A Put-Call Transformation of the Exchange Option Problem under Stochastic Volatility and Jump Diffusion Dynamics
AA Put-Call Transformation of the Exchange Option Problem underStochastic Volatility and Jump-Diffusion Dynamics
Len Patrick Dominic M. Garces a,b and Gerald H. L. Cheang a a Centre for Industrial and Applied Mathematics, School of Information Technology andMathematical Sciences, University of South Australia, Mawson Lakes SA 5095, Australia; b Department of Mathematics, School of Science and Engineering, Ateneo de ManilaUniversity, Quezon City 1108, Metro Manila, Philippines
ARTICLE HISTORY
Compiled February 25, 2020
ABSTRACT
We price European and American exchange options where the underlying asset pricesare modelled using a Merton (1976) jump-diffusion with a common Heston (1993)stochastic volatility process. Pricing is performed under an equivalent martingalemeasure obtained by setting the second asset yield process as the num´eraire asset,as suggested by Bjerskund and Stensland (1993). Such a choice for the num´erairereduces the exchange option pricing problem, a two-dimensional problem, to pricinga call option written on the ratio of the yield processes of the two assets, a one-dimensional problem. The joint transition density function of the asset yield ratioprocess and the instantaneous variance process is then determined from the corre-sponding Kolmogorov backward equation via integral transforms. We then deter-mine integral representations for the European exchange option price and the earlyexercise premium and state a linked system of integral equations that characterizesthe American exchange option price and the associated early exercise boundary.Properties of the early exercise boundary near maturity are also discussed.
KEYWORDS
Exchange options; change-of-num´eraire; jump diffusion processes; put-calltransformation; stochastic volatility
1. Introduction
We investigate the pricing of European and American exchange options written onassets with prices driven by stochastic volatility and jump-diffusion (SVJD) dynamics.The earliest analysis of European exchange options was that of Margrabe (1978) whoassumed that the underlying non-dividend-paying stocks are modelled with correlatedgeometric Brownian motions. Assuming that the European exchange option price islinear homogeneous in the stock prices, Margrabe (1978) transformed the problemto the classical European call option pricing problem and computed the exchangeoption price using the solution of Black and Scholes (1973). Fischer (1978) considereda closely related problem of determining call option prices when the exercise price
Corresponding Author; E-mail: len patrick [email protected]; ORCiD: https://orcid.org/0000-0002-2737-7348
E-mail: [email protected]; ORCiD: https://orcid.org/0000-0003-3786-0285 a r X i v : . [ q -f i n . M F ] F e b s also a diffusion process. Bjerskund and Stensland (1993) considered the pricing ofAmerican exchange options as an optimal stopping problem in a pure diffusion setting.They suggested that by choosing one of the stocks as the num´eraire and by a change ofmeasure to the corresponding equivalent martingale measure, the American exchangeoption pricing problem may be simplified to the problem of pricing an American callor put option. Bjerskund and Stensland (1993) refer to this technique as the “put-calltransformation.”With well-established evidence pointing to the deficiencies of the geometric Brown-ian motion in accurately modelling asset price returns, there has since been a movementto study option prices (including exchange options) under alternative asset price mod-els. Cheang, Chiarella, and Ziogas (2006) used the put-call transformation techniqueto price European exchange options where the underlying assets are modelled us-ing correlated Merton (1976) jump-diffusion models. With the same asset price modelspecification, Cheang and Chiarella (2011) and Caldana et al. (2015) studied Europeanand American exchange option prices using the risk-neutral approach. Cufaro-Petroniand Sabino (2018) priced European exchange options assuming that underlying assetprice jumps are correlated and considered applications in energy markets. Antonelliand Scarlatti (2010), Al`os and Rheinlander (2017), and Kim and Park (2017) pricedEuropean exchange options where underlying assets are driven by stochastic volatilitymodels. Notably, Al`os and Rheinlander (2017) also employed the put-call transfor-mation and discussed hedging under the resulting martingale measure. Fajardo andMordecki (2006) also used a similar transformation, which they called the “dual mar-ket method”, to price options (including perpetual exchange options) when underlyingprices are driven by L´evy processes. Ma, Pan, and Wang (2020) analyzed Europeanexchange options when asset prices are modelled using Hawkes jump-diffusion pro-cesses, allowing for jump-contagion in individual assets and jump interdependenceamong multiple assets. More recently, Cheang and Garces (2020), derived analyti-cal representations for the European and American exchange option prices assumingthat stock prices are modelled using a pair of Bates (1996) stochastic volatility andjump-diffusion dynamics.In this paper, we derive integral representations of the price of European and Amer-ican exchange options when underlying asset prices are modelled using Merton (1976)jump-diffusion dynamics with a common underlying Heston (1993) stochastic volatil-ity process. To do so, we follow the suggested approach of Bjerskund and Stensland(1993) and assign the yield process of the second stock as the num´eraire asset, incontrast to using the money market account as the num´eraire. Doing so simplifies theoriginal two-dimensional problem to a one-dimensional problem of pricing an ordinarycall option written on the ratio of the yield processes of the two underlying assets. This allows us to follow the techniques of Cheang, Chiarella, and Ziogas (2013) whoanalyzed single-asset American call options under the Bates (1996) stochastic volatilityand jump-diffusion dynamics.Our main contribution is to extend the Bjerskund and Stensland (1993) strategy forvaluing American exchange options in a pure-diffusion setting into the SJVD frame- The empirical literature addressing the limitations of the Black and Scholes (1973) is extremely rich and willnot be reviewed in its totality here. Instead, we invite the reader to consult Bakshi, Cao, and Chen (1997),Duffie, Pan, and Singleton (2000), Cont (2001), Andersen, Benzoni, and Lund (2002), Chernov et al. (2003),Eraker, Johannes, and Polson (2003), Kou (2008), and the references therein. While the title of this paper mentions a “put-call transformation”, our main analysis consists of transformingthe exchange option pricing problem to the problem of pricing a call option on the asset yield ratio by choosingthe second asset yield process as the num´eraire. The exchange option problem can be reduced to a put optionon the asset yield ratio if one alternatively chooses the yield process of the first asset as the num´eraire. and jump-diffusion model.In this analysis, pricing takes place under the equivalent martingale measure ˆ Q corresponding to setting the second asset yield process as the num´eraire. Under ˆ Q , wefind that the the no-arbitrage price of the European exchange option can be written asa function of only the asset yield ratio ˜ s and the instantaneous variance v . Furthermore,we verify an early exercise representation of the discounted American exchange optionprice, which can also be written as a function of only ˜ s and v .To evaluate expectations under ˆ Q , we require the joint transition density functionof ˜ s and v . To do so, we determine the Kolmogorov equation for the transition densityand solve this using Fourier and Laplace transforms. With the joint transition densityfunction, we then provide integral representations for the European exchange optionprice and the early exercise premium and present a linked system of integral equationsthat characterize the price of the American exchange option and the unknown earlyexercise boundary. We also note that the joint transition density function we obtain inthis analysis enables us to price any European-type option written on the two assetsprovided that its payoff function can be written in terms of the terminal asset yieldratio and instantaneous variance.Our analysis also serves as an alternative of that of Cheang and Garces (2020) inthe following aspects.(1) Our SVJD model specification allows us to incorporate the possibility of corre-lation between the asset price returns and between each individual asset priceprocess and the instantaneous variance process.(2) We take one of the asset yield processes as the num´eraire instead of the moneymarket account used by Cheang and Garces (2020).(3) We express option prices in terms of the transition density function of the un-derlying stochastic processes under ˆ Q .(4) We provide a more in-depth analysis of the early exercise boundary (particularlyits behavior near maturity) and the early exercise premium for the Americanexchange option, and thus extending the earlier work of Chiarella and Ziogas(2009) who investigated the limit of the earliy exercise boundary for Americancall options under jump-diffusion dynamics.We are primarily concerned with obtaining analytical representations for exchangeoption prices under our SVJD model. As such, it is not the goal of this paper to dis-cuss the numerical solution of the option pricing problem or the calibration of modelparameters of observable market data as these matters warrant their own dedicatedexposition. However, we discuss how our results set the stage for numerical imple-mentation.Note that while the succeeding analysis focuses on exchange options written on This method has been used by Chiarella and Ziogas (2009) to price American call options under jump-diffusion dynamics, by Chiarella, Ziogas, and Ziveyi (2010) to price American call options under stochasticvolatility, by Cheang, Chiarella, and Ziogas (2013) to price American call options under SVJD dynamics, andby Chiarella and Ziveyi (2014) to price American spread call options under pure diffusion dynamics. There is considerable extant work on calibrating the parameters of stochastic volatility and/or jump-diffusionmodels in respect to option pricing. These results are discussed in the works cited in the first footnote of thispaper. , energy market options (surveyed in Benth andZdanowicz 2015), and the option to enter/exit an emerging market (Miller 2012),among others. Ma, Pan, and Wang (2020) provide additional examples of financialcontracts which can be priced under the exchange option framework.The rest of the paper is organized as follows. Section 2 specifies the stochasticvolatility and jump-diffusion model for underlying asset prices, discusses the construc-tion of the measure ˆ Q , and presents the dynamics of the asset yield ratio and varianceprocesses under ˆ Q . Section 3 discusses the integro-partial differential equation (IPDE)for the discounted European exchange option price and the free-boundary IPDE forthe American exchange option. Section 4 uses probabilistic arguments to verify theearly exercise representation of the American exchange option price and to express theAmerican exchange option price as a solution of an inhomogeneous IPDE to be solvedover a domain unrestricted by the early exercise boundary. Section 5 discusses someproperties of the early exercise boundary near the maturity of the option. Section 6shows the solution of the Kolmogorov equation for the joint transition density func-tion using integral transform methods. With the transition density function, Sections7 and 8 present the integral representations for the European and American exchangeoption prices, respectively. Section 9 concludes the paper. Proofs of some results whichinvolve lengthy, but otherwise rather elementary, calculations are given as appendices.
2. A Stochastic Volatility Jump-Diffusion Model
In this section, we discuss the model specification for the underlying stock prices. Let(Ω , F , P ) be a probability space equipped with a filtration {F t } ≤ t ≤ T satisfying theusual conditions. Here, T > { W ( t ) } , { W ( t ) } , and { Z ( t ) } be standard P -Brownian motions with instantaneouscorrelations given by d W ( t ) d W ( t ) = ρ w d t and d W j ( t ) d Z ( t ) = ρ j d t , for j = 1 , . Denote by Σ the correlation matrix of the random vector B ( t ) = ( W ( t ) , W ( t ) , Z ( t )) (cid:62) .Let p (d y j , d t ) ( j = 1 ,
2) be the counting measure associated to a marked Poissonprocess with P -local characteristics ( λ j , m P (d y j )). Underlying p (d y j , d t ) is a sequenceof ordered pairs { ( T i,n , Y i,n ) } where Y i,n is the “mark” of the n th occurrence of anevent that occurs at a non-explosive time T i,n . The marks Y j, , Y j, , . . . are i.i.d. real-valued random variables with non-atomic density m P (d y j ). Associated to the eventtimes, we define a Poisson counting process { N j ( t ) } given by N j ( t ) = (cid:80) ∞ n =1 ( T j,n ≤ t ) ( Y j,n ∈ R ) , where ( · ) is the indicator function. For simplicity, we assume that theintensities λ j and jump-size densities m P (d y j ) are constant through time, although thesubsequent analysis can be extended to the case where the intensities and densitiesare deterministic functions of time.We assume that the counting measures are independent of the Brownian motions Cheang and Chiarella (2011) assumed that only one asset price process had jumps while the other wasmodelled as a pure-diffusion process. Quittard-Pinon and Randrianarivony (2010) discuss in greater detail theEuropean exchange option pricing problem under a similar model specification. See Runggaldier (2003) for more details. {F t } is the natural filtration generatedby the Brownian motions and the counting measures, augmented with the collectionof P -null sets.Denote by { S ( t ) } and { S ( t ) } the price processes of two assets that pay a constantdividend yield of q and q , respectively, per annum. As stock prices may jump, welet S ( t ) and S ( t ) denote the stock prices prior to any jumps occurring at time t . Let { v ( t ) } be the instantaneous variance process that governs the volatility of both stockprice processes. We assume that the dynamics of the stock prices and the instantaneousvariance satisfy the stochastic differential equationsd S j ( t ) S j ( t ) = ( µ j − λ j κ j ) d t + σ j (cid:112) v ( t ) d W j ( t ) + (cid:90) R ( e y j − p (d y j , d t ) , j = 1 , , (1)d v ( t ) = ξ (cid:0) η − v ( t ) (cid:1) d t + ω (cid:112) v ( t ) d Z ( t ) . (2)Here, κ j ≡ E P [ e Y j −
1] = (cid:82) R ( e y j − m P (d y j ) is the mean jump size of the price of asset j under P , and µ j , σ j , ξ , η , and ω are positive constants. It is also assumed that theinitial values of these stochastic processes are positive. We refer to this model as the proportional stochastic volatility and jump-diffusion (SVJD) model. As described above, the model features a common instantaneous variance processand independent jump terms for each asset. The individual jump processes may betaken to model idiosyncratic risk factors in each asset that cause sudden changes inreturns. Although extremely rare, it is possible that jumps for both stocks arriveat the same time, representing market shocks or sudden events that may affect bothassets. In addition, the common variance process models systematic market volatilityor volatility at the macroeconomic level. As such, individual asset prices may providefeedback to each other via the correlation between the diffusion components and thedependence on a common stochastic volatility.We also assume the existence of a money market account whose value process isdenoted by { M ( t ) } , with M ( t ) = e rt for t ≥
0, where r > { v ( t ) } remains strictly positiveand finite for all 0 ≤ t ≤ T under P and any other probability measure equivalent to P (Andersen and Piterbarg 2007). Assumption 2.1.
