ ), C χ ( τ ) (cid:44) C (cid:32) λ ,µ λ ,µ λ ,µ (cid:33) χ ( τ ), D χ ( τ ) (cid:44) D (cid:32) λ ,µ λ ,µ λ ,µ (cid:33) χ ( τ ) satisfy the following ODEs ∂A χ ( τ ) ∂τ = B χ ( τ ) α v m v + C χ ( τ ) α θ m θ + D χ ( τ ) ( α rv m v + m r ) − C χ ( τ ) (cid:0) σ θr δ r + ζ (cid:1) − D χ ( τ ) (cid:0) σ rθ ζ + δ r (cid:1) − C χ ( τ ) D χ ( τ ) (cid:0) σ rθ ζ + σ θr δ r (cid:1) ,∂B χ ( τ ) ∂τ = − B χ ( τ ) α v + C χ ( τ ) α θv − D χ ( τ ) α rv − B χ ( τ ) η − C χ ( τ ) (cid:0) β θ + σ θv η + σ θr (cid:1) − D χ ( τ ) (cid:0) σ rv η + σ rθ β θ + 1 (cid:1) − B χ ( τ ) C χ ( τ ) σ θv η − B χ ( τ ) D χ ( τ ) σ rv η − C χ ( τ ) D χ ( τ ) (cid:0) σ θv σ rv η + σ rθ β θ + σ θr (cid:1) + µ ,∂C χ ( τ ) ∂τ = − C χ ( τ ) α θ + D χ ( τ ) α r + µ ,∂D χ ( τ ) ∂τ = − D χ ( τ ) α r + µ , (IA15)subject to the boundary condition A χ = 0, B χ (0) = λ , C χ (0) = λ , D χ (0) = λ . A χ ( τ ) and5 χ ( τ ) can be solved easily using numerical integration, and C χ ( τ ) and D χ ( τ ) are given as C χ ( τ ) = λ e − α θ τ + µ + µ α θ (cid:0) − e − α θ τ (cid:1) + λ α r − µ α θ − α r (cid:0) e − α r τ − e − α θ τ (cid:1) ,D χ ( τ ) = λ e − α r τ + µ α r (cid:0) − e − α r τ (cid:1) . Similarly, A χ ∗ ( τ ) (cid:44) A (cid:32) λ ,µ λ ,µ λ ,µ (cid:33) χ ∗ ( τ ), B χ ∗ ( τ ) (cid:44) B (cid:32) λ ,µ λ ,µ λ ,µ (cid:33) χ ∗ ( τ ), C χ ∗ ( τ ) (cid:44) C (cid:32) λ ,µ λ ,µ λ ,µ (cid:33) χ ∗ ( τ ), D χ ∗ ( τ ) (cid:44) D (cid:32) λ ,µ λ ,µ λ ,µ (cid:33) χ ∗ ( τ )are obtained as in equation (IA15) with the characteristic χ replaced with the characteristic χ ∗ . ID Expected Bond Prices under the Quadratic Term StructureModel As in Ahn et al. (2002), an N -factor quadratic term structure model (QTSM) satisfies thefollowing assumptions. First, the nominal instantaneous interest rate is a quadratic functionof the state variables r s = α + β (cid:48) Y s + Y s (cid:48) Ψ Y s , (IA16)where α is a constant, β is an N -dimensional vector, and Ψ is an N × N matrix of constants.We assume that α − β (cid:48) Ψ − β ≥ 0, and Ψ is a positive semidefinite matrix.The SDEs of the state variables Y are characterized as multivariate Gaussian processeswith mean reverting properties:d Y s = ( µ + ξY s ) d s + Σd W P s , (IA17)where µ is an N -dimensional vector of constants and ξ and Σ are N -dimensional squarematrices. We assume that ξ is “diagonalizable” and has negative real components of eigen-values. W P is an N -dimensional vector of standard Brownian motions that are mutuallyindependent.Similar to Ahn et al. (2002), the market price of risks γ s is assumed to be γ s = Σ − ( γ + γ Y s ) , (IA18)where γ is an N -dimensional vector, γ is an N × N matrix satisfying that Σ − (cid:48) Σ − γ issymmetric. Hence, using Theorem 1 and Appendix B.1, the state variables Y under the R Y s = (cid:2) µ + ξY s − { s ≥ H } ( γ + γ Y s ) (cid:3) d s + Σd W R s . (IA19)Hence, the time t expectation of the future price of a T -maturity default-free zero-couponbond at time H is given under the N -factor QTSM model as follows. For all t ≤ H ≤ T , E t [ P ( H, T )] = E R t (cid:20) exp (cid:18) − (cid:90) TH r u d u (cid:19)(cid:21) = E R t (cid:20) exp (cid:18) − (cid:90) TH ( α + β (cid:48) Y u + Y u (cid:48) Ψ Y u ) d u (cid:19)(cid:21) = exp (cid:32) − α ( T − H ) − A (cid:16) ,β , Ψ (cid:17) µ ∗ ,ξ ∗ ( T − H ) − A (cid:16) b , c , (cid:17) µ,ξ ( H − t ) − B (cid:16) b , c , (cid:17) µ,ξ ( H − t ) (cid:48) Y t − Y t (cid:48) C (cid:16) b , c , (cid:17) µ,ξ ( H − t ) Y t (cid:33) , (IA20)where b = B (cid:16) ,β , Ψ (cid:17) µ ∗ ,ξ ∗ ( T − H ) , c = C (cid:16) ,β , Ψ (cid:17) µ ∗ ,ξ ∗ ( T − H ) ,µ ∗ = µ − γ , ξ ∗ = ξ − γ . For any well-defined matrices b, c, d, q , µ and ξ , A µ,ξ ( τ ) (cid:44) A (cid:16) b,dc,q (cid:17) µ,ξ ( τ ), B µ,ξ ( τ ) (cid:44) B (cid:16) b,dc,q (cid:17) µ,ξ ( τ ), and C µ,ξ ( τ ) (cid:44) C (cid:16) b,dc,q (cid:17) µ,ξ ( τ ) satisfy the following ODEsd A µ,ξ ( τ )d τ = µ (cid:48) B µ,ξ ( τ ) − B µ,ξ ( τ ) (cid:48) ΣΣ (cid:48) B µ,ξ ( τ ) + tr (ΣΣ (cid:48) C µ,ξ ( τ )) , d B µ,ξ ( τ )d τ = ξ (cid:48) B µ,ξ ( τ ) + 2 C µ,ξ ( τ ) µ − C µ,ξ ( τ )ΣΣ (cid:48) B µ,ξ ( τ ) + d, d C µ,ξ ( τ )d τ = − C µ,ξ ( τ )ΣΣ (cid:48) C µ,ξ ( τ ) + C µ,ξ ( τ ) ξ + ξ (cid:48) C µ,ξ ( τ ) + q, (IA21)with the terminal conditions A µ,ξ (0) = 0, B µ,ξ (0) = b , C µ,ξ (0) = c . These ODEs can be easilysolved numerically. A µ ∗ ,ξ ∗ ( τ ) (cid:44) A (cid:16) b,dc,q (cid:17) µ ∗ ,ξ ∗ ( τ ), B µ ∗ ,ξ ∗ ( τ ) (cid:44) B (cid:16) b,dc,q (cid:17) µ,ξ ( τ ), and C µ ∗ ,ξ ∗ ( τ ) (cid:44) C (cid:16) b,dc,q (cid:17) µ ∗ ,ξ ∗ ( τ ) areobtained similarly as in equation (IA21) with the drift coefficients ( µ, ξ ) replaced with driftcoefficient ( µ ∗ , ξ ∗ ). Example: The QTSM3 model. Since the empirical literature suggests that three factorsare required to describe the term structure (see, e.g., Knez et al., 1996; Dai and Singleton,2000), we fix N = 3 and explore the particular QTSM3 model defined in Ahn et al. (2002).According to the parameter restrictions by Ahn et al. (2002), in this case, µ ≥ α > β =0, and Σ is a diagonal matrix. In addition, ξ and γ are diagonal, and this assumption results7n orthogonal state variables under the P measure as well as the R measure. Furthermore, Ψis the 3 × I , such that there are no interactions among the state variablesin the determination of the interest rate. The number of parameters in QTSM3 is 