The parameters ξ , η , and ω and the correlation coefficients ρ and ρ satisfy 2 ξη ≥ ω and − < ρ j < min (cid:8) ξ/ω, (cid:9) , j = 1 , S j ( t ) = S j (0) exp (cid:40) ( µ j − λ j κ j ) t − σ j (cid:90) t v ( s ) d s + σ (cid:90) t (cid:112) v ( s ) d W j ( s ) + N j ( t ) (cid:88) n =1 Y j,n (cid:41) , for 0 < t ≤ T . Assumption 2.1 and the non-explosion assumption on the point pro-cesses imply that the integrals and summation that appear above are well-defined. Italso follows that S j ( t ) > P -a.s. for all t ∈ [0 , T ], and hence either asset can be used In contrast, Cheang and Chiarella (2011) introduced an additional compound Poisson process appearing inboth asset return processes which capture macroeconomic shocks or systematic risk factors which may introducesudden jumps in returns.
5s a num´eraire.Instead of the money market account, we take { S ( t ) e q t } , the second asset yieldprocess, as the num´eraire and define the probability measure ˆ Q , equivalent to P , suchthat the first asset yield process and the money market account, when discountedby S ( t ) e q t , are martingales under ˆ Q . With the second asset yield process as thenum´eraire, the discounted price of any other asset with price process { X ( t ) } is definedby ˜ X ( t ) = X ( t ) / ( S ( t ) e q t ). In the absence of arbitrage opportunities, the appropriatediscounting factor for the period [ t, t (cid:48) ] for any t (cid:48) ∈ [ t, T ] is given by DF ( t, t (cid:48) ) = S ( t ) e q t / ( S ( t (cid:48) ) e q t (cid:48) ).Next, we discuss the construction of the equivalent probability measure ˆ Q . Thefollowing standard proposition specifies the form of the Radon-Nikod´ym derivative dˆ Q d P . Proposition 2.2.
Suppose θ ( t ) = (cid:0) ψ ( t ) , ψ ( t ) , ζ ( t ) (cid:1) (cid:62) is a vector of F t -adapted pro-cesses and let γ , γ , ν , ν be constants. Define the process { L t } by L ( t ) = exp (cid:40) − (cid:90) t (cid:16) Σ − θ ( s ) (cid:17) (cid:62) d B ( s ) − (cid:90) t θ ( s ) (cid:62) Σ − θ ( s ) d s (cid:41) × exp N ( t ) (cid:88) n =1 ( γ Y ,n + ν ) − λ t (cid:16) e ν E P ( e γ Y ) − (cid:17) × exp N ( t ) (cid:88) n =1 ( γ Y ,n + ν ) − λ t (cid:16) e ν E P ( e γ Y ) − (cid:17) (3) and suppose that { L ( t ) } is a strictly positive P -martingale such that E P [ L ( t )] = 1 forall t ∈ [0 , T ] . Then L ( T ) is the Radon-Nikod´ym derivative of some probability measure ˆ Q equivalent to P and the following hold:(1) Under ˆ Q , the vector process B ( t ) has drift − θ ( t ) ;(2) The Poisson process N j ( t ) has a new intensity ˜ λ j = λ j e ν j E P [ e γ j Y j ] , j = 1 , under ˆ Q ; and(3) The moment generating function of jump sizes random variable Y j under ˆ Q isgiven by M ˆ Q ,Y j ( u ) = M P ,Y j ( u + γ j ) /M P ,Y j ( γ j ) , j = 1 , . Proof.
See e.g. Runggaldier (2003, Theorem 2.4) and Cheang and Teh (2014, Theorem1).The Radon-Nikod´ym derivative L ( T ) = dˆ Q d P can be used to characterize any prob-ability measure ˆ Q equivalent to P as parameterized by the vector process { θ ( t ) } andthe constants γ , γ , ν , ν . We assume that γ , γ , ν , ν are constant to preserve thetime-homogeneity of the intensity and the jump size distribution. As the market underthe SVJD is generally incomplete, one can construct multiple equivalent martingalemeasures consistent with the no-arbitrage assumption.We now specify the parameters of L ( T ) so that ˆ Q becomes an equivalent martingalemeasure corresponding to the num´eraire { S ( t ) e q t } . Let { ˜ S ( t ) } and { ˜ M ( t ) } , where˜ S ( t ) = S ( t ) e q t / ( S ( t ) e q t ) and ˜ M ( t ) = e rt / ( S ( t ) e q t ) , be the first asset yield processand the money market account when discounted using the second stock’s yield process.In particular, we will refer to { ˜ S ( t ) } as the asset yield ratio process . If we choose6 ψ ( t ) } , { ψ ( t ) } , and { ζ ( t ) } as ψ ( t ) = µ + q − r − ρ w σ σ v ( t ) − λ κ + ˜ λ ˜ κ σ (cid:112) v ( t ) (4) ψ ( t ) = µ + q − r − σ v ( t ) − λ κ − ˜ λ ˜ κ − σ (cid:112) v ( t ) (5) ζ ( t ) = Λ ω (cid:112) v ( t ) for some constant Λ ≥ , (6)where ˜ κ = E ˆ Q [ e Y −
1] and ˜ κ − = E ˆ Q [ e − Y − { ˜ S ( t ) } and { ˜ M ( t ) } are ˆ Q -martingales on [0 , T ]. With this choice of parameters for ˆ Q , the dynamics of the instantaneous variancebecomes d v ( t ) = (cid:2) ξη − ( ξ + Λ) v ( t ) (cid:3) d t + ω (cid:112) v ( t ) d ¯ Z ( t ) . (7)where { ¯ Z ( t ) } is a ˆ Q -Wiener process. The choice of ζ ( t ) preserves the structure ofthe instantaneous variance as a square-root process. Assumption 2.1 ensures that thisprocess is strictly positive and finite ˆ Q -a.s.Under ˆ Q , ˜ S ( t ) satisfies the equationd ˜ S ( t ) = − ˜ S ( t ) (cid:16) ˜ λ ˜ κ + ˜ λ ˜ κ − (cid:17) d t + σ (cid:112) v ( t ) ˜ S ( t ) d ¯ W ( t )+ (cid:90) R ( e y −
1) ˜ S ( t ) p (d y , d t ) + (cid:90) R (cid:16) e − y − (cid:17) ˜ S ( t ) p (d y , d t ) . (8)where we define σ d ¯ W ( t ) ≡ σ d ¯ W ( t ) − σ d ¯ W ( t ) with standard ˆ Q -Wiener processes { ¯ W ( t ) } and { ¯ W ( t ) } and σ = σ + σ − ρ w σ σ . This equation admits a solution˜ S ( t ) given by˜ S ( t ) = ˜ S (0) exp (cid:40) − (˜ λ ˜ κ + ˜ λ ˜ κ − ) t − σ (cid:90) t v ( s ) d s + σ (cid:90) t (cid:112) v ( s ) d ¯ W ( s )+ N ( t ) (cid:88) m =1 Y ,m − N ( t ) (cid:88) n =1 Y ,n (cid:41) Lastly, we note that the instantaneous correlation between the ˆ Q -Brownian motions { ¯ W ( t ) } and { ¯ Z ( t ) } is given by E ˆ Q (cid:2) d ¯ W ( t ) d ¯ Z ( t ) (cid:3) = 1 σ ( σ ρ − σ ρ ) d t = σ ρ − σ ρ (cid:112) σ + σ − ρ w σ σ d t. This assertion can be proved using Itˆo’s Lemma on ˜ S ( t ) and ˜ M ( t ) and eliminating the resulting drift term asrequired by the martingale representation for jump-diffusion processes (see Runggaldier 2003, Theorem 2.3). In view of Proposition 2.2, we note that d ¯ W j ( t ) = ψ j ( t ) d t + d W j ( t ). . The Exchange Option Pricing IPDE Now we derive the integro-partial differential equation (IPDE) for the price of an ex-change option written on S and S . Denote by C ( t, S ( t ) , S ( t ) , v ( t )) the price of a Eu-ropean exchange option whose terminal payoff is given by C (cid:0) T, S ( T ) , S ( T ) , v ( T ) (cid:1) = (cid:0) S ( T ) − S ( T ) (cid:1) + , where x + ≡ max { x, } . A rearrangement of terms expresses thediscounted terminal payoff as C (cid:0) T, S ( T ) , S ( T ) , v ( T ) (cid:1) S ( T ) e q T = e − q T (cid:16) ˜ S ( T ) − e ( q − q ) T (cid:17) + . Let ˜ C ( t, S ( t ) , S ( t ) , v ( t )) ≡ C (cid:0) t, S ( t ) , S ( t ) , v ( t ) (cid:1) / (cid:0) S ( t ) e q t (cid:1) denote the discountedEuropean exchange option price. Then, assuming that no arbitrage opportunities exist,˜ C ( t, S ( t ) , S ( t ) , v ( t )) is given by˜ C (cid:0) t, S ( t ) , S ( t ) , v ( t ) (cid:1) = E ˆ Q (cid:20) ˜ C (cid:0) T, S ( T ) , S ( T ) , v ( T ) (cid:1)(cid:12)(cid:12)(cid:12) F t (cid:21) = e − q T E ˆ Q (cid:34) (cid:16) ˜ S ( T ) − e ( q − q ) T (cid:17) + (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) . (9)In other words, the price at any time t < T of the European exchange option measuredin units of the second asset yield process is the expected value, under the probabilitymeasure ˆ Q , of the terminal payoff measured in units of the second asset yield process(Geman, El Karoui, and Rochet 1995). From the last equation, we also note thatthe terminal payoff is variable only in the asset yield ratio ˜ S ( t ). Thus, we assumethat the discounted European exchange option price is represented by the process˜ V ( t, ˜ S ( t ) , v ( t )) ≡ ˜ C (cid:0) t, S ( t ) , S ( t ) , v ( t ) (cid:1) , and so˜ V ( t, ˜ S ( t ) , v ( t )) = e − q T E ˆ Q (cid:34) (cid:16) ˜ S ( T ) − e ( q − q ) T (cid:17) + (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) . (10)At this point, we have shown that, by taking the second stock’s yield process asthe num´eraire asset, the exchange option pricing problem is equivalent to pricing aEuropean call option on the asset yield price ratio ˜ S ( t ) with maturity date T and strikeprice e ( q − q ) T . In the succeeding analysis, we shall take advantage of this simplificationand employ techniques in pricing European call options under stochastic volatility andjump-diffusion dynamics (e.g. Bates 1996; Cheang, Chiarella, and Ziogas 2013). Remark 1.
If we choose the first asset yield process { S ( t ) e q t } as the num´eraire,then the exchange option pricing problem simplifies to the valuation of a put option written on the asset yield ratio ( S ( t ) e q t ) / ( S ( t ) e q t ).The following technical assumption is required to implement Itˆo’s formula for jump-diffusion processes. Assumption 3.1.
For t ∈ [0 , T ], ˜ V ( t, ˜ s, v ) is (at least) twice-differentiable in ˜ s and v and differentiable in t with continuous partial derivatives.In the following proposition, we derive the IPDE that characterizes the discounted8uropean exchange option price. Proposition 3.2.
The price at time t ∈ [0 , T ) of the European exchange option isgiven by C ( t, S ( t ) , S ( t ) , v ( t )) = S ( t ) e q t ˜ V ( t, ˜ S ( t ) , v ( t )) , (11) where ˜ V , satisfying Assumption 3.1, is the solution of the terminal value problem ∂ ˜ V∂t + L ˜ s,v (cid:104) ˜ V ( t, ˜ S ( t ) , v ( t )) (cid:105) , ( t, ˜ S ( t ) , ˜ v ( t )) ∈ [0 , T ] × R (12)˜ V ( T ) = e − q T (cid:16) ˜ S ( T ) − e ( q − q ) T (cid:17) + , (13) with R = (0 , ∞ ) × (0 , ∞ ) and the IPDE operator L ˜ s,v defined as L ˜ s,v (cid:104) ˜ V ( t, ˜ S, v ) (cid:105) = − ˜ S (cid:16) ˜ λ ˜ κ + ˜ λ ˜ κ − (cid:17) ∂ ˜ V∂ ˜ s + (cid:2) ξη − ( ξ + Λ) v (cid:3) ∂ ˜ V∂v + 12 σ v ˜ S ∂ ˜ V∂ ˜ s + 12 ω v ∂ ˜ V∂v + ω ( σ ρ − σ ρ ) v ˜ S ∂ ˜ V∂ ˜ s∂v + ˜ λ E Y ˆ Q (cid:20) ˜ V (cid:16) t, ˜ Se Y , v (cid:17) − ˜ V ( t, ˜ S, v ) (cid:21) + ˜ λ E Y ˆ Q (cid:20) ˜ V (cid:16) t, ˜ Se − Y , v (cid:17) − ˜ V ( t, ˜ S, v ) (cid:21) , (14) where E Y i ˆ Q is the expectation with respect to the r.v. Y i ( i = 1 , ) under the measure ˆ Q .Note that all partial derivatives are evaluated at ( t, ˜ S ( t ) , v ( t )) . Proof.
The tower property for conditional expectations imply that { ˜ V ( t ) } is a ˆ Q -martingale, with integrability guaranteed by Assumption 3.1. With equations (7) and(8) in mind, an application of Itˆo’s formula shows that ˜ V ( t ) satisfies the stochasticdifferential equationd ˜ V ( t ) = (cid:40) ∂ ˜ V∂t + L ˜ s,v (cid:104) ˜ V ( t, ˜ S ( t ) , v ( t )) (cid:105)(cid:41) d t + σ (cid:112) v ( t ) ˜ S ( t ) ∂ ˜ V∂ ˜ s d ¯ W ( t ) + ω (cid:112) v ( t ) ∂ ˜ V∂v d ¯ Z ( t )+ (cid:90) R (cid:20) ˜ V (cid:16) t, ˜ S ( t ) e y , v ( t ) (cid:17) − ˜ V ( t, ˜ S ( t ) , v ( t )) (cid:21) q (d y , d t )+ (cid:90) R (cid:20) ˜ V (cid:16) t, ˜ S ( t ) e − y , v ( t ) (cid:17) − ˜ V ( t, ˜ S ( t ) , v ( t )) (cid:21) q (d y , d t ) , (15)where L ˜ s,v is the IPDE operator defined by equation (14). Since { ˜ V ( t ) } is a ˆ Q -martingale, the drift must be equal to zero, giving us the equation (12). Terminalcondition (13) follows from the discussion at the start of this section.Let C A ( t, S ( t ) , S ( t ) , v ( t )) be the price at time t of an American exchange optionwritten on S and S . After a rearrangement of terms, standard theory on American9ption pricing (see e.g. Myneni 1992) dictates that the discounted American exchangeoption price ˜ V A ( t, ˜ S ( t ) , v ( t )) is given by˜ V A ( t, ˜ S ( t ) , v ( t )) ≡ C A ( t, S ( t ) , S ( t ) , v ( t ) S ( t ) e q t = ess sup u ∈ [ t,T ] e − q u E ˆ Q (cid:34) (cid:16) ˜ S ( u ) − e ( q − q ) u (cid:17) + (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) , (16)where the supremum is taken over all ˆ Q -stopping times u ∈ [ t, T ]. From here, we seethat the change of num´eraire reduces the problem to pricing an American call option onthe asset yield price ratio ˜ S ( t ) with maturity date T and strike price e ( q − q ) T , similarto our observation for the European exchange option. The price of the Americanexchange option also hedges against the exchange option payoff in the sense that˜ V A ( t, ˜ S ( t ) , v ( t )) ≥ e − q t (cid:16) ˜ S ( t ) − e ( q − q ) t (cid:17) + ∀ t ∈ [0 , T )˜ V A ( T, ˜ S ( T ) , v ( T )) = e − q T (cid:16) ˜ S ( T ) − e ( q − q ) T (cid:17) + . Before prescribing additional boundary conditions to IPDE (12) for the Americanexchange option, we first define the continuation and stopping regions, denoted by C and S , respectively, that divide the domain [0 , T ] × R of IPDE (12). These regionsare given by S = (cid:26) ( t, ˜ S, v ) ∈ [0 , T ] × R : ˜ V A ( t, ˜ S, v ) = e − q t (cid:16) ˜ S − e ( q − q ) t (cid:17) + (cid:27) C = (cid:26) ( t, ˜ S, v ) ∈ [0 , T ] × R : ˜ V A ( t, ˜ S, v ) > e − q t (cid:16) ˜ S − e ( q − q ) t (cid:17) + (cid:27) . (17)Denote by S ( t ) and C ( t ) the stopping and continuation regions at a fixed t ∈ [0 , T ].From Broadie and Detemple (1997), there exists a critical stock price ratio B ( t, v ) ≥
1, dependent on the current variance level (Touzi 1999), such that the stopping andcontinuation regions can be written as S = (cid:110) ( t, ˜ S, v ) ∈ [0 , T ] × R : ˜ S ≥ B ( t, v ) e ( q − q ) t (cid:111) C = (cid:110) ( t, ˜ S, v ) ∈ [0 , T ] × R : ˜ S < B ( t, v ) e ( q − q ) t (cid:111) . (18)The line s = B ( t, v ) s on the s s -plane is known as the early exercise boundary. For a fixed t ∈ [0 , T ] and v ∈ (0 , ∞ ), the early exercise boundary and the continuationand stopping regions are illustrated in Figure 1. It is known that in the continuationregion the American exchange option behaves like its live European counterpart, andso ˜ V A satisfies IPDE (12) for ( t, ˜ S, v ) ∈ C .We require value-matching and smooth-pasting conditions on IPDE (12) to enforcethe no-arbitrage assumption and to ensure that the discounted exchange option price Mishura and Shevchenko (2009) analyzed, in further detail, the properties of the exercise region of thefinite-maturity American exchange option in a pure diffusion setting. In the same setting, Villeneuve (1999) es-tablished the nonemptiness of exercise regions of American rainbow options, which include spread and exchangeoptions as special cases. igure 1. The early exercise boundary and the continuation and stopping regions for the transformed Amer-ican exchange option. ˜ V A and its partial derivative ∂ ˜ V A /∂ ˜ s are both continuous across the early exerciseboundary A ( t, v ) ≡ B ( t, v ) e ( q − q ) t . Specifically, the required value-matching conditionis ˜ V A ( t, A ( t, v ) , v ( t )) = e − q t (cid:16) A ( t, v ) − e ( q − q ) t (cid:17) , (19)and the smooth-pasting conditions arelim ˜ S → A ( t,v ) ∂ ˜ V A ∂ ˜ s ( t, ˜ S ( t ) , v ( t )) = e − q t lim ˜ S → A ( t,v ) ∂ ˜ V A ∂v ( t, ˜ S ( t ) , v ( t )) = 0lim ˜ S → A ( t,v ) ∂ ˜ V A ∂t ( t, ˜ S ( t ) , v ( t )) = − q e − q t ˜ S ( t ) + q e − q t . (20)Therefore, the discounted American exchange option price is a solution to IPDE (12)over the domain 0 ≤ t ≤ T , 0 < ˜ S < A ( t, v ), 0 < v < ∞ . The IPDE has terminal andboundary conditions˜ V ( T, ˜ S ( T ) , v ( T )) = e − q T (cid:16) ˜ S ( T ) − e ( q − q ) T (cid:17) + ˜ V ( t, , v ( t )) = 0 , (21)value-matching condition (19) and smooth-pasting condition (20).11 . An Early Exercise Representation In this section, we show that ˜ V A ( t, ˜ S ( t ) , v ( t )) can be decomposed into the sum ofthe discounted European exchange option price ˜ V ( t, ˜ S ( t ) , v ( t )) and an early exercisepremium. Proposition 4.1.
Suppose Assumption 3.1 also holds for ˜ V A ( t, ˜ S, v ) . Assume furtherthat the smooth pasting conditions (20) across the early exercise boundary hold. Then ˜ V A ( t, ˜ S ( t ) , v ( t )) can be expressed as ˜ V A ( t, ˜ S ( t ) , v ( t )) = ˜ V ( t, ˜ S ( t ) , v ( t )) + ˜ V P ( t, ˜ S ( t ) , v ( t )) , (22) where ˜ V ( t, ˜ S ( t ) , v ( t )) is the discounted European exchange option price given by equa-tion (10) and ˜ V P ( t, ˜ S ( t ) , v ( t )) is the early exercise premium given by ˜ V P ( t, ˜ S ( t ) , v ( t )) = − E ˆ Q (cid:90) Tt (cid:40) ∂ ˜ V A ∂t + L ˜ s,v (cid:104) ˜ V A ( s, ˜ S ( s ) , v ( s )) (cid:105)(cid:41) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t . (23) Note that all partial derivatives in ∂ ˜ V A /∂t + L ˜ s,v [ ˜ V A ( s, ˜ S ( s ) , v ( s ))] are all evaluatedat ( s, ˜ S ( s ) , v ( s )) . Proof.
Given that Assumption 3.1 also applies to ˜ V A ( t, ˜ S ( t ) , v ( t )) and equipped withthe smooth-pasting conditions discussed above, an application of Itˆo’s formula verifiesthat ˜ V A ( t ) ≡ ˜ V A ( t, ˜ S ( t ) , v ( t )) satisfies equation (15). Integrating over [ t , t ], where0 ≤ t ≤ t ≤ T , we find that˜ V A ( t ) = ˜ V A ( t ) + (cid:90) tt (cid:40) ∂ ˜ V A ∂t + L ˜ s,v (cid:104) ˜ V A ( s, ˜ S ( s ) , v ( s )) (cid:105)(cid:41) d s + (cid:90) tt σ (cid:112) v ( s ) ˜ S ( s ) ∂ ˜ V A ∂ ˜ s d ¯ W ( s ) + (cid:90) tt ω (cid:112) v ( s ) ∂ ˜ V A ∂v d ¯ Z ( s )+ (cid:90) tt (cid:90) R (cid:20) ˜ V A (cid:16) s, ˜ S ( s ) e y , v ( s ) (cid:17) − ˜ V A ( s, ˜ S ( s ) , v ( s )) (cid:21) q (d y , d s )+ (cid:90) tt (cid:90) R (cid:20) ˜ V A (cid:16) s, ˜ S ( s ) e − y , v ( s ) (cid:17) − ˜ V A ( s, ˜ S ( s ) , v ( s )) (cid:21) q (d y , d s ) . Next, we take the ˆ Q -expectation of the above equation conditional on F t . We notethat the integrals with respect to the Wiener processes and the compensated countingmeasures are all independent of F t and that the unconditional expectation of theintegrals with respect to the Wiener processes is zero. These observations, combinedwith the martingale representation theorem for marked point processes (see Br´emaud1981), imply that E ˆ Q (cid:20) ˜ V A ( t ) (cid:12)(cid:12)(cid:12) F t (cid:21) = ˜ V A ( t ) + E ˆ Q (cid:90) tt (cid:40) ∂ ˜ V A ∂t + L ˜ s,v (cid:104) ˜ V A ( s, ˜ S ( s ) , v ( s )) (cid:105)(cid:41) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t . t = T and t = t . Terminal condition (21) and equation (10) imply that theleft-hand side of this equation is equal to ˜ V ( t, ˜ S ( t ) , v ( t )), the price of the Europeanexchange option. Rearranging yields the result stated in the proposition.For a fixed t ∈ [0 , T ], let A ( t ) be the event that ˜ S ( t ) and v ( t ) are in the stoppingregion S ( t ); that is, A ( t ) ≡ { ( ˜ S ( t ) , v ( t )) ∈ S ( t ) } . The complement event A c ( t ) denotesthe event that ˜ S ( t ) and v ( t ) are in the continuation region C ( t ).We now seek to evaluate the expectation defining the early exercise premium inequation (23). This is discussed in the next proposition. Proposition 4.2.
The early exercise premium ˜ V P ( t, ˜ S ( t ) , v ( t )) is given by ˜ V P ( t, ˜ S ( t ) , v ( t ))= E ˆ Q (cid:90) Tt (cid:16) q e − q s ˜ S ( s ) − q e − q s (cid:17) ( A ( s )) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t − ˜ λ E ˆ Q (cid:90) Tt E Y ˆ Q (cid:34)(cid:18) ˜ V A (cid:16) s, ˜ S ( s ) e Y , v ( s ) (cid:17) − (cid:16) e − q s ˜ S ( s ) e Y − e − q s (cid:17)(cid:19) ( A ( s )) (cid:35) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t − ˜ λ E ˆ Q (cid:90) Tt E Y ˆ Q (cid:34)(cid:18) ˜ V A (cid:16) s, ˜ S ( s ) e − Y , v ( s ) (cid:17) − (cid:16) e − q s ˜ S ( s ) e − Y − e − q s (cid:17)(cid:19) ( A ( s )) (cid:35) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t , (24) where ( · ) is the indicator function, A ( s ) is the event { ( ˜ S ( t ) , v ( t )) ∈ S ( t ) } and A ( s ) = (cid:110) B ( s, v ) e ( q − q ) s ≤ ˜ S ( s ) < B ( s, v ) e ( q − q ) s e − Y (cid:111) A ( t ) = (cid:110) B ( s, v ) e ( q − q ) s ≤ ˜ S ( s ) < B ( s, v ) e ( q − q ) s e Y (cid:111) . Proof.