16. Animportant advantage is that the QTSM3 model allows for a fully closed-form solution forthe expected bond price: E t [ P ( H, T )] = E R t (cid:20) exp (cid:18) − (cid:90) TH r u d u (cid:19)(cid:21) = E R t (cid:20) exp (cid:18) − (cid:90) TH ( α + Y u (cid:48) Ψ Y u ) d u (cid:19)(cid:21) = exp − α ( T − H ) − (cid:88) i =1 A (cid:16) , , (cid:17) µ ∗ i ,ξ ∗ ii ( T − H ) − (cid:88) i =1 A (cid:18) ˆ b i , c i , (cid:19) µ i ,ξ ii ( H − t ) − (cid:88) i =1 B (cid:18) ˆ b i , c i , (cid:19) µ i ,ξ ii ( H − t ) Y it − (cid:88) i =1 C (cid:18) ˆ b i , c i , (cid:19) µ i ,ξ ii ( H − t ) Y it , (IA22)where ˆ b i = B (cid:16) , , (cid:17) µ ∗ i ,ξ ∗ ii ( T − H ) , ˆ c i = C (cid:16) , , (cid:17) µ ∗ i ,ξ ∗ ii ( T − H ) ,µ ∗ i = µ i − γ i , ξ ∗ ii = ξ ii − γ ii , and for any well-defined b i , c i , q i , and the coefficients µ i , ξ ii , we have A (cid:16) b i , c i ,q i (cid:17) µ i ,ξ ii ( τ ) = (cid:18) µ i β i (cid:19) q i τ − b i β i Σ ii (cid:0) e β i τ − (cid:1) − b i µ i (cid:0) e β i τ − (cid:1) (cid:2) ( β i − ξ ii ) (cid:0) e β i τ − (cid:1) + 2 β i (cid:3) β i [(2 c i Σ ii + β i − ξ ii ) (e β i τ − 1) + 2 β i ] − µ i (cid:0) e β i τ − (cid:1) (cid:2) (( β i − ξ ii ) q i − c i ξ ii ) (cid:0) e β i τ − (cid:1) + 2 β i q i (cid:3) β i [(2 c i Σ ii + β i − ξ ii ) (e β i τ − 1) + 2 β i ] − 12 ln 2 β i e ( β i − ξ ii ) τ (2 c i Σ ii + β i − ξ ii ) (e β i τ − 1) + 2 β i ,B (cid:16) b i , c i ,q i (cid:17) µ i ,ξ ii ( τ ) = 2 b i β i e β i τ + 2 µ i (cid:104) ( q i + c i ( ξ ii + β i )) (cid:0) e β i τ − (cid:1) + 2 c i β i (cid:0) e β i τ − (cid:1)(cid:105) β i [(2 c i Σ ii + β i − ξ ii ) (e β i τ − 1) + 2 β i ] ,C (cid:16) b i , c i ,q i (cid:17) µ i ,ξ ii ( τ ) = c i (cid:2) β i + (cid:0) e β i τ − (cid:1) ( β i + ξ ii ) (cid:3) + q i (cid:0) e β i τ − (cid:1) (2 c i Σ ii + β i − ξ ii ) (e β i τ − 1) + 2 β i , (IA23)where β i = (cid:112) ξ ii + 2Σ ii q i . A (cid:16) b i , c i ,q i (cid:17) µ ∗ i ,ξ ∗ ii ( τ ), B (cid:16) b i , c i ,q i (cid:17) µ ∗ i ξ ∗ ii ( τ ) and C (cid:16) b i , c i ,q i (cid:17) µ ∗ i ,ξ ∗ ii ( τ ) are obtained similarly as inequation (IA23) with the coefficients ( µ i , ξ ii ) replaced with the coefficients ( µ ∗ i , ξ ∗ ii ).8 E Expected Prices of Interest Rate Derivatives This section demonstrates the application of the R measure for obtaining the expected pricesof a wide range of interest rate derivatives. For expositional purposes, this section uses thegeneral exponentially affine-jump framework of Chacko and Das (2002). In the broadestsense, the exponentially affine models include not only the ATSMs presented in section 3.1.2but also the quadratic term structure models presented in the Internet Appendix ID, sincethe quadratic functions of the state variables can be treated as a new set of state variablesin the exponentially affine class. To obtain the expected prices of interest rate derivativesusing this framework, we present an extended R -transform similar to that given in equation(55), which converges to an extended form of the specific Q -transform given by Chacko andDas (2002) when t = H . The extended Q -transform that we obtain as a special case when t = H is similar to the extended Q -transform given by equation (2.13) in Duffie et al. (2000).Consider the short rate and the state variable processes under the physical measure P ,given as follows: d r s = µ ( r u , Y u ) d s + σ ( r u , Y u ) d W P s + d (cid:32) N s (cid:88) i =1 J r,i (cid:33) , d Y s = α ( Y s ) d s + δ ( Y s ) d W P s + d (cid:32) N s (cid:88) i =1 J Y,i (cid:33) , (IA24)where Y is an M × W P is a vector of N -dimensional Brownian motions, and N is a vector L oforthogonal Poisson processes under P with jump intensities given by λ i , i = 1 , , ..., L . Thejump sizes of the Poisson processes under P are defined by the L × J r,i and the M × L matrix J Y,i of correlated random variables. The instantaneous diffusion covariancematrix of the state variables is given by Λ( Y ), while the vector of instantaneous diffusioncovariances between the state variables Y i , i = 1 , , ..., M , and r is given by ρ ( Y ). Thetechnical regularity conditions for the drift, diffusion, and jump coefficients are assumed tobe the same as in Chacko and Das (2002).Assume that the short rate r and the state variable Y processes are given under the See section 7 in Chacko and Das (2002). Cheng and Scaillet (2002); Gourieroux and Sufana (2003) alsoconsider QTSMs as a special case of affine models by stacking the factors and their squares. Q , as follows:d r s = µ ∗ ( r u , Y u )d s + σ ( r u , Y u ) d W Q s + d (cid:32) N s (cid:88) i =1 J r,i (cid:33) , d Y s = α ∗ ( Y s ) d s + δ ( Y s ) d W Q s + d (cid:32) N s (cid:88) i =1 J Y,i (cid:33) , (IA25)where µ ∗ ( r u , Y u ) = µ ( r u , Y u ) − γ r ( r u , Y u ), and α ∗ ( Y s ) = α ( Y s ) − γ Y ( Y s ), γ r and γ Y aremarket prices of risks of the short rate and the state variables. W Q is a vector of N Brownianmotions under Q . The Poisson process N under Q has jump intensities λ ∗ i , i = 1 , , ..., L .Applying Theorem 1, Appendix B.1 and B.3, the short rate and the state variables areexpressed under the R measure as follows:d r s = (cid:2) µ ( r u , Y u ) − { s ≥ H } γ r ( r u , Y u ) (cid:3) d s + σ ( r u , Y u ) d W R s + d (cid:32) N s (cid:88) i =1 J r,i (cid:33) , d Y s = (cid:2) α ( Y s ) − { s ≥ H } γ Y ( Y s ) (cid:3) d s + δ ( Y s ) d W R s + d (cid:32) N s (cid:88) i =1 J Y,i (cid:33) , (IA26)where W R is a vector of N -dimensional Brownian motions under R , the jump processes (cid:80) N s i =1 J r,i and (cid:80) N s i =1 J Y,i under R are defined similarly to that in footnote 20.It is convenient to define an extended R -transform based upon equation (55) for com-puting the expected prices of various types of interest rate derivatives. To do so, we firstpropose an extended Q -transform, similar to equation (2.13) in Duffie et al. (2000), definedwith respect to the short rate and state variable processes given in equation (IA25), asfollows: ϕ ( b , c , d ; t, T ) (cid:44) E Q t (cid:20) exp (cid:18) − b (cid:48) g T − (cid:90) Tt c (cid:48) g u d u (cid:19) d (cid:48) g T (cid:21) = E Q t (cid:20) exp (cid:18) − b − b r T − b (cid:48) Y T − (cid:90) Tt c d u − (cid:90) Tt c r u d u − (cid:90) Tt c (cid:48) Y u d u (cid:19) × (cid:16) d + d r T + d (cid:48) Y T (cid:17)(cid:21) = exp (cid:16) − A χ ∗ ( T − t ; b , c , d ) − B χ ∗ ( T − t ; b , c , d ) r t − C χ ∗ ( T − t ; b , c , d ) (cid:48) Y t (cid:17) × (cid:16) D χ ∗ ( T − t ; b , c , d ) + E χ ∗ ( T − t ; b , c , d ) r t + F χ ∗ ( T − t ; b , c , d ) (cid:48) Y t (cid:17) , (IA27) Notably, γ r and γ Y incorporate the MPRs of uncertainties from both the diffusion term and the jumpterm. g = (1 , r, Y ), b = ( b , b , b ), c = ( c , c , c ), d = ( d , d , d ), and the characteristic χ ∗ = ( µ ∗ , α ∗ , λ ∗ , σ, δ ) captures the distribution of r and Y under Q . As shown later throughan example, the vectors of constants b , c , and d , can be chosen appropriately depending onthe payoff structure of the specific interest rate derivative.The above transform extends the specific Q -transform given by equation (4) in Chackoand Das (2002) to allow for greater flexibility in the derivation of analytical solutions ofthe current prices of interest rate derivatives. Similar to the above transform, we define anextended R -transform as a special case of equation (55) for the derivation of the analyticalsolutions of the expected prices of interest rate derivatives, as follows: ϕ R ( b , c , d ; t, T, H ) (cid:44) E R t (cid:20) exp (cid:18) − b (cid:48) g T − (cid:90) TH c (cid:48) g u d u (cid:19) d (cid:48) g T (cid:21) = E R t (cid:20) exp (cid:18) − b − b r T − b (cid:48) Y T − (cid:90) TH c d u − (cid:90) TH c r u d u − (cid:90) TH c (cid:48) Y u d u (cid:19) × (cid:16) d + d r T + d (cid:48) Y T (cid:17)(cid:21) = exp (cid:16) − A χ (cid:16) H − t ; ˆ b , , ˆ d (cid:17) − B χ (cid:16) H − t ; ˆ b , , ˆ d (cid:17) r t − C χ (cid:16) H − t ; ˆ b , , ˆ d (cid:17) (cid:48) Y t (cid:17) × (cid:16) D χ (cid:16) H − t ; ˆ b , , ˆ d (cid:17) + E χ (cid:16) H − t ; ˆ b , , ˆ d (cid:17) r t + F χ (cid:16) H − t ; ˆ b , , ˆ d (cid:17) (cid:48) Y t (cid:17) , (IA28)with ˆ b = (cid:16) A χ ∗ ( T − H ; b , c , d ) , B χ ∗ ( T − H ; b , c , d ) , C χ ∗ ( T − H ; b , c , d ) (cid:17) , ˆ d = (cid:16) D χ ∗ ( T − H ; b , c , d ) , E χ ∗ ( T − H ; b , c , d ) , F χ ∗ ( T − H ; b , c , d ) (cid:17) . where χ and χ ∗ are the characteristics under P and Q , respectively. We demonstrate the useof the above transform with a specific example given below. Example: A Two-Jump Model. Now, we obtain the expected prices of different classesof interest rate derivatives using the two-jump affine model of Chacko and Das (2002). Thestochastic process for the short rate, which is the sole state variable under this model, isgiven as d r s = α r ( m r − r s ) d s + σ r d W P s + d (cid:32) N u,s (cid:88) i =1 J u,i (cid:33) − d N d,s (cid:88) i =1 J d,i , (IA29)where α r , m r , and σ r are constants. The number of Poisson jumps N u,s and N d,s under P λ u and λ d , respectively, and the corresponding jump sizes J u,i and J d,i under P are exponentially distributed random variables with positive means µ u and µ d ,respectively.Similarly, the risk-neutral short rate process is given asd r s = [ α r ( m r − r s ) − γ r σ r ] d s + σ r d W Q s + d (cid:32) N u,s (cid:88) i =1 J u,i (cid:33) − d N d,s (cid:88) i =1 J d,i = α ∗ r ( m ∗ r − r s ) d s + σ r d W Q s + d (cid:32) N u,s (cid:88) i =1 J u,i (cid:33) − d N d,s (cid:88) i =1 J d,i , (IA30)where α ∗ r = α r , m ∗ r = m r − γ r σ r /α r . The number of Poisson jumps N u,s and N d,s under Q arrive with intensities λ ∗ u and λ ∗ d , respectively, and the corresponding jump sizes J u,i and J d,i are exponentially distributed random variables with positive means µ ∗ u and µ ∗ d , respectively.Similar to the general interest rate model given earlier in this section, the short rateprocess under the R measure is given asd r s = [ α r ( m r − r s ) − s ≥ H γ r σ r ] d s + σ r d W R s + d (cid:32) N u,s (cid:88) i =1 J u,i (cid:33) − d N d,s (cid:88) i =1 J d,i , (IA31)where the jump processes under the R measure are defined in a manner similar to that infootnote 20.For g = (1 , r ), b = ( b , b ), c = ( c , c ), d = ( d , d ), the extended R -transform of thistwo-jump model is ϕ R ( b , c , d ; t, T, H ) (cid:44) E R t (cid:20) exp (cid:18) − b (cid:48) g T − (cid:90) TH c (cid:48) g u d u (cid:19) d (cid:48) g T (cid:21) = E R t (cid:20) exp (cid:18) − b − b r T − (cid:90) TH c d u − (cid:90) TH c r u d u (cid:19) ( d + d r T ) (cid:21) = exp (cid:16) − A ( ˆ b , , ˆ d ) χ ( H − t ) − B ( ˆ b , , ˆ d ) χ ( H − t ) r t (cid:17) × (cid:16) D ( ˆ b , , ˆ d ) χ ( H − t ) + E ( ˆ b , , ˆ d ) χ ( H − t ) r t (cid:17) , (IA32)where ˆ b = (cid:16) A ( b , c , d ) χ ∗ ( T − H ) , B ( b , c , d ) χ ∗ ( T − H ) (cid:17) , ˆ d = (cid:16) D ( b , c , d ) χ ∗ ( T − H ) , E ( b , c , d ) χ ∗ ( T − H ) (cid:17) , with the characteristics χ = ( α r , m r , λ u , λ d , µ u , µ d , σ r ), and χ ∗ = ( α ∗ r , m ∗ r , λ ∗ u , λ ∗ d , µ ∗ u , µ ∗ d , σ r ),12nd A ( b , c , d ) χ ( τ ) = b + c τ + α r m r (cid:20) b B α r ( τ ) + c τ − B α r ( τ ) α r (cid:21) − ( λ u + λ d ) τ − σ r (cid:34) b − e − α r τ α r + b c B α r ( τ ) + c τ − B α r ( τ ) − α r B α r ( τ ) α r (cid:35) + λ u α r + c µ u ln (cid:18) (1 + u µ u ) e α r τ + u µ u b µ u (cid:19) + λ d α r − c µ d ln (cid:18) (1 − u µ d ) e α r τ − u µ d − b µ d (cid:19) ,B ( b , c , d ) χ ( τ ) = b e − α r τ + c B α r ( τ ) ,D ( b , c , d ) χ ( τ ) = d + d m r (e α r τ − − d σ r (cid:20) u τ + u α r (e α r τ − (cid:21) + d λ u µ u α r (1 + u µ u ) (cid:20) e α r τ − − u µ u u µ u ln (cid:18) (1 + u µ u ) e α r τ + u µ u b µ u (cid:19) + u µ u u µ u (cid:18) 11 + b µ u − u µ u ) e α r τ + u µ u (cid:19)(cid:21) − d λ d µ d α r (1 − u µ d ) (cid:20) e α r τ − u µ d − u µ d ln (cid:18) (1 − u µ d ) e α r τ − u µ d − b µ d (cid:19) + u µ d − u µ d (cid:18) − b µ d − − u µ d ) e α r τ − u µ d (cid:19)(cid:21) ,E ( b , c , d ) χ ( τ ) = d e α r τ , (IA33)with B α r ( τ ) = (1 − e − α r τ ) /α r , and u = b − c /α r , u = c /α r . A ( b , c , d ) χ ∗ ( τ ), B ( b , c , d ) χ ∗ ( τ ) and C ( b , c , d ) χ ∗ ( τ ) are obtained similarly as in equation (IA33) with the characteristic χ replacedwith the characteristic χ ∗ .The following considers three types of interest rate derivatives based upon the types ofterminal payoff functions, given as • Payoffs that are linear functions of the state variables. These may be used to pricecaps, floors, yield options, and slope options. • Payoffs that are exponential in the state variables, used to price bond options, forwards,and futures options. • Payoffs that are integrals of the state variables, as in the case of average rate optionson the short rate and Asian options on yields. Linear Payoffs. The payoff function is given by a linear function of the short rate and13he state variables. Specifically, C T = ( k + k r T − K ) + , where k and k are constants.The expected price of a European call option for this payoff function is given by E t [ C H ] = Π ,t Π ,t − K E t [ P ( H, T )]Π ,t , (IA34)whereΠ ,t = ϕ R ( , c ∗ , k ∗ ; t, T, H ) , E t [ P ( H, T )] = ϕ R ( , c ∗ , ; t, T, H ) , Φ ( u ) = 1Π ,t ϕ R ( − iuk ∗ , c ∗ , k ∗ ; t, T, H ) , Φ ( u ) = 1 E t [ P ( H, T )] ϕ R ( − iuk ∗ , c ∗ , ; t, T, H ) , Π j,t = 12 + 1 π (cid:90) ∞ Re (cid:20) e − iuK Φ j ( u ) iu (cid:21) d u, j = 1 , , with k ∗ = ( k , k ), and c ∗ = (0 , Example: Interest Rate Cap. Consider a cap on the short rate with a strike price of K .The expected price of the cap is given by equation (IA34) with k = 0 and k = 1. Exponential Linear Payoffs. The payoff function is given by an exponential linearfunction of the short rate and the state variables. Specifically, C T = (exp ( k + k r T ) − K ) + , where k and k are constants.The expected price of a European call option for this payoff function is given by E t [ C H ] = Π ,t Π ,t − K E t [ P ( H, T )]Π ,t , (IA35)whereΠ ,t = ϕ R ( − k ∗ , c ∗ , ; t, T, H ) , E t [ P ( H, T )] = ϕ R ( , c ∗ , ; t, T, H ) , Φ ( u ) = 1Π ,t ϕ R ( − ( iu + 1) k ∗ , c ∗ , ; t, T, H ) , Φ ( u ) = 1 E t [ P ( H, T )] ϕ R ( − iuk ∗ , c ∗ , ; t, T, H ) , Π j,t = 12 + 1 π (cid:90) ∞ Re (cid:20) e − iu ln K Φ j ( u ) iu (cid:21) d u, j = 1 , , k ∗ = ( k , k ), and c ∗ = (0 , Example: Bond Option. Consider a bond option maturing at time T , written on adiscount bond maturing at time S with S ≥ T , and a strike price of K . The expectedprice of the bond option is given by equation (IA35) with k = A χ ∗ ( S − T ; , c ∗ , ) and k = B χ ∗ ( S − T ; , c ∗ , ). Integro-Linear Payoffs. The payoff function is given by a path integral of a linearfunction of the interest rate and the state variables. Specifically, C T = (cid:18)(cid:90) Tt ( k + k r u ) d u − K (cid:19) + , where k and k are constants.The expected price of a European call option for this payoff function is given by E t [ C H ] = Π ,t Π ,t − K E t [ P ( H, T )]Π ,t , (IA36)whereΠ ,t = ∂ϕ R ( , c , ; t, T, H ) ∂z (cid:12)(cid:12)(cid:12) z =0 , E t [ P ( H, T )] = ϕ R ( , c ∗ , ; t, T, H ) , Φ ( u ) = 1Π ,t ∂ϕ R ( , c , ; t, T, H ) ∂z (cid:12)(cid:12)(cid:12) z = iu , Φ ( u ) = 1 E t [ P ( H, T )] ϕ R ( , c , ; t, T, H ) (cid:12)(cid:12)(cid:12) z = iu , Π j,t = 12 + 1 π (cid:90) ∞ Re (cid:20) e − iuK Φ j ( u ) iu (cid:21) d u, j = 1 , , with c = ( − zk , − zk + 1), and c ∗ = (0 , Example: Asian Option on the Short Rate. Consider an Asian option on the short ratewith the strike price of K . The expected price of the Asian option is given by equation(IA36) with k = 0 and k = 1. IF Expected Option Prices under the CGMY L´evy Model Another set of models used widely in the equity option literature are the infinite-activityL´evy models (see, e.g., Madan and Seneta, 1990; Barndorff-Nielsen, 1997; Madan et al.,1998; Eberlein and Prause, 2002; Carr et al., 2002; Carr and Wu, 2003). The CGMY model(Carr, Geman, Madan, and Yor, 2002) is