Note first that for any time s ∈ [ t, T ], ∂ ˜ V A ∂t + L ˜ s,v (cid:104) ˜ V A ( s, ˜ S ( s ) , v ( s )) (cid:105) = (cid:32) ∂ ˜ V A ∂t + L ˜ s,v (cid:104) ˜ V A ( s, ˜ S ( s ) , v ( s )) (cid:105)(cid:33) ( A ( t ))+ (cid:32) ∂ ˜ V A ∂t + L ˜ s,v (cid:104) ˜ V A ( s, ˜ S ( s ) , v ( s )) (cid:105)(cid:33) ( A c ( t ))= (cid:32) ∂ ˜ V A ∂t + L ˜ s,v (cid:104) ˜ V A ( s, ˜ S ( s ) , v ( s )) (cid:105)(cid:33) ( A ( t )) , since in the continuation region, the American exchange option behaves like its live13uropean counterpart and the integro-partial differential terms vanish (see equation(12)). In the stopping region (i.e. if ( A ( s )) = 1), we note that ˜ V A ( s, ˜ S ( s ) , v ( s )) = e − q s ˜ S ( s ) − e − q s (see equation (17)). Applying the integro-partial differential opera-tors, recalling the definition of ˜ κ and ˜ κ − , and rearranging the terms, we find that (cid:32) ∂ ˜ V A ∂t + L ˜ s,v (cid:104) ˜ V A ( s, ˜ S ( s ) , v ( s )) (cid:105)(cid:33) ( A ( s ))= (cid:104) − q e − q s ˜ S ( s ) + q e − q s (cid:105) ( A ( s ))+ ˜ λ E Y ˆ Q (cid:20) ˜ V A (cid:16) s, ˜ S ( s ) e Y , v ( s ) (cid:17) − (cid:16) e − q s ˜ S ( s ) e Y − e − q s (cid:17)(cid:21) ( A ( s ))+ ˜ λ E Y ˆ Q (cid:20) ˜ V A (cid:16) s, ˜ S ( s ) e − Y , v ( s ) (cid:17) − (cid:16) e − q s ˜ S ( s ) e − Y − e − q s (cid:17)(cid:21) ( A ( s )) . (25)Observe that the expectations above contain option prices determined after the jump in ˜ S ( s ) occurring at time s .As stated in the proposition, define A ( s ) and A ( s ) as the events in which theasset price ratio is initially in the stopping region but is sent back into the continuationregion after a jump by a factor e Y and e − Y , respectively, at time s . From the stoppingand continuation criteria in equation (17), we note that˜ V A (cid:16) s, ˜ S ( s ) e Y , v ( s ) (cid:17) ≥ e − q s ˜ S ( s ) e Y − e − q s , (26)with the strict inequality occurring when A ( s ) is true, and˜ V A (cid:16) s, ˜ S ( s ) e − Y , v ( s ) (cid:17) ≥ e − q s ˜ S ( s ) e − Y − e − q s , (27)with the strict inequality occurring when A ( t ) is true. Therefore, we have (cid:32) ∂ ˜ V A ∂t + L ˜ s,v (cid:104) ˜ V A ( s, ˜ S ( s ) , v ( s )) (cid:105)(cid:33) ( A ( s ))= − (cid:104) q e − q s ˜ S ( s ) − q e − q s (cid:105) ( A ( s ))+ ˜ λ E Y ˆ Q (cid:34)(cid:18) ˜ V A (cid:16) s, ˜ S ( s ) e Y , v ( s ) (cid:17) − (cid:16) e − q s ˜ S ( s ) e Y − e − q s (cid:17)(cid:19) ( A ( s )) (cid:35) + ˜ λ E Y ˆ Q (cid:34)(cid:18) ˜ V A (cid:16) s, ˜ S ( s ) e − Y , v ( s ) (cid:17) − (cid:16) e − q s ˜ S ( s ) e − Y − e − q s (cid:17)(cid:19) ( A ( s )) (cid:35) . Using the above expression in the early exercise premium given in equation (23), weobtain (24).
Remark 2. If G ( y ) and G ( y ) are the probability density functions (pdfs) of Y and14 , respectively under ˆ Q , then the early exercise premium may be written as ˜ V P ( t, ˜ S ( t ) , v ( t ))= E ˆ Q (cid:90) Tt (cid:16) q e − q s ˜ S ( s ) − q e − q s (cid:17) ( A ( s )) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t − ˜ λ E ˆ Q (cid:34) (cid:90) Tt (cid:90) b ( s, ˜ S ( s ) ,v ( s )) −∞ (cid:18) ˜ V A (cid:16) s, ˜ S ( s ) e y , v ( s ) (cid:17) − (cid:16) e − q s ˜ S ( s ) e y − e − q s (cid:17)(cid:19) × G ( y ) ( A ( s )) d y d s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) − ˜ λ E ˆ Q (cid:34) (cid:90) Tt (cid:90) ∞− b ( s, ˜ S ( s ) ,v ( s )) (cid:18) ˜ V A (cid:16) s, ˜ S ( s ) e − y , v ( s ) (cid:17) − (cid:16) e − q s ˜ S ( s ) e − y − e − q s (cid:17)(cid:19) × G ( y ) ( A ( s )) d y d s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) , (28) where b ( s, ˜ S ( s ) , v ( s )) ≡ ln (cid:16) B ( s, v ( s )) e ( q − q ) s / ˜ S ( s ) (cid:17) .Similar to the findings of Gukhal (2001), Chiarella and Ziogas (2004), and Cheang,Chiarella, and Ziogas (2013), the early exercise premium (23) for our transformedproblem can be further decomposed into a diffusion component (the positive term) anda jump component (the negative terms). However, unlike the early exercise premiumderived by Cheang, Chiarella, and Ziogas (2013) for an American call option underSVJD dynamics, our early exercise premium representation contains two jump terms.This is because ˜ S ( t ) has two sources of jumps: the jumps in the price of the first asset(given by the counting measure p (d y , d t )) and the jumps in the num´eraire process(given by the counting measure p (d y , d t )). Nonetheless, the interpretation remainsthe same: the diffusion term captures the discounted expected value of cash flows dueto dividends when asset prices are in the stopping region and the jump terms capturethe rebalancing costs incurred by the holder of the American exchange option when ajump instantaneously occurs in the price of either asset, causing ˜ S ( t ) to jump back intothe continuation region immediately after the option is exercised. Figure 2 illustratesthe loss (captured by the difference in option value and the exercise value) incurred bythe option holder when the asset yield ratio jumps back into the continuation regiondue to a jump, by a factor e Y , in the price of the first asset. A similar graphicalanalysis holds if the price of the num´eraire asset instantaneously jumps instead of theprice of the first asset (in this case, the new asset yield ratio is ˜ S ( t ) e − Y ).Recall that the discounted American exchange option ˜ V A ( t, ˜ S ( t ) , v ( t )) is a solutionof the homogeneous IPDE ∂ ˜ V A ∂t + L ˜ s,v (cid:104) ˜ V A ( t, ˜ S ( t ) , v ( t )) (cid:105) = 0over the restricted domain 0 ≤ t ≤ T , 0 < ˜ S ( t ) < B ( t, v ) e ( q − q ) t , and 0 < v < ∞ sub- In this situation, the investor is unable to adjust the decision to exercise in response to the instantaneousjump in asset prices and is therefore vulnerable to the rebalancing cost described earlier. A similar phenomenonin the context of consumption-investment problems with transaction costs in a L´evy-driven market is exploredin greater technical detail by De Valli`ere, Kabanov, and L´epinette (2016). igure 2. Loss incurred by the option holder when the asset ratio instantaneously jumps from ˜ S ( t ) in thestopping region to ˜ S ( t ) e Y back in the continuation region. ject to the value-matching condition (19), the smooth-pasting condition (20), andboundary conditions (21). Following Jamshidian (1992) and Chiarella and Ziogas(2004), the restriction on the domain can be lifted by adding the appropriate inho-mogeneous term to the IPDE such that the equation holds for ˜ S ( t ) > V A ( t, ˜ S ( t ) , v ( t )) and its first-order partial derivative withrespect to ˜ S are continuous, but the value-matching and smooth-pasting conditionsare sufficient to meet this requirement. Proposition 4.3.
The discounted American exchange option price ˜ V A ( t, ˜ S, v ) is asolution to the inhomogeneous IPDE ∂ ˜ V A ∂t + L ˜ s,v (cid:104) ˜ V A ( t, ˜ S ( t ) , v ( t )) (cid:105) + Ξ( t, ˜ S ( t ) , v ( t )) , (29) where the inhomogeneous term Ξ is given by Ξ( t, ˜ S ( t ) , v ( t ))= (cid:16) q e − q t ˜ S ( t ) − q e − q t (cid:17) ( A ( t )) − ˜ λ ( A ( t )) (cid:90) b ( t, ˜ S ( t ) ,v ( t )) −∞ (cid:20) ˜ V A (cid:16) t, ˜ S ( t ) e y , v ( t ) (cid:17) − (cid:16) e − q t ˜ S ( t ) e y − e − q t (cid:17)(cid:21) G ( y ) d y − ˜ λ ( A ( t )) (cid:90) ∞− b ( t, ˜ S ( t ) ,v ( t )) (cid:20) ˜ V A (cid:16) t, ˜ S ( t ) e − y , v ( t ) (cid:17) − (cid:16) e − q t ˜ S ( t ) e − y − e − q t (cid:17)(cid:21) G ( y ) d y, (30) where G and G are the pdfs of Y and Y , respectively, under ˆ Q , and b ( t, ˜ S ( t ) , v ( t )) ≡ B ( t, v ( t )) e ( q − q ) t / ˜ S ( t )] . This equation is to be solved for ( t, ˜ S ( t ) , v ( t )) ∈ [0 , T ] × R ,subject to terminal and boundary conditions (21) . Proof.
Observe that for all ( t, ˜ S ( t ) , v ( t )) ∈ [0 , T ] × R , the equation0 = ∂ ˜ V A ∂t + L ˜ s,v (cid:104) ˜ V A ( t, ˜ S ( t ) , v ( t )) (cid:105) − (cid:32) ∂ ˜ V A ∂t + L ˜ s,v (cid:104) ˜ V A ( t, ˜ S ( t ) , v ( t )) (cid:105)(cid:33) ( A ( t ))holds. Equation (30) is obtained by expanding the negative term in the above equa-tion, as was done to obtain equation (25) and using the G and G to rewrite theexpectations as integrals.
5. Limit of the Early Exercise Boundary at Maturity
Of particular interest is the behavior of the unknown early exercise boundary nearthe maturity of the option and the conditions on model parameters under which theboundary is continuous at maturity. The next proposition presents the limit of theearly exercise boundary, which we obtain following the method of Chiarella and Ziogas(2009).
Proposition 5.1.
The limit B ( T − , v ) ≡ lim t → T − B ( t, v ) is a solution of the equation B ( T − , v ) = max , q + ˜ λ (cid:82) − ln B ( T − ,v ) −∞ G ( y ) d y + ˜ λ (cid:82) ∞ ln B ( T − ,v ) G ( y ) d yq + ˜ λ (cid:82) − ln B ( T − ,v ) −∞ e y G ( y ) d y + ˜ λ (cid:82) ∞ ln B ( T − ,v ) e − y G ( y ) d y . (31) Proof.
The method of Chiarella and Ziogas (2009), adapted to our situation, is asfollows. First, we set the inhomogeneous term Ξ( t, ˜ S, v ) (given by equation (30))to zero and evaluate the result at t = T and ˜ S = B ( T − , v ) e ( q − q ) T . The resultingexpression is then rearranged to yield equation (31).Performing the first step yields the equation0 = e − q T (cid:16) q B ( T − , v ) − q (cid:17) − ˜ λ (cid:90) − ln (cid:20) B ( T,v ) e ( q − q TB ( T − ,v ) e ( q − q T (cid:21) −∞ (cid:20) ˜ V A (cid:16) T, B ( T − , v ) e ( q − q ) T e y , v ( T ) (cid:17) − e − q T (cid:16) B ( T − , v ) e y − (cid:17)(cid:21) G ( y ) d y − ˜ λ (cid:90) ∞ ln (cid:20) B ( T,v ) e ( q − q TB ( T − ,v ) e ( q − q T (cid:21) (cid:20) ˜ V A (cid:16) T, B ( T − , v ) e ( q − q ) T e − y , v ( T ) (cid:17) − e − q T (cid:16) B ( T − , v ) e − y − (cid:17)(cid:21) G ( y ) d y. (32) Chiarella and Ziogas (2009) proposed this method as an alternative to the local analysis of the option PDEfor small time-to-maturity options as was done by Wilmott, Dewynne, and Howison (1993) in the pure diffusioncase.
17t maturity t = T , the option will be exercised if ˜ S ( T ) ≥ e ( q − q ) T , and so B ( T, v ) = 1.Thus in the above calculation, setting ˜ S = B ( T − , v ) e ( q − q ) T induces the stoppingcriterion in equation (18) for S ( T ) since B ( T − , v ) ≥
1. This implies that ( A ( T )) = 1in the inhomogeneous term (30). Furthermore, terminal condition (21) implies that˜ V A (cid:16) T, B ( T − , v ) e ( q − q ) T e y , v ( T ) (cid:17) = max (cid:26) , e − q T (cid:16) B ( T − , v ) e y − (cid:17)(cid:27) ˜ V A (cid:16) T, B ( T − , v ) e ( q − q ) T e − y , v ( T ) (cid:17) = max (cid:26) , e − q T (cid:16) B ( T − , v ) e − y − (cid:17)(cid:27) . Thus, the first integral in equation (32) will be zero if B ( T − , v ) e y − ≥ y ≥ − ln B ( T − , v ). Likewise, the second integral will be zero if B ( T − , v ) e − y − ≥ y ≤ ln B ( T − , v ). In other words, the integral terms will vanish if the maximumfunctions yield the nonzero alternative. Following this analysis, equation (32) simplifiesto 0 = q B ( T − , v ) − q + ˜ λ (cid:90) − ln B ( T − ,v ) −∞ (cid:104) B ( T − , v ) e y − (cid:105) G ( y ) d y + ˜ λ (cid:90) ∞ ln B ( T − ,v ) (cid:104) B ( T − , v ) e − y − (cid:105) G ( y ) d y. Rearranging the terms yields the equation B ( T − , v ) = q + ˜ λ (cid:82) − ln B ( T − ,v ) −∞ G ( y ) d y + ˜ λ (cid:82) ∞ ln B ( T − ,v ) G ( y ) d yq + ˜ λ (cid:82) − ln B ( T − ,v ) −∞ e y G ( y ) d y + ˜ λ (cid:82) ∞ ln B ( T − ,v ) e − y G ( y ) d y . We note lastly from Broadie and Detemple (1997) that B ( t, v ) ≥ t ∈ [0 , T ]and v ∈ (0 , ∞ ). Therefore, we must enforce a lower bound of 1 on B ( T − , v ) via themaximum function. The result stated in the proposition thus holds.Equation (31) must be solved implicitly for B ( T − , v ), which can be done usingstandard root-finding techniques. From our analysis, we find that the limit is dependenton the asset dividend yields q and q , the jump intensities ˜ λ and ˜ λ , and the jumpsize densities G and G . These dependencies highlight the influence of jumps in assetprices on the limiting behavior of the early exercise boundary. In the succeedingdiscussion, we investigate the more specific effects of these parameters on the limitof the early exercise boundary. We note further that equation (31) does not dependon the instantaneous variance v since the option payoff is independent of v . However,equation (31) is true for all v ∈ (0 , ∞ ).In the absence of jumps (i.e. when ˜ λ = ˜ λ = 0), the limit reduces to max { , q /q } .This is consistent with the result of Broadie and Detemple (1997) for American ex-change options in the pure diffusion case. In the pure diffusion case, B ( T − , v ) = q /q > q > q , implying that the early exercise boundary many not be continu-ous in t at maturity. When jumps are present, the analysis of continuity becomes morecomplicated, as shown below.First, we present some conditions under which equation (31) has a solution. This is in contrast to the proposition of Carr and Hirsa (2003), in their analysis of the one-asset Americanput option where the log-price is driven by a L´evy process, that the limit of the early exercise boundary is onlydependent on the dividend yield and the risk-free rate. roposition 5.2. Suppose q , q ≥ and ˜ λ , ˜ λ > are given and let G and G becontinious probability density functions. The equation x = q + ˜ λ (cid:82) − ln x −∞ G ( y ) d y + ˜ λ (cid:82) ∞ ln x G ( y ) d yq + ˜ λ (cid:82) − ln x −∞ e y G ( y ) d y + ˜ λ (cid:82) ∞ ln x e − y G ( y ) d y (33) has a unique solution x ∗ ∈ (0 , ∞ ) if q > . Furthermore, x ∗ > if and only if q − q + ˜ λ (cid:90) −∞ (1 − e y ) G ( y ) d y + ˜ λ (cid:90) ∞ (1 − e − y ) G ( y ) d y > . Proof.