quite flexible in capturing both the fat tails and15kewness in asset returns and nests the variance gamma (Madan and Seneta, 1990; Madanet al., 1998) model. Here, we derive the closed-form solution of the expected future priceof an equity option under the CGMY model. Assume that the L´evy density of the CGMYprocess ν ( x ) is given by ν ( x ) = C exp ( − G | x | ) | x | Y , for x < ,C exp ( − M | x | ) | x | Y , for x > , (IA37)where C > G ≥ M ≥ 0, and Y < 2. We denote by X ( C, G, M, Y ) the infinitely divisibleprocess of independent increments with L´evy density given by equation (IA37). Notably,this CGMY process has a completely monotone L´evy density for Y > − 1. In addition, it isa process of infinite activity for Y > Y > ψ ( z ; C, G, M, Y, T − t ) (cid:44) E P t [exp ( z ( X T ( C, G, M, Y ) − X t ( C, G, M, Y )))]= exp (cid:2) ( T − t ) C Γ( − Y ) (cid:0) ( M − z ) Y − M Y + ( G + z ) Y − G Y (cid:1)(cid:3) . (IA38)We assume that the martingale component of the movement in the logarithm of prices isgiven by a Brownian motion and the above CGMY process. Hence, the asset price processunder the physical measure P is assumed to be given by S T = S t exp (cid:20) (cid:18) µ + ω − σ (cid:19) ( T − t ) + σ ( W T − W t )+ ( X T ( C, G, M, Y ) − X t ( C, G, M, Y )) (cid:21) , (IA39)where W is a standard Brownian motion independent of the process X ( C, G, M, Y ), and ω is a “convexity correction” defined byexp( − ω ( T − t )) (cid:44) E P t [exp ( X T ( C, G, M, Y ) − X t ( C, G, M, Y ))]= ψ (1; C, G, M, Y, T − t ) . (IA40)Letting C ∗ , G ∗ , M ∗ , and Y ∗ denote the risk-neutral parameters, the asset price process16nder the risk-neutral measure Q is given as S T = S t exp (cid:20) (cid:18) r + ω ∗ − σ (cid:19) ( T − t ) + σ ( W ∗ T − W ∗ t )+ ( X T ( C ∗ , G ∗ , M ∗ , Y ∗ ) − X t ( C ∗ , G ∗ , M ∗ , Y ∗ )) (cid:21) , (IA41)and exp( − ω ∗ ( T − t )) = ψ (1; C ∗ , G ∗ , M ∗ , Y ∗ , T − t ) . (IA42)Hence, the asset process under the R measure is given by S T = S t exp (cid:20) (cid:18) µ + ω R − σ (cid:19) ( H − t ) + (cid:18) r + ω ∗ R − σ (cid:19) ( T − H ) + σ ( W R T − W R t )+ ( X T ( C , G , M , Y ) − X t ( C , G , M , Y )) (cid:21) , (IA43)where C = ( C, C ∗ ), and G , M , and Y are similarly defined. ω R and ω ∗ R are defined byexp( − ω R ( H − t )) = ψ (1; C, G, M, Y, H − t ) , exp( − ω ∗ R ( T − H )) = ψ (1; C ∗ , G ∗ , M ∗ , Y ∗ , T − H ) . (IA44)The CGMY process X ( C , G , M , Y ) under R is defined by the compensator of the jumpprocess’ random measure as follows: ν XR (d s, d x ) = { s 17 exp (cid:18) − r ( T − H ) + z (cid:18) ln S t − σ T − t ) + ( µ + ω R )( H − t ) + ( r + ω ∗ R )( T − H ) (cid:19) + 12 z σ ( T − t ) (cid:19) × ψ R ( z ; C , G , M , Y ) , (IA46)where ψ R ( z ; C , G , M , Y ) (cid:44) E R t [exp ( z ( X T ( C , G , M , Y ) − X t ( C , G , M , Y )))]= exp (cid:104) ( H − t ) C Γ( − Y ) (cid:0) ( M − z ) Y − M Y + ( G + z ) Y − G Y (cid:1) + ( T − H ) C ∗ Γ( − Y ∗ ) (cid:16) ( M ∗ − z ) Y ∗ − M ∗ Y ∗ + ( G ∗ + z ) Y ∗ − G ∗ Y ∗ (cid:17)(cid:105) . (IA47)By substituting φ R ( z ) given in equation (IA46) in equations (66) through (67) givesthe expected future prices of the call option and the put option under the CGMY model.These equations can also be extended to obtain the expected future prices of multiple optionswritten on the same underlying asset with different strike prices in a computationally efficientmanner using the fast Fourier transform of Carr and Madan (1999). IG Expected Return Simulation The results of this paper have a range of empirical applications for studying the cross-section of “finite-horizon” returns (both parametrically and non-parametrically) in differentsecurities markets, including equity derivatives, Treasury bonds, corporate bonds, interestrate derivatives, credit derivatives, and others. This section provides an empirical simulationof expected returns of equity options using the constant volatility model of Black and Scholes(1973) (BS), the stochastic volatility model of Heston (1993) (SV), the stochastic volatilityjump model of Pan (2002) (SVJ), and the stochastic volatility double-jump model of Duffieet al. (2000) (SVJJ). Expected Return Definition. Since we consider the expected returns of a variety of con-tingent claims with finite expiration/maturity dates in this paper, we need a return measurebased upon “expected returns” that applies to all contingent claims and is also meaningfulfor comparisons across different horizons . Let F t be the current time t price of a nondivi-dend paying claim, and let F H be its expected future value at time H . Define R t ( H ), as the annualized log expected return (ALER) over the time interval t to H , so that the expectedprice can be expressed as E t [ F H ] = F t exp ( R t ( H ) · ( H − t )) , ≤ t < H ≤ T. (IA48)18he ALER is obtained by taking the log of the expected (gross) return and then annualizingit as follows: R t ( H ) = 1 H − t ln (cid:18) E t [ F H ] F t (cid:19) . (IA49)Though for various econometric reasons, it is convenient to use the expected log return(ELR) in the Treasury bond market, the ALER is more intuitive for comparing expectedreturns both intertemporally and cross-sectionally for different types of finite-maturity con-tingent claims, such as equity options, Treasury bonds, corporate bonds, and others. Simulation Approach. In our simulations, we report the three-dimensional surface ofexpected returns of equity options under the BS, SV, SVJ, and SVJJ models. The parametersand MPRs are borrowed from Tables I and IV in Broadie et al. (2007) and Eraker et al.