Our proof adapts the arguments made by Chiarella, Kang, and Meyer (2015,pp. 34-35). For x ∈ (0 , ∞ ), define the function f ( x ) = q + ˜ λ (cid:90) − ln x −∞ G ( y ) d y + ˜ λ (cid:90) ∞ ln x G ( y ) d y − x (cid:32) q + ˜ λ (cid:90) − ln x −∞ e y G ( y ) d y + ˜ λ (cid:90) ∞ ln x e − y G ( y ) d y (cid:33) . Denote by x ∗ a zero of f (i.e. f ( x ∗ ) = 0) on (0 , ∞ ), if any exist.Differentiating f with respect to x yields f (cid:48) ( x ) = − (cid:32) q + ˜ λ (cid:90) − ln x −∞ e y G ( y ) d y + ˜ λ (cid:90) ∞ ln x e − y G ( y ) d y (cid:33) . The integrals above are nonnegative. Hence, f is strictly decreasing on (0 , ∞ ) if q > x → + f ( x ) = q + ˜ λ (cid:90) ∞−∞ G ( y ) d y + ˜ λ (cid:90) ∞−∞ G ( y ) d y = q + ˜ λ + ˜ λ > x →∞ f ( x ) = q + ˜ λ lim x →∞ (cid:90) − ln x −∞ G ( y ) d y + ˜ λ lim x →∞ (cid:90) ∞ ln x G ( y ) d y − lim x →∞ x (cid:32) q + ˜ λ (cid:90) − ln x −∞ e y G ( y ) d y + ˜ λ (cid:90) ∞ ln x e − y G ( y ) d y (cid:33) = q − lim x →∞ xq . Thus, if q >
0, then lim x →∞ f ( x ) < f strictly decreases from positive tonegative values as x increases on (0 , ∞ ). Therefore, there exists a unique x ∗ ∈ (0 , ∞ )such that f ( x ∗ ) = 0.Now suppose q >
0. Evaluating f at x = 1 gives us f (1) = q − q + ˜ λ (cid:90) −∞ (1 − e y ) G ( y ) d y + ˜ λ (cid:90) ∞ (1 − e − y ) G ( y ) d y. f (1) ≤
0, then x ∗ must be in the interval (0 ,
1] since f is strictly decreasing. Other-wise, x ∗ > f (1) >
0, which is the condition stated in the proposition.If a solution x ∗ of equation (33) exists, then the limit of the early exercise boundaryis B ( T − , v ) = max { , x ∗ } . Remark 3. If q = 0 and q >
0, then f (cid:48) ( x ) ≤ x ∈ (0 , ∞ ) and lim x →∞ f ( x ) = q >
0. That is, f is non-increasing and remains positive as x increases in (0 , ∞ ). Thus, f does not have any zeros on (0 , ∞ ) and hence it is not optimal to exercise the optionprior to maturity.An immediate result from Proposition 5.2 is a condition for the continuity of B ( t, v )at maturity. Proposition 5.3.
Suppose q > . For any fixed v ∈ (0 , ∞ ) , B ( t, v ) is continuous atmaturity t = T if q ≥ q + ˜ λ (cid:90) −∞ (1 − e y ) G ( y ) d y + ˜ λ (cid:90) ∞ (1 − e − y ) G ( y ) d y. (34) Proof.
Suppose q > x ∗ to equation (33) lies in the interval(0 , B ( T − , v ) = 1, which is also the value of B ( T, v ). Thus, B ( t, v ) iscontinuous at the option maturity.We briefly discuss the behavior of B ( T − , v ) with respect to changes in q . Notethat ∂f /∂q = − x <
0, so when q decreases, f ( x ) increases. In particular, for agiven q >
0, there exists x ∗ ∈ (0 , ∞ ) such that f ( x ∗ ) = 0. If q decreases, then f ( x ∗ ) increases away from zero, thereby moving the unique zero of f to some othernumber x (cid:48) ∈ (0 , ∞ ) such that x (cid:48) > x ∗ . In other words, the solution x ∗ of equation(33) increases without bound, and consequently B ( T − , v ) → ∞ , as q → + . Thus,when the first asset bears no dividend yield, it is not optimal to exercise the Americanexchange option early or at least immediately prior to the option maturity.
6. The Transition Density Function
To determine the price of the American exchange option using equation (22), we needto solve for the price of the European exchange option ˜ V and evaluate the early exercisepremium ˜ V P in equation (24). To do so, we need to solve for the joint transition densityfunction of the asset yield ratio process ˜ S and the variance process v under Q . Withthe joint transition density function, we may evaluate the expectations in equations(10) and (22).Let Q ( T, u, b ; t, s, v ) denote the joint transition density function of ( ˜ S, v ) under theprobability measure ˆ Q : Q ( T, ˜ s T , v T ; t, ˜ s, v ) = ˆ Q (cid:18) ˜ S ( T ) = ˜ s T , v ( T ) = v T (cid:12)(cid:12)(cid:12) ˜ S ( t ) = ˜ s, v ( t ) = v (cid:19) . This denotes the probability of passage from ( ˜ S ( t ) , v ( t )) = (˜ s, v ) at time t to (˜ s T , v T )20t time T . The Kolmogorov backward equation associated to Q is given by ∂Q∂t + L ˜ s,v (cid:2) Q ( T, ˜ s T , v T ; t, ˜ s, v ) (cid:3) = 0 , (35)which is to be solved for t ∈ [0 , T ] and (˜ s, v ) ∈ R subject to the terminal condition Q ( T, ˜ s T , v T ; T, ˜ s, v ) = δ (˜ s − ˜ s T ) δ ( v − v T ) , where δ ( · ) is the Dirac-delta function. Since T , ˜ s T , and v T are specific constants, we will denote Q by Q ( t, ˜ s, v ) in the succeedingcalculations for notational brevity.To solve equation (35), let x = ln ˜ s and define the function H by H ( T, x T , v T ; t, x, v ) = Q ( T, e x T , v t ; t, e x , v ) . (36)Thus, when written in terms of H and its partial derivatives, equation (35) becomes0 = ∂H∂t − (cid:18) ˜ λ ˜ κ + ˜ λ ˜ κ − + 12 σ v (cid:19) ∂H∂x + (cid:2) ξη − ( ξ + Λ) v (cid:3) ∂H∂v + 12 σ v ∂ H∂x + 12 ω v ∂ H∂v + ω ( σ ρ − σ ρ ) v ∂ H∂x∂v + ˜ λ E Y ˆ Q (cid:2) H ( t, x + Y , v ) − H ( t, x, v ) (cid:3) + ˜ λ E Y ˆ Q (cid:2) H ( t, x − Y , v ) − H ( t, x, v ) (cid:3) . (37)The associated terminal condition is H ( T, x T , v T ; T, x, v ) = δ ( x − x T ) δ ( v − v T ) . Movingforward, we shall denote H by H ( t, x, v ) to emphasize that we are solving equation(37) in t , x , and v and that x T and v T are given terminal values of these variables.The function H ( t, x, v ) may be interpreted as the joint transition density functionof the process ( X ( t ) , v ( t )) indicating the probability of passage from ( x, v ) at time t to ( x T , v T ) at time t = T . In symbols, we have H ( T, x T , v T ; t, x, v ) = ˆ Q (cid:16) X ( T ) = x T , v ( T ) = v T (cid:12)(cid:12) X ( t ) = x, v ( t ) = v (cid:17) . Since the coefficients of equation (37) no longer contain x , we can take its Fouriertransform with respect to x to further simplify the equation. The following technicalassumption is required to be able to take the Fourier transform of the partial derivativesof H . This assumption is reasonable to impose on H as it is a transition density functionand is expected to vanish in the extremities of its domain (Chiarella, Ziogas, and Ziveyi2010; Cheang, Chiarella, and Ziogas 2013). Assumption 6.1. As x → ±∞ , H ( t, x, v ) → ∂H/∂x →
0, and ∂H/∂v → H , we now take the Fourier transform of equation (37)in x . Proposition 6.2.
Let ˆ H ( t, φ, v ) denote the Fourier transform of H ( t, x, v ) with re-spect to x , ˆ H ( t, φ, v ) = F x (cid:8) H ( t, x, v ) (cid:9) ( φ ) = (cid:90) ∞−∞ e iφx H ( t, x, v ) d x. (38)21 hen ˆ H satisfies the equation ∂ ˆ H∂t + (cid:18) εv − iφ Ψ (cid:19) ˆ H + ( α − Θ v ) ∂ ˆ H∂v + 12 ω v ∂ ˆ H∂v , (39) where α ≡ ξη Θ = Θ( φ ) ≡ ξ + Λ + iφω ( σ ρ − σ ρ ) ε = ε ( φ ) ≡ σ (cid:16) iφ − φ (cid:17) Ψ = Ψ( φ ) ≡ − ˜ λ ˜ κ − ˜ λ ˜ κ − − ˜ λ iφ (cid:0) ϕ ( φ ) − (cid:1) − ˜ λ iφ (cid:0) ϕ ( − φ ) − (cid:1) , (40) and ϕ j ( φ ) = (cid:82) R e − iφy G j ( y ) d y is the characteristic function of Y j under ˆ Q , j = 1 , .The associated terminal condition is ˆ H ( T, φ, v ) = e iφx T δ ( v − v T ) . (41) Proof.
See Appendix A.Save for some minor notational differences, PDE (39) is identical to the PDE pre-sented in Cheang, Chiarella, and Ziogas (2013, Proposition 4.1) for the transitiondensity function of a single-asset stochastic volatility jump-diffusion model. This re-semblance is expected since the underlying asset yield ratio process ˜ S ( t ) is modelledsimilarly with a Heston-type stochastic volatility process but with two jump compo-nents. The notational difference is pronounced in the definition of Ψ( φ ) in equation(40) where we have two terms corresponding to the jump size variables Y and Y ,in contrast to that of Cheang, Chiarella, and Ziogas (2013) who only have one jumpterm. Due to these similarities, we follow the method of Cheang, Chiarella, and Ziogas(2013) in the succeeding calculations to determine the solution of equation (39).At this point, we have reduced IPDE (37) to a second-order PDE (39) in t and v .Solutions of second-order PDEs such as equation (39) have been obtained by Feller(1951) using a Laplace transform with respect to v . Thus we may further simplifyequation (39) by taking its Laplace transform with respect to v . First, additionaltechnical assumptions must be imposed on ˆ H ( t, φ, v ) to ensure that all required Laplacetransforms are well-defined. Assumption 6.3. As v → + ∞ , e − ϑv ˆ H ( t, φ, v ) → e − ϑv ∂ ˆ H/∂v → H is a transi-tion density function. In particular, the Fourier transform ˆ H defined in equation (38)and its partial derivative in v both decay to zero since H ( t, x, v ) decays to zero as v → ∞ . Assumption 6.3 also implies that the growth of ˆ H and ∂ ˆ H/∂v is dominatedby the growth of the exponential term e ϑv as v → ∞ for any ϑ >
0. Taking Assump-tion 6.3 to be true, we now reduce equation (39) to a first-order PDE in the followingproposition.
Proposition 6.4.
Let ¯ H ( t, φ, ϑ ) be the Laplace transform of ˆ H ( t, φ, v ) with respect to , ¯ H ( t, φ, ϑ ) = L v (cid:110) ˆ H ( t, φ, v ) (cid:111) ( ϑ ) = (cid:90) ∞ e − ϑv ˆ H ( t, φ, v ) d v. (42) Then ¯ H satisfies the equation − ∂ ¯ H∂t + (cid:20) ω ϑ − Θ ϑ + 12 ε (cid:21) ∂ ¯ H∂ϑ = (cid:104) ( α − ω ) + Θ − iφ Ψ (cid:105) ¯ H + f ( t ) , (43) where f ( t ) ≡ ( ω / − α ) ˆ H ( t, φ, must be determined such that lim ϑ →∞ ¯ H ( t, φ, ϑ ) = 0 . (44) Equation (43) has terminal condition ¯ H ( T, φ, ϑ ) = exp { iφx T − ϑv T } . (45) Proof.
See Appendix B.In the next proposition, we present the solution of equation (43).
Proposition 6.5.
The solution ¯ H ( t, φ, ϑ ) of equation (43) is given by ¯ H ( t, φ, ϑ ) = exp (cid:34) ( α − ω )(Θ − (cid:122) ) ω + Θ − iφ Ψ (cid:35) ( T − t ) × (cid:34) (cid:122) ( ω ϑ − Θ + (cid:122) )( e (cid:122) ( T − t ) −
1) + 2 (cid:122) (cid:35) − αω × exp (cid:40) iφx T − (cid:18) Θ − (cid:122) ω (cid:19) v T (cid:41) × exp (cid:40) − (cid:122) v T ( ω ϑ − Θ + (cid:122) ) e (cid:122) ( T − t ) ω (cid:2) ( ω ϑ − Θ + (cid:122) )( e (cid:122) ( T − t ) −
1) + 2 (cid:122) (cid:3) (cid:41) × Γ (cid:18) αω − β ( φ, ϑ ; v T ) (cid:19) , (46) where (cid:122) = (cid:122) ( φ ) ≡ (cid:112) Θ ( φ ) − ω ε ( φ ) β ( φ, ϑ ; v T ) ≡ (cid:122) v T e (cid:122) ( T − t ) ω ( e (cid:122) ( T − t ) − × (cid:122) ( ω ϑ − Θ + (cid:122) )( e (cid:122) ( T − t ) −
1) + 2 (cid:122) , (47)Γ( u ; β ) is the (lower) incomplete gamma function Γ( u ; β ) = u ) (cid:82) β e − x x u − d x, and Γ( u ) is the gamma function Γ( u ) = (cid:82) ∞ e − x x u − d x . quation (43) [CZZ-2010] [CCZ-2013] ω σ σ ε Λ Λ ϑ s s − iφ Ψ − iφ ( r − q ) − iφ Ψ T − t τ T − t Table 1.
Comparison of notation and coefficients of equation (43), Chiarella, Ziogas, and Ziveyi (2010, equa-tion 29) [CZZ-2010] and Cheang, Chiarella, and Ziogas (2013, equation 46) [CCZ-2013].
Proof.
PDE (43) is an inhomogeneous first-order equation which can be solved via themethod of characteristics and the method of variation of parameters. This procedureyields a solution ¯ H in terms of the unknown function f ( t ). Condition (44) is thenapplied to determine f ( t ) and characterize ¯ H completely in terms of the parametersintroduced in Propositions 6.2 and 6.4.Due to close resemblances in the form of the PDE, the derivation of equation (46)follows the proof presented in Chiarella, Ziogas, and Ziveyi (2010, Appendix 4). Foreasy comparison between our analysis and the proofs presented in Chiarella, Ziogas,and Ziveyi (2010) and Cheang, Chiarella, and Ziogas (2013), we show in Table 1 theequivalence of notations used in the PDE.At this point, we now recover the original transition density function Q ( t, ˜ s, v ) byinverting the Laplace and Fourier transforms on ¯ H ( t, φ, ϑ ) given in equation (46). Tothis end, we first solve for the inverse Laplace transform of ¯ H . Proposition 6.6.
The inverse Laplace transform of ¯ H ( t, φ, ϑ ) in equation (46) is ˆ H ( t, φ, v ) = exp (cid:26) (Θ − (cid:122) ) ω ( v − v T + α ( T − t )) (cid:27) exp (cid:8) iφx T − iφ Ψ( T − t ) (cid:9) × (cid:122) e (cid:122) ( T − t ) ω ( e (cid:122) ( T − t ) − (cid:34) v T e (cid:122) ( T − t ) v (cid:35) αω − exp (cid:40) − (cid:122) ( v T e (cid:122) ( T − t ) + v ) ω ( e (cid:122) ( T − t ) − (cid:41) × I αω − (cid:32) (cid:122) (cid:112) v T ve (cid:122) ( T − t ) ω ( e (cid:122) ( T − t ) − (cid:33) , (48) where I k ( u ) is the modified Bessel function of the first kind I k ( u ) = ∞ (cid:88) n =0 ( u/ n + k n !Γ( n + k + 1) . (49) Proof.
Refer to Chiarella, Ziogas, and Ziveyi (2010, Appendix 5), keeping in mindthe notational equivalence established in Table 1. Note however that Chiarella, Ziogas, and Ziveyi (2010) uses time-to-maturity τ ≡ T − t instead of calendartime t , as what was done in our analysis and by Cheang, Chiarella, and Ziogas (2013). H ( t, φ, v ), we now invert the Fourier transform to recover H ( t, x, v ) using the inversion formula H ( t, x, v ) = F − x (cid:110) ˆ H ( t, φ, v ) (cid:111) ( x ) = 12 π (cid:90) ∞−∞ e − iφx ˆ H ( t, φ, v ) d φ (50)that corresponds to the Fourier transform defined by equation (38). The original tran-sition density function Q ( t, ˜ s, v ) is then obtained by reversing the substitution x = ln ˜ s made in equation (36). The result is presented in the proposition below. Proposition 6.7.
The transition density function Q ( t, ˜ s, v ) ≡ Q ( T, ˜ s T , v T ; t, ˜ s, v ) isgiven by Q ( t, ˜ s, v ) = ∞ (cid:88) m =0 ∞ (cid:88) n =0 (˜ λ ( T − t )) m (˜ λ ( T − t )) n e − (˜ λ +˜ λ )( T − t ) m ! n ! × E ( m,n )ˆ Q (cid:34) π (cid:90) ∞−∞ exp (cid:40) − iφ ln (cid:18) ˜ s ˜ s T e − (˜ λ ˜ κ +˜ λ ˜ κ − )( T − t ) e Υ ,m − Υ ,n (cid:19)(cid:41) × h ( T − t, φ, v ; v T ) d φ (cid:35) , (51) where h ( τ, φ, v ; v T ) = exp (cid:26) (Θ − (cid:122) ) ω ( v − v T + ατ ) (cid:27) (cid:122) e (cid:122) τ ω ( e (cid:122) τ − (cid:34) v T e (cid:122) τ v (cid:35) αω − × exp (cid:40) − (cid:122) ( v T e (cid:122) τ + v ) ω ( e (cid:122) τ − (cid:41) × I αω − (cid:32) (cid:122) √ v T ve (cid:122) τ ω ( e (cid:122) τ − (cid:33) . (52) Here, Υ ,m and Υ ,n are given by Υ ,m = m (cid:88) k =1 Y ,k and Υ ,n = n (cid:88) l =1 Y ,l , where { Y , , . . . , Y ,m } and { Y , , . . . , Y ,n } are collections of i.i.d. random variablessampled from populations with ˆ Q -density functions G ( y ) and G ( y ) , respectively, of Y and Y , and E ( m,n )ˆ Q [ · ] is the expectation operator with respect to Υ ,m and Υ ,n only. Proof.
See Appendix C.
7. The European Exchange Option Price
Given the transition density function Q ( T, ˜ s T , v T ; t, ˜ s, v ), we can now compute theprice of any European-style option on the two assets which matures at time T andwhose payoff, when discounted by the second asset yield process, can be written asa function ˜ F ( T, ˜ s T , v T ) of the terminal asset yield ratio ˜ S ( T ) = ˜ s T and the terminal25nstantaneous variance v ( T ) = v T . Following equation (9) and the results stated inProposition 3.2, the no-arbitrage price at time t ∈ [0 , T ) of such a claim is given by P ( t, ˜ s, v ) = S ( t ) e q t (cid:90) ∞ (cid:90) ∞ ˜ F ( T, ˜ s T , v T ) Q ( T, ˜ s T , v T ; t, ˜ s, v ) d v T d x T . In this section, we calculate the price of the European exchange option following thevaluation formula stated above.For our calculations, it is more convenient to use the log-price variable x T = ln ˜ s T and the corresponding transition density function H ( T, x T , v T ; t, ˜ s T , v T ). This impliesthat the option price ˜ V ( t, ˜ s, v ), as given by equation (10), can be written as ˜ V ( t, ˜ s, v ) = e − q T (cid:90) ∞−∞ (cid:90) ∞ (cid:16) e x T − e ( q − q ) T (cid:17) + H ( T, x T , v T ; t, ln ˜ s, v ) d v T d x T . (53)Since the payoff is independent of the terminal variance level v T , we may evaluate theintegral with respect to v T by evaluating (cid:82) ∞ H ( T, x T , v T ; t, x, v ) d v T . We present thisintegral in the following lemma. Lemma 7.1.
The integral (cid:82) ∞ H ( T, x T , v T ; t, ln ˜ s, v ) d v T is given by (cid:90) ∞ H ( T, x T , v T ; t, ln ˜ s, v ) d v T = ∞ (cid:88) m =0 ∞ (cid:88) n =0 (˜ λ ( T − t )) m (˜ λ ( T − t )) n e − (˜ λ +˜ λ )( T − t ) m ! n ! × E ( m,n )ˆ Q (cid:34) π (cid:90) ∞−∞ e iφx T f (cid:16) T − t, ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( T − t ) e Υ ,m − Υ ,n , v ; − φ (cid:17) d φ (cid:35) , (54) where f ( τ, z, v ; φ ) = exp (cid:8) iφ ln z + B ( τ, − φ ) + D ( τ, − φ ) v (cid:9) B ( τ, φ ) = αω (Θ + (cid:122) ) τ − (cid:32) − χe (cid:122) τ − χ (cid:33) D ( τ, φ ) = Θ + (cid:122) ω (cid:32) − e (cid:122) τ − χe (cid:122) τ (cid:33) , (55) with χ = (Θ + (cid:122) ) / (Θ − (cid:122) ) , α and Θ as defined in Proposition 6.2, and (cid:122) as definedin Proposition 6.5. Proof.
Most of the proof deals with the evaluation of (cid:82) ∞ h ( T − t, φ, v ; v T ) d v T , where h is given by equation (52). The steps in this integration closely follow those discussedin Chiarella, Ziogas, and Ziveyi (2010, Appendix 6). From this point forward, we make the simplifying notation ˜ S ( t ) = ˜ s and v ( t ) = v to be consistent with ournotation for the transition density function.
26e are now ready to solve for the European exchange option price. Our results areshown in the next proposition.
Proposition 7.2.
The discounted price ˜ V ( t, ˜ s, v ) of the European exchange option attime t is given by ˜ V ( t, ˜ s, v )= ∞ (cid:88) m =0 ∞ (cid:88) n =0 (˜ λ ( T − t )) m (˜ λ ( T − t )) n e − (˜ λ +˜ λ )( T − t ) m ! n ! × E ( m,n )ˆ Q (cid:34) e − q T ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( T − t ) e Υ ,m − Υ ,n × P E (cid:16) T − t, ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( T − t ) e Υ ,m − Υ ,n , v ; ( q − q ) T (cid:17) − e − q T P E (cid:16) T − t, ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( T − t ) e Υ ,m − Υ ,n , v ; ( q − q ) T (cid:17) (cid:35) (56) where P E and P E are defined as P E ( τ, z, v ; K ) = 12 + 12 π (cid:90) ∞ f ( τ, z, v ; φ ) e − iφK − f ( τ, z, v ; − φ ) e iφK iφ d φP E ( τ, z, v ; K ) = 12 + 12 π (cid:90) ∞ f ( τ, z, v ; φ ) e − iφK − f ( τ, z, v ; − φ ) e iφK iφ d φ, (57) with f ( τ, z, v ; φ ) given in equation (55) , f ( τ, z, v ; φ ) given by f ( τ, z, v ; φ ) = exp (cid:8) iφ ln z + B ( τ, − φ ) + D ( τ, − φ ) (cid:9) B ( τ, φ ) = αω (cid:40) (Θ + (cid:122) ) τ − (cid:34) − χ e (cid:122) τ − χ (cid:35) (cid:41) D ( τ, φ ) = Θ + (cid:122) ω (cid:34) − e (cid:122) τ − χ e (cid:122) τ (cid:35) , (58)Θ ( φ ) ≡ Θ( φ − i ) , (cid:122) ( φ ) ≡ (cid:122) ( φ − i ) , and χ ( φ ) = χ ( φ − i ) . Proof.