(2003). Broadie et al. (2007) use a dataset of S&P 500 futures options from January 1987to March 2003 to estimate risk premia or MPRs of diffusive volatility ( γ v ), price jump ( γ J ),and volatility jump ( γ vJ ). However, Broadie et al. (2007) do not report the diffusive MPRs( γ S ), so we first borrow the diffusive drift values from Eraker et al. (2003), since Broadieet al.’s (2007) parameters under P measures are also borrowed from Eraker et al. (2003).Then, we calibrate the diffusive MPRs ( γ S ) by using constant risk-free rate 6.83% ( r ), whichis the time-series average of 3-month Treasury bill rates from January 2, 1980 to December31, 1999—the same sample period in Eraker et al. (2003).Table IA1 and Table IA2 give the parameters and MPRs we use in our simulations.Broadie et al. (2007) constrained several parameters for each model; we, nevertheless, alwaysuse their most general model parameters. In other words, for each model, we always usetheir estimates with most flexibility. We also set the initial value of volatility v t to be equalto the long-term mean m v . While Broadie et al. (2007) do not report the parameters forthe BS model, we estimate the BS model parameters using the same sample period as in For example, see Dai and Singleton (2002); Bansal and Zhou (2002); Cochrane and Piazzesi (2005);Eraker et al. (2015). The ELR is not so intuitive even in the Treasury bond market. For example, the instantaneous ELRsof default-free zero-coupon bonds of different maturities are not the same, even when the local expectationshypothesis (L-EH) holds (see Cox et al., 1981), but the instantaneous ALERs of all default-free bonds equalthe riskless short rate when the L-EH holds. The ELR of a portfolio of securities cannot be obtained fromthe ELRs of the securities in the portfolio; however, the ALER of a portfolio of securities can be obtainedeasily using the ALERs of the securities in the portfolio.Moreover, since traders and fund managers are rewarded for the portfolio’s simple expost return achievedby their asset allocation and security picking skills, they care about knowing the expected simple returns,which can be obtained easily from the ALERs, but not so easily from the ELRs without knowing more aboutthe return distributions.Finally, for securities that may be worth zero in any future states, the ELR decreases to negative infinity,so it cannot be used for corporate bonds and most derivatives that have truncated payoffs. [Insert Table IA1 about here.][Insert Table IA2 about here.][Insert Table IA3 about here.] Simulation Analysis. Figure IA1 depicts the horizon structure of annualized log expectedreturns (ALERs) on the market index under the BS, SV, SVJ, and SVJJ models. The BSmodel has the largest ALER of 14.3%, and the SV model has the second largest ALER of12.3%. The ALERs of these two models mainly reflect the rewards from the diffusive marketprice of risk (MPR) γ S . The SVJ model has the smallest ALER of 9.8%, and the SVJJmodel has a slightly higher ALER of 10.8%. The ALERs of the SVJ and SVJJ models arelower than those of the BS and SV models, because the former models depend upon theMPRs related to volatility of price jump risk and volatility jump risk. [Insert Figure IA1 about here.] Figure IA2 and Figure IA3 depict the surfaces of ALERs for at-the-money (ATM) callsand at-the-money (ATM) puts, respectively, under the BS, SV, SVJ, and SVJJ models. Thegrey shaded areas depict the corresponding ALERs when all of the MPRs for the respectivemodels equal zero. As expected, the ALERs equal the risk-free rate of 6.83% ( r ) under thiscase for all models. [Insert Figure IA2 about here.][Insert Figure IA3 about here.] Figure IA2(a) and Figure IA3(a) consider the BS model, which is the most basic modelusing only the diffusive MPR to generate the expected returns. The ALERs for calls arepositive and exceed the risk-free rate, while the ALERs for puts are negative and less than therisk-free rate under all combinations of holding periods and option maturities. The length ofthe holding period has less of an effect on the ALER than the length of the option maturity.Finally, shorter holding periods and shorter option maturities lead to higher (lower) ALERsfor call (put) options. Some of these patterns remain the same, while others change as weconsider more complex models with the MPRs related to volatility risk and jumps risks.Figure IA2(b) and Figure IA3(b) illustrate the ALER patterns under the SV model. TheALERS are very similar to those under the BS model, except that they are slightly lower forboth calls and puts due to the negative effect of the MPR for volatility risk under the SVmodel. 