See Appendix D.The integrals that appear in equations (56) and (57) can be evaluated using stan-dard numerical integration schemes. Details of this implementation will no longer bediscussed in this paper.Equation (56) expresses the discounted European exchange option price as a sum ofPoisson-weighted expectations of Heston-Bates-type formulas which arise when assetprice dynamics have stochastic volatility. Our results are similar to those of Cheang,Chiarella, and Ziogas (2013), except that in the present analysis, there are two jumpcomponents considered. Equation (56) may also be read as an expected value of theseHeston-Bates-type expressions conditional on the number of jumps in the prices ofthe two assets (denoted by m and n ) observed over the remaining life of the option.27iven the number of jumps m and n , the Heston-Bates expressions are then averagedwith respect to the accumulated jump sizes Υ ,m and Υ ,n in the prices of stocks 1and 2, respectively. Our representation for the discounted European exchange optionprice depends on the current asset yield ratio ˜ s , the dividend yields of the two assets,the jump parameters under ˆ Q , the accumulation of jumps Υ ,m and Υ ,n , and theinstantaneous variance level v . Notably absent is the risk-free rate r , which is consistentwith the original result of Margrabe (1978). The expectation E ( m,n )ˆ Q may be evaluatedfurther by specifying the jump-size distribution (e.g. a log-normal distribution as inMerton (1976) or an asymmetric double exponential distribution `a la Kou (2002)),but we leave this unspecified for now.The time t price of the European exchange option price C ( t, S , S , v ) may be ob-tained by multiplying ˜ V ( t, ˜ s, v ) by S e q t . Doing so, we can write C ( t, S , S , v ) = S e − q ( T − t ) ˆ Q − S e − q ( T − t ) ˆ Q , (59)whereˆ Q = ∞ (cid:88) m =0 ∞ (cid:88) n =0 (˜ λ ( T − t )) m (˜ λ ( T − t )) n e − (˜ λ +˜ λ )( T − t ) m ! n ! × E ( m,n )ˆ Q (cid:34) e − (˜ λ ˜ κ +˜ λ ˜ κ − )( T − t ) e Υ ,m − Υ ,n × P E (cid:16) T − t, ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( T − t ) e Υ ,m − Υ ,n , v ; ( q − q ) T (cid:17) (cid:35) ˆ Q = ∞ (cid:88) m =0 ∞ (cid:88) n =0 (˜ λ ( T − t )) m (˜ λ ( T − t )) n e − (˜ λ +˜ λ )( T − t ) m ! n ! × E ( m,n )ˆ Q (cid:34) P E (cid:16) T − t, ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( T − t ) e Υ ,m − Υ ,n , v ; ( q − q ) T (cid:17) (cid:35) . (60)This form emphasizes the similarity of our result to the original Margrabe (1978)result and to the Cheang and Chiarella (2011) extension to the jump-diffusion case.Furthermore, this representation is also consistent with the form obtained in Cheangand Garces (2020, equation 27) which was obtained using the change-of-num´erairetechniqueof Geman, El Karoui, and Rochet (1995). In their analysis, ˆ Q and ˆ Q wereinterpreted as probabilities that the exchange option is in-the-money at maturity,taken under probability measures Q and Q equivalent to the risk-neutral proba-bility measure Q (corresponding to the choice of the money-market account as thenum´eraire). The alternative measures Q and Q were obtained by taking the processes { S ( t ) e − ( r − q ) t /S (0) } and { S ( t ) e − ( r − q ) t /S (0) } , respectively, as the num´eraire pro-cess. We refer to the process { S i ( t ) e − ( r − q i ) t } as the discounted yield process of stock i , i = 1 ,
2. As such, theprobability measure ˆ Q used in this analysis is different from Q as the former corresponds to the asset yieldprocess S ( t ) e q t as the num´eraire. However, ˆ Q , Q , and Q are equivalent probability measures as they areall equivalent to the objective market measure P . In this paper, we chose to perform our calculations under ˆ Q as it reduces the exchange option pricing problem to the pricing of an ordinary call option on the asset yieldprocess ˜ S ( t ). . The American Exchange Option Price Knowledge of the transition density function also allows us to evaluate the expectationsoccurring in the early exercise premium ˜ V P ( t, ˜ s, v ) given in equation (24). The startingpoint of our calculations is equation (28) which expresses the early exercise premium interms of the jump size density functions. In the succeeding calculations, let ˜ S ( u ) = ˜ s u , v ( u ) = v u , and x u = ln ˜ s u , u ∈ [ t, T ]. The following proposition provides an integralrepresentation for the early exercise premium. Proposition 8.1.
The discounted American exchange option price is given by ˜ V A ( t, ˜ s, v ) = ˜ V ( t, ˜ sv ) + ˜ V P ( t, ˜ s, v ) , where ˜ V is the European exchange option price given in Proposition 7.2 and ˜ V P is theearly exercise premium given by ˜ V P ( t, ˜ s, v ) = ˜ V PD ( t, ˜ s, v ) − ˜ λ ˜ V PJ ( t, ˜ s, v ) − ˜ λ ˜ V PJ ( t, ˜ s, v ) . (61) Here, ˜ V PD is given by ˜ V PD ( t, ˜ s, v )= ∞ (cid:88) m =0 ∞ (cid:88) n =0 ˜ λ m ˜ λ n m ! n ! E ( m,n )ˆ Q (cid:34) (cid:90) T (cid:90) ∞ ( u − t ) m + n e − (˜ λ +˜ λ )( u − t ) × (cid:32) q e − q u ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( u − t ) e Υ ,m − Υ ,n × P A (cid:16) u − t, ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( u − t ) e Υ ,m − Υ ,n , v ; v u , A ( u, v u ) (cid:17) − q e − q u × P A (cid:16) u − t, ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( u − t ) e Υ ,m − Υ ,n , v ; v u , A ( u, v u ) (cid:17) (cid:33) d v u d u (cid:35) , (62) where A ( u, v u ) ≡ B ( u, v u ) e ( q − q ) u is the critical asset yield ratio associated to thestopping region S ( u ) (see equation (18) ), and P A and P A are defined as P A ( τ, z, v ; v u , K ) = 12 + 12 π (cid:90) ∞ f ( τ, z, v ; φ, v u ) e − iφ ln K − f ( τ, z, v ; − φ, v u ) e iφ ln K iφ d φP A ( τ, z, v ; v u , K ) = 12 + 12 π (cid:90) ∞ f ( τ, z, v ; φ, v u ) e − iφ ln K − f ( τ, z, v ; − φ, v u ) e iφ ln K iφ d φ, (63) with f ( τ, z, v ; φ, v u ) = e iφ ln z h ( τ, − φ, v ; v u ) , f given by f ( τ, z, v ; φ, v u ) = e iφ ln z h ( τ, − ( φ − i ) , v ; v u ) , and h ( τ, φ, v ; v u ) given by equation (52) . Furthermore, ˜ V PJ ( t, ˜ s, v ) and ˜ V PJ ( t, ˜ s, v ) are iven by ˜ V PJ ( t, ˜ s, v )= ∞ (cid:88) m =0 ∞ (cid:88) n =0 ˜ λ m ˜ λ n m ! n ! E ( m,n )ˆ Q (cid:34) π (cid:90) T (cid:90) ∞ ( u − t ) m + n e − (˜ λ +˜ λ )( u − t ) × (cid:90) −∞ G ( y ) (cid:90) ln A ( u,v u ) − y ln A ( u,v u ) (cid:18) V A ( u, e x u + y , v u ) − (cid:16) e − q u e x u + y − e − q u (cid:17)(cid:19) × (cid:90) ∞−∞ e iφx u exp (cid:26) − iφ ln (cid:16) ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( u − t ) e Υ ,m − Υ ,n (cid:17)(cid:27) × h ( u − t, φ, v ; v u ) d φ d x u d y d v u d u (cid:35) , (64) and ˜ V PJ ( t, ˜ s, v )= ∞ (cid:88) m =0 ∞ (cid:88) n =0 ˜ λ m ˜ λ n m ! n ! E ( m,n )ˆ Q (cid:34) π (cid:90) T (cid:90) ∞ ( u − t ) m + n e − (˜ λ +˜ λ )( u − t ) × (cid:90) ∞ G ( y ) (cid:90) ln A ( u,v u )+ y ln A ( u,v u ) (cid:18) V A ( u, e x u − y , v u ) − (cid:16) e − q u e x u − y − e − q u (cid:17)(cid:19) × (cid:90) ∞−∞ e iφx u exp (cid:26) − iφ ln (cid:16) ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( u − t ) e Υ ,m − Υ ,n (cid:17)(cid:27) × h ( u − t, φ, v ; v u ) d φ d x u d y d v u d u (cid:35) . (65) Proof.
See Appendix E.Because of the possibility of jumps in the prices of the two underlying assets, theearly exercise premium, as seen from its jump components, remains dependent onthe (yet unknown) discounted American exchange option price ˜ V A ( t, ˜ s, v ). The earlyexercise premium components also require knowledge of the unknown critical assetyield ratio B ( t, v ) for the term A ( t, v ) = B ( t, v ) e ( q − q ) t that appears in equations(62), (64), and (65). In this regard, we require another equation that, when coupledwith equation (22), characterizes both ˜ V A and B ( t, v ). Proposition 8.2.
The discounted American exchange option ˜ V A ( t, ˜ s, v ) and the crit-ical asset yield ratio B ( t, v ) are the solution of the linked system of integral equations ˜ V A ( t, ˜ s, v ) = ˜ V ( t, ˜ s, v ) + ˜ V P ( t, ˜ s, v ) e − q t (cid:16) A ( t, v ) − e ( q − q ) t (cid:17) = ˜ V ( t, A ( t, v ) , v ) + ˜ V P ( t, A ( t, v ) , v ) , (66) where A ( t, v ) = B ( t, v ) e ( q − q ) t , ˜ V ( t, ˜ s, v ) is the price of the European exchange op-tion given in Proposition 7.2, and ˜ V P ( t, ˜ s, v ) is the early exercise premium given inProposition 8.1. roof. The first equation is the early exercise representation for ˜ V A ( t, ˜ s, v ) whereasthe second equation is obtained by evaluating the first equation at ˜ s = A ( t, v ) andinvoking the value-matching condition (19) to rewrite the left-hand side. The linkedsystem is supplemented with expressions for ˜ V ( t, ˜ s, v ) and ˜ V P ( t, ˜ s, v ) given in Propo-sitions 7.2 and 8.1, respectively.The discounted American exchange option price ˜ V A ( t, ˜ s, v ), once determined, ismeasured in units of the second asset yield process. To obtain the nominal price,˜ V A ( t, ˜ s, v ) is simply multipled by S ( t ) e − q t .While our analysis covers only the application of the change-of-num´eraire techniqueto reduce the dimensionality of the problem and producing integral representations ofthe option prices and the early exercise premium, our results may be linked to existingliterature that grapple with the numerical aspects of the option pricing problem. Mostnotably, the method-of-lines (MOL) and the component-wise splitting approaches usedby Chiarella et al. (2009) may be used to solve the IPDE obtained in Proposition 3.2and the corresponding free-boundary problem (19) to (21) for the American exchangeoption since the structure of the IPDE operator we defined in our analysis is similarto that tackled by Chiarella et al. (2009). Our IPDE, however, slightly differs due tothe presence of a second integral component corresponding to the jumps in the secondasset price. One may also extend the numerical methods explored by Adolfsson et al.(2013) who priced single-asset American options under stochastic volatility dynamicsand by Kang and Meyer (2014) who priced American calls under stochastic volatilityand interest rates. Moreover, the transition density function that we obtained may alsobe used in Monte Carlo simulation schemes to price options for which an analyticalrepresentation may not be readily available. Lastly, numerical integration schemes maybe used to obtain option prices from our integral representations, provided that valuesfor the model parameters are available.
9. Summary and Conclusion
We considered the problem of pricing European and American exchange options whenthe underlying asset prices are modelled with jump-diffusion dynamics and a commonHeston-type stochastic volatility process. Our results are thus extensions of those ob-tained by Margrabe (1978) and Bjerskund and Stensland (1993) for European andAmerican exchange options under pure diffusion dynamics and those by Cheang andChiarella (2011) who analyzed exchange options under jump-diffusion dynamics.The results of this paper complement those presented by Cheang and Garces (2020).In the SVJD model of Cheang and Garces (2020), the assumption that asset priceswere uncorrelated (among other simplifying assumptions on the correlation structureof the model) had to be enforced to obtain analytical representation of exchange optionprices. In this paper, we used a proportional stochastic volatility jump-diffusion modelto include the possibility of correlation between asset returns in our representationof exchange option prices. We showed that the representations we obtained in thisanalysis are of similar form to those obtained by Cheang and Garces (2020), as wellas those obtained in earlier studies by Margrabe (1978), Fischer (1978), and Cheangand Chiarella (2011).Given the proportional SVJD model specification, the two-dimensional exchangeoption pricing problem was reduced to a one-dimensional call option pricing prob-lem by taking the second asset yield process as the num´eraire, following the put-call31ransformation method described by Bjerskund and Stensland (1993). We defined theequivalent martingale measure ˆ Q , corresponding to our choice of num´eraire, which wasthen used to express the exchange option prices as expectations under ˆ Q . Then, withusual martingale arguments, we derived the IPDE for the European option price andthe Kolmogorov backward equation for the joint transition density function of ˜ S ( t )and v ( t ).Having reduced the problem to pricing a call option on the asset yield ratio ˜ S ( t ),which also has stochastic volatility and jump-diffusion dynamics, we then adapted themethodology of Cheang, Chiarella, and Ziogas (2013) to obtain an early exercise rep-resentation for the American exchange option via probabilistic arguments and to solvefor the joint transition density function via Fourier and Laplace transforms. Equippedwith the joint transition density function, we then obtained integral representationsfor the price of the European exchange option and the early exercise premium and alinked system of integral equations for the American exchange option price and theunknown early exercise boundary associated to the American option. We also analyzedthe limiting behavior of the early exercise boundary immediately before maturity andhow the dividend yields and the jump parameters affected this limit.With the simplification of the exchange option pricing problem, our results can thenbe linked to a number of existing numerical and simulation techniques for single-assetoptions (see e.g. Adolfsson et al. 2013; Chiarella et al. 2009) to obtain option pricesgiven the model parameters. Acknowledgments
The first author is supported by a Research Training Program International (RPTi)scholarship awarded by the Australian Commonwealth Government and by a FacultyDevelopment Grant from the Loyola Schools of Ateneo de Manila University.
Disclosure Statement
The authors report no potential conflict of interest arising from the results of thispaper.
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Appendix A. Fourier Transform of the Transition Density Function IPDE
A straightforward application of equation (38) yields F x (cid:26) ∂H∂t (cid:27) = ∂ ˆ H∂t F x (cid:26) ∂H∂v (cid:27) = ∂ ˆ H∂v F x (cid:40) ∂ H∂v (cid:41) = ∂ ˆ H∂v . Furthermore, with Assumption 6.1, integration by parts gives us F x (cid:26) ∂H∂x (cid:27) = − iφ ˆ H, F x (cid:40) ∂ H∂x (cid:41) = − φ ˆ H, F x (cid:40) ∂ H∂x∂v (cid:41) = − iφ ∂ ˆ H∂v .