20igure IA2(c) and Figure IA3(c) illustrate the ALER patterns under the SVJ model,which incorporates two additional MPRs related to the mean price jump and the volatilityof price jump . The MPR for the mean price jump increases the call ALERs but decreases theput ALERs, while the MPR for the volatility of price jump decreases both the call and theput ALERs. Based upon parameters in Table IA1 and Table IA2, the effect of the MPR forthe volatility of price jump dominates the effect of the MPR for the mean price jump, and asa result the ALERs of both calls and puts are significantly lower under the SVJ model thanunder the BS model and the SV model. Curiously, the ALERs of the “holding-to-maturity”call options are also negative and lower than the risk-free rate suggesting risk-loving behavioron the upside (see Coval and Shumway, 2001).Figure IA2(d) and Figure IA3(d) illustrate the ALER patterns under the SVJJ model,which allows contemporaneous jump arrivals in both the volatility and the price. While theALER surface shapes are similar under the SVJ model and the SVJJ model, the ALERs arelower under the SVJJ model. This is because the additional MPR related to the volatilityjump risk reduces the ALERs of both calls and puts even further. Similar to the caseof the SVJ model, the ALERs of the “holding-to-maturity” call options are negative andlower than the risk-free rate suggesting risk-loving behavior on the upside under the SVJJmodel, as well. The risk-loving behavior on the upside under both the SVJ model and theSVJJ model is consistent with the well-known “pricing kernel puzzle” (see, e.g., Jackwerth,2000; A¨ıt-Sahalia and Lo, 2000; Rosenberg and Engle, 2002) as well as the application ofthe cumulative prospect theory by Baele et al. (2019) for explaining the negative expectedoption returns of call options.Figure IA4 and Figure IA5 illustrate the ALER patterns of the out-of-the-money (OTM)calls and the OTM puts, respectively, under the BS, SV, SVJ, and SVJJ models. Both theOTM calls and the OTM puts lead to more extreme patterns for the ALERs than the ATMcalls and the ATM puts. For example, the ALERs of the OTM calls and the OTM puts aresignificantly more negative than the corresponding ALERs of the ATM calls and the ATMputs, under the SVJ and SVJJ models. Very high negative ALERs are obtained using theshortest holding periods and shortest maturities, under the SVJ and SVJJ models using theOTM calls and the OTM puts. [Insert Figure IA4 about here.][Insert Figure IA5 about here.] Figure IA6 illustrates the surface of ALERs of straddles constructed by holding a longposition in the ATM call and a long position in the ATM put. The high negative ALERson straddles accrue due to the market prices of risk related to volatility risks (related to21he MPRs of diffusion volatility, volatility in price jumps, and jumps in volatility). Since allof these risks are captured by the SVJJ model, and since the MPRs related to these riskalways decrease the ALERs for both calls and puts, the most negative ALERs for straddlesare obtained using the SVJJ model. [Insert Figure IA6 about here.] IH A Procedure to Extract the Expected FSPD To obtain the expected FSPD from expected future prices of options using Theorem 2, weconsider extending the methods used for obtaining the SPD (defined as the second derivativeof the call price function with respect to the strike price in Breeden and Litzenberger (1978))from current option prices. Both parametric estimation methods and non-parametric esti-mation methods can be used for extracting the SPD from option prices (Figlewski, 2018).The parametric methods include composite distributions based on the normal/lognormal(see, e.g., Jarrow and Rudd, 1982; Rubinstein and Mark, 1998; Madan and Milne, 1994)and mixture models (see, e.g., Melick and Thomas, 1997; Soderlind and Svensson, 1997;Gemmill and Saflekos, 2000). The non-parametric estimation methods can be classified intothree categories: maximum entropy (see, e.g., Stutzer, 1996; Buchen and Kelly, 1996), kernel(see, e.g., A¨ıt-Sahalia and Lo, 1998; Pritsker, 1998), and curve-fitting (see, e.g., Jackwerth,2000, 2004; Rosenberg and Engle, 2002; Figlewski, 2009) methods. We extend the fast andstable curve-fitting method of Jackwerth (2004) that fits the implied volatilities using theBlack and Scholes (1973) model, and then numerically approximates the SPD. .We briefly recall the main steps for using the Jackwerth (2004) method. The firststep inverts the option prices into Black-Scholes implied volatilities, ¯ σ ( K/S t ), using a fi-nite number of options with different strike prices. This allows greater stability since im-plied volatilities for different strike prices are much closer in size than the correspondingoption prices. Let { K , K , ..., K N } represent a set of strike prices ordered from the low-est to the highest. Using a “smoothness” parameter as in Jackwerth (2004), a smooth BSimplied volatility (IV) curve ˆ σ ( K/S t ) is fitted by optimization. In the next step, each op-timized IV ˆ σ ( K i /S t ) is converted back into a call price using the Black-Scholes equation,i.e., C i = C BS ( S t , K i , t, T, r ; ˆ σ ( K i /S t )), and the SPD (discounted Q T density) is estimated For an extensive review of the curve-fitting methods, see Jackwerth (2004) and Figlewski (2018). < i < N , f t ( S T = K i ) = P ( t, T ) p Q T t ( S T = K i )= ∂ C BS ( S t , K i , t, T, r ; ˆ σ ( K i /S t )) ∂K i ≈ C i +1 − C i + C i − (∆ K ) . (IA50)We extend the fast and stable curve-fitting method of Jackwerth (2004) for obtaining theexpected FSPD from the expected future option prices by not only fitting implied volatilitiesusing the current call price formula of the Black and Scholes (1973) model, but also the im-plied drifts