The Fourier transform of the first expectation in equation (37) is the Fourier transformof a convolution-type integral and is calculated as follows: F x (cid:110) E Y ˆ Q (cid:2) H ( t, x + Y , v ) (cid:3)(cid:111) = (cid:90) ∞−∞ (cid:90) ∞−∞ e iφx H ( t, x + y, v ) G ( y ) d y d x = (cid:90) ∞−∞ e − iφy G ( y ) (cid:32)(cid:90) ∞−∞ e iφz H ( t, z, v ) d z (cid:33) d y = ϕ ( φ ) ˆ H ( t, φ, v ) . A similar calculation shows that F x { E Y ˆ Q [ H ( t, x − Y , v )] } = ϕ ( − φ ) ˆ H ( t, φ, v ) . Thus, the Fourier transform of equation (37) is0 = ∂ ˆ H∂t + iφ (cid:18) ˜ λ ˜ κ + ˜ λ ˜ κ − + 12 σ v (cid:19) ˆ H + (cid:2) ξη − ( ξ + Λ) v (cid:3) ∂ ˆ H∂v − σ vφ ˆ H + 12 ω v ∂ ˆ H∂v − ω ( σ ρ − σ ρ ) viφ ∂ ˆ H∂v
35 ˜ λ (cid:0) ϕ ( φ ) − (cid:1) ˆ H + ˜ λ (cid:0) ϕ ( − φ ) − (cid:1) ˆ H. Factoring the above expression and using the notation introduced in equation (40)yields equation (39).The associated terminal condition is obtained by a straightforward application ofequation (38) and the definition of the Dirac-delta function.
Appendix B. Laplace Transform of the Transformed PDE
Given that Assumption 6.3 holds, we find that L v (cid:40) ∂ ˆ H∂t (cid:41) = ∂ ¯ H∂t L v (cid:40)(cid:18) εv − iφ Ψ (cid:19) ˆ H (cid:41) = − ε ∂ ¯ H∂ϑ − iφ Ψ ¯ H L v (cid:40) ( α − Θ v ) ∂ ˆ H∂v (cid:41) = − α ˆ H ( t, φ,
0) + ( αϑ + Θ) ¯ H + Θ ϑ ∂ ¯ H∂ϑ L v (cid:40) ω v ∂ ˆ H∂v (cid:41) = 12 ω ˆ H ( t, φ, − ω ϑ ∂ ¯ H∂ϑ − ω ϑ ¯ H. The Laplace transform of equation (39) is therefore0 = ∂ ¯ H∂t − ε ∂ ¯ H∂ϑ − iφ Ψ ¯ H − α ˆ H ( t, φ,
0) + ( αϑ + Θ) ¯ H + Θ ϑ ∂ ¯ H∂ϑ + 12 ω ˆ H ( t, φ, − ω ϑ ∂ ¯ H∂ϑ − ω ϑ ¯ H. Factoring and defining f ( t ) ≡ ( ω / − α ) ˆ H ( t, φ,
0) yields equation (43).Note that, at this point, ˆ H ( t, φ,
0) is unknown but it has to be determined so thatthe solution ¯ H to equation (43) is finite for all ϑ >
0. A sufficient condition for this isequation (44) (Cheang, Chiarella, and Ziogas 2013).The terminal condition (45) is a consequence of the definition (42) and the Dirac-delta function.
Appendix C. Recovering the Transition Density Function
Using the inversion formula (50) and the expressions for Ψ and ˆ H ( t, φ, v ) in equations(40) and (48), respectively, H ( t, x, v ) is given by H ( t, x, v ) = 12 π (cid:90) ∞−∞ exp (cid:26) − iφ (cid:104) x − x T − ˜ λ ˜ κ ( T − t ) − ˜ λ ˜ κ − ( T − t ) (cid:105)(cid:27) × exp (cid:110) ˜ λ ( T − t )[ ϕ ( φ ) − (cid:111) exp (cid:110) ˜ λ ( T − t )[ ϕ ( − φ ) − (cid:111) × h ( T − t, φ, v ; v T ) d φ, h is the function given by equation (52).We note that e ˜ λ ( T − t ) ϕ ( φ ) = ∞ (cid:88) m =0 (˜ λ ( T − t )) m m ! [ ϕ ( φ )] m . If Y , , . . . , Y ,m are i.i.d. random variables with density function G ( y ), then[ ϕ ( φ )] m = E ( m )ˆ Q [ e − iφ Υ ,m ], where Υ ,m = (cid:80) mk =1 Y ,k , since ϕ ( · ) is the characteris-tic function of each of the Y ,k ’s. Here, the expectation operator E ( m )ˆ Q acts only onΥ ,m . It follows thatexp (cid:110) ˜ λ ( T − t )[ ϕ ( φ ) − (cid:111) = e − ˜ λ ( T − t ) ∞ (cid:88) m =0 (˜ λ ( T − t )) m m ! E ( m )ˆ Q (cid:104) e − iφ Υ ,m (cid:105) . A similar analysis calculation yieldsexp (cid:110) ˜ λ ( T − t )[ ϕ ( − φ ) − (cid:111) = e − ˜ λ ( T − t ) ∞ (cid:88) n =0 (˜ λ ( T − t )) n n ! E ( n )ˆ Q (cid:104) e iφ Υ ,n (cid:105) , where Υ ,n = (cid:80) nl =1 Y ,l , { Y , , . . . , Y ,n } is a sample of i.i.d. random variables from apopulation with density function G ( y ), and E ( n )ˆ Q is the expectation operator actingonly on Υ ,n . We then obtain H ( t, x, v ) = ∞ (cid:88) m =0 ∞ (cid:88) n =0 (˜ λ ( T − t )) m (˜ λ ( T − t )) n e − (˜ λ +˜ λ )( T − t ) m ! n ! × E ( m,n )ˆ Q (cid:34) π (cid:90) ∞−∞ exp (cid:26) − iφ (cid:104) x − x T − ˜ λ ˜ κ ( T − t ) − ˜ λ ˜ κ − ( T − t ) (cid:105)(cid:27) × exp (cid:110) − iφ (cid:0) Υ ,m − Υ ,n (cid:1)(cid:111) h ( T − t, φ, v ; v T ) d φ (cid:35) . (C1)Here, E ( m,n )ˆ Q is the expectation acting only on Υ ,m and Υ ,n .Reversing the substitutions x = ln ˜ s and x T = ln ˜ s T and combining the resultingexponent under one natural logarithm yields the desired result. Appendix D. Integral Representation of the Discounted EuropeanExchange Option Price
From equation (53) and Lemma 7.1, ˜ V ( t, ˜ s, v ) may be written as˜ V ( t, ˜ s, v ) = e − q T ∞ (cid:88) m =0 ∞ (cid:88) n =0 (˜ λ ( T − t )) m (˜ λ ( T − t )) n e − (˜ λ +˜ λ )( T − t ) m ! n !37 E ( m,n )ˆ Q (cid:34) π (cid:90) ∞−∞ (cid:90) ∞ ( q − q ) T (cid:16) e x T − e ( q − q ) T (cid:17) e iφx T × f (cid:16) T − t, ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( T − t ) e Υ ,m − Υ ,n , v ; − φ (cid:17) d x T d φ (cid:35) . To simplify notation, temporarily set f ( − φ ) ≡ f ( T − t, ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( T − t ) e Υ ,m − Υ ,n , v ; − φ )and K ≡ ( q − q ) T . Furthermore, let I denote the double integral inside the aboveexpectation. It follows that I = (cid:90) ∞−∞ (cid:90) ∞ K e i ( φ − i ) x T f ( − φ ) d x T d φ − e K (cid:90) ∞−∞ (cid:90) ∞ K e iφx T f ( − φ ) d x T d φ. Denote by I and I the first and second terms, respectively, of I . Integrating withrespect to x T , we have I = (cid:90) ∞−∞ f ( − φ ) (cid:90) ∞ K e i ( φ − i ) x T d x T d φ = lim b →∞ (cid:90) ∞−∞ f ( − φ ) (cid:34) e i ( φ − i ) b − e i ( φ − i ) K i ( φ − i ) (cid:35) d φ = lim b →∞ (cid:90) ∞−∞ f ( − u − i ) (cid:34) e iub − e iuK iu (cid:35) d u = lim b →∞ (cid:90) ∞−∞ f ( φ − i ) (cid:34) e − iφK − e − iφb iφ (cid:35) d φ = (cid:90) ∞ f ( φ − i ) e − iφK − f ( − φ − i ) e iφK iφ d φ − lim b →∞ (cid:90) ∞ f ( φ − i ) e − iφb − f ( − φ − i ) e iφb iφ d φ, where the last equality is a result of splitting the domain of integration on φ . Observethat f ( φ − i ) = f ( T − t, ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( T − t ) e Υ ,m − Υ ,n , v ; φ − i )= ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( T − t ) e Υ ,m − Υ ,n f (cid:16) T − t, ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( T − t ) e Υ ,m − Υ ,n , v ; φ (cid:17) , where f and its components are defined in equation (58). Thus, we may write I = ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( T − t ) e Υ ,m − Υ ,n × (cid:34) (cid:90) ∞ f ( φ ) e − iφK − f ( − φ ) e iφK iφ d φ − lim b →∞ (cid:90) ∞ f ( φ ) e − iφb − f ( − φ ) e iφb iφ d φ (cid:35) , where f ( φ ) ≡ f ( T − t, ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( T − t ) e Υ ,m − Υ ,n , v ; φ ).The limit above may be evaluated using the results in Shephard (1991). Let F ( x )be the cumulative distribution function of a random variable whose mean exists andwhose characteristic function is f ( φ ). Then by Shephard (1991, Theorem 3), we have F ( b ) = 12 − π (cid:90) ∞ f ( φ ) e − iφb − f ( − φ ) e iφb iφ d φ. F ( b ) → b → ∞ , it follows that I = ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( T − t ) e Υ ,m − Υ ,n (cid:34) π + (cid:90) ∞ f ( φ ) e − iφK − f ( − φ ) e iφK iφ d φ (cid:35) . The same calculations apply for evaluating I . Evaluating the integral with respectto x T and splitting the domain of integration in φ , we have I = lim b →∞ (cid:90) ∞−∞ f ( − φ ) (cid:34) e iφb − e iφK iφ (cid:35) d φ = (cid:90) ∞ f ( φ ) e − iφK − f ( − φ ) e iφK iφ d φ − lim b →∞ (cid:90) ∞ f ( φ ) e − iφb − f ( − φ ) e iφb iφ d φ. Using Shephard (1991, Theorem 3) to evaluate the limit, we find that I = π + (cid:90) ∞ f ( φ ) e − iφK − f ( − φ ) e iφK iφ d φ. Using these results to rewrite the double integral in ˜ V ( t, ˜ s, v ), putting back ( q − q ) T in place of K , and writing the resulting expression in terms of P E and P E yield theexpression presented in the proposition. Appendix E. Integral Representation of the Early Exercise Premium
Equation (61) for the early exercise premium is written such that ˜ V PD , ˜ V PJ , and ˜ V PJ denote the first, second, and third expectations, respectively, that appear in equation(28). These terms may be interpreted as the diffusion and jump components, respec-tively, of the early exercise premium. In each term, the indicator function ( A ( u ))appears, where u ∈ [ t, T ]. In terms of x u and v u , the event A ( u ) may be written as A ( u ) = { x u ≥ ln A ( u, v u ) } . Evaluating ˜ V PD : From equation (28), ˜ V PD is given by˜ V PD ( t, ˜ s, v ) = E ˆ Q (cid:34) (cid:90) Tt e − q u (cid:16) q ˜ s u − q e ( q − q ) u (cid:17) ( A ( u )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ S ( t ) = ˜ s, v ( t ) = v (cid:35) , and in terms of the transition density function H in equation (C1), we have˜ V PD ( t, ˜ s, v ) = (cid:90) Tt (cid:90) ∞ (cid:90) ∞ ln A ( u,v u ) e − q u (cid:16) q e x u − q e K ( u ) (cid:17) H ( u, x u , v u ; t, ln ˜ s, v ) d x u d v u d t, where K ( u ) ≡ ( q − q ) u . Using equation (C1), we find that˜ V PD ( t, ˜ s, v ) = ∞ (cid:88) m =0 ∞ (cid:88) n =0 ˜ λ m ˜ λ n m ! n ! E ( m,n )ˆ Q (cid:34) (cid:90) Tt (cid:90) ∞ ( u − t ) m + n e − (˜ λ +˜ λ )( u − t ) e − q u × π (cid:90) ∞ ln A ( u,v u ) (cid:16) q e x u − q e K ( u ) (cid:17) (cid:90) ∞−∞ e iφx u exp (cid:26) − iφ ln (cid:16) ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( u − t ) e Υ ,m − Υ ,n (cid:17)(cid:27) × h ( u − t, φ, v ; v u ) d φ d x u d v u d u (cid:35) . In terms of the function f introduced in the proposition, we can write˜ V PD ( t, ˜ s, v ) = ∞ (cid:88) m =0 ∞ (cid:88) n =0 ˜ λ m ˜ λ n m ! n ! E ( m,n )ˆ Q (cid:34) (cid:90) Tt (cid:90) ∞ ( u − t ) m + n e − (˜ λ +˜ λ )( u − t ) e − q u × π (cid:90) ∞−∞ f ( − φ ) (cid:90) ∞ ln A ( u,v u ) (cid:16) q e x u − q e K ( u ) (cid:17) e iφx u d x u d φ d v u d u (cid:35) , (E1)where we set f ( − φ ) ≡ f ( u − t, ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( u − t ) e Υ ,m − Υ ,n , y ; − φ, v u ).We next integrate the innermost double integral, which we denote by I ∗ , using thetechniques employed in the proof