using the expected future call price formula of the Black and Scholes (1973) modelgiven in equation (21), and then numerically approximate the expected FSPD using Theorem2 (equation (68)). This extension is outlined in the following steps. The expected returns ofoptions are first estimated using historical data, and then expected option prices are obtainedusing equation (73) for a finite number of strike prices given by the set { K , K , ..., K N } . Asin Jackwerth (2004), the current option prices are inverted into BS IV ¯ σ ( K i /S t ) and usinga “smoothness” parameter, a smooth BS IV curve ˆ σ ( K/S t ) is fitted by optimization. Then,using fixed optimized IV ˆ σ ( K/S t ), we invert the expected future option price into the BSimplied drift ¯ µ ( K/S t ) using the Black-Scholes expected future call price equation (21). Next,following the optimization method of Jackwerth (2004), a smooth ID (implied drift) curveˆ µ ( K/S t ) is fitted by solving a similar optimization problem that balances smoothness againstthe fit of the IDs. For each strike price, the optimized IV ˆ σ ( K i /S t ) and ID ˆ µ ( K i /S t ), areconverted back into the expected call price using the Black-Scholes expected price equation(21), i.e., EC i = ˆ C BS ( S t , K i , t, T, H, r ; ˆ σ ( K i /S t ) , ˆ µ ( K i /S t )). Finally, the expected FSPD isnumerically approximated using Theorem 2 (equation (68)), as follows: For any 1 < i < N , g ( H, S T = K i ) = E P t [ P ( H, T )] p R T t ( S T = K i )= ∂ ˆ C BS ( S t , K i , t, T, H, r ; ˆ σ ( K i /S t ) , ˆ µ ( K i /S t )) ∂K i ≈ EC i +1 − EC i + EC i − (∆ K ) . (IA51)23 igures Figure IA1: Horizon Structure of Annualized Log Expected Returns on Stocks (a) BS (b) SV(c) SVJ (d) SVJJ Notes : This figure shows the horizon structure of annualized log expected returns (ALERs) onthe market index under the BS, SV, SVJ, and SVJJ models. The models’ common parametersand MPRs are given in Table IA1 and Table IA2. (a) BS (b) SV(c) SVJ (d) SVJJ Notes : This figure shows the surfaces of ALERs for at-the-money (ATM) calls under the BS,SV, SVJ, and SVJJ models. The models’ common parameters and MPRs are given in TableIA1 and Table IA2. The grey shaded area represents the ALERs when all of the MPRs arezero. (a) BS (b) SV(c) SVJ (d) SVJJ Notes : This figure shows the surfaces of ALERs for at-the-money (ATM) puts under the BS,SV, SVJ, and SVJJ models. The models’ common parameters and MPRs are given in TableIA1 and Table IA2. The grey shaded area represents the ALERs when all of the MPRs arezero. (a) BS (b) SV(c) SVJ (d) SVJJ Notes : This figure shows the surfaces of ALERs for out-of-the-money (OTM) calls under theBS, SV, SVJ, and SVJJ models. The models’ common parameters and MPRs are given inTable IA1 and Table IA2. The grey shaded area represents the ALERs when all of the MPRsare zero. (a) BS (b) SV(c) SVJ (d) SVJJ Notes : This figure shows the surfaces of ALERs for out-of-the-money (OTM) puts under theBS, SV, SVJ, and SVJJ models. The models’ common parameters and MPRs are given inTable IA1 and Table IA2. The grey shaded area represents the ALERs when all of the MPRsare zero. (a) BS (b) SV(c) SVJ (d) SVJJ Notes : This figure shows the surfaces of ALERs for straddles constructed by holding a longposition in the ATM call and a long position in the ATM put under the BS, SV, SVJ, andSVJJ models. The models’ common parameters and MPRs are given in Table IA1 and TableIA2. The grey shaded area represents the ALERs when all of the MPRs are zero. ables Table IA1: Parameters in Simulations α v m v = v t σ v ρ λ = λ ∗ µ S σ S µ v µ ∗ S σ ∗ S µ ∗ v BS – – – – – – – – 0.1588 ( σ )SV 5.7960 0.0227 0.3528 -0.40 – – – – –SVJ 3.2760 0.0204 0.2520 -0.47 1.5120 -0.0259 0.0407 – –-0.0491 0.0994SVJJ 6.5520 0.0136 0.2016 -0.48 1.5120 -0.0263 0.0289 0.0373 0 ( ρ J )-0.0539 0.0578 0.2213 Notes : This table gives the parameters used in the simulations. The parameters are borrowed from TablesI and IV in Broadie et al. (2007) and Eraker et al. (2003). We also set the initial value of volatility v t to beequal to the long-term mean m v . Since Broadie et al. (2007) do not report the parameters for the BSmodel, we estimate the BS model parameters using the same sample period as in Eraker et al. (2003). γ S γ v γ J γ vJ BS 0.4691 – – –SV 2.4219 -1.2600 – –SVJ 0.0775 -1.5120 0.0279 –SVJJ 0.0858 -7.8120 0.0383 -0.2781 Notes : This table gives MPRs used in the simulations. The MPRs are borrowed from Tables I and IV inBroadie et al. (2007) and Eraker et al. (2003), where γ J = λ ¯ µ − λ ∗ ¯ µ ∗ , with ¯ µ = e µ S + σ S − µ ∗ = e µ ∗ S + σ ∗ S − 1, and γ vJ = λµ v − λ ∗ µ ∗ v . While Broadie et al. (2007) do not report γ S , hence we firstborrow the diffusive drift values from Eraker et al. (2003), since Broadie et al.’s (2007) parameters under P measures are also borrowed from Eraker et al. (2003). Then, we calibrate γ S by using constant risk-freerate 6.83% ( r ), which is the time-series average of 3-month Treasury bill rates from January 2, 1980 toDecember 31, 1999—the same sample period in Eraker et al. (2003). α v = 252 α d v m v = 252 m d v / σ v = 252 σ d v / ρ = ρ d λ = 252 λ d µ s = µ d s σ s = σ d s µ v = 252 µ d v / ρ J = ρ d J γ v = 252 γ v d Notes : This table gives the scaling rules of parameters. For annual decimal parameter Θ, its daily decimalcounterpart is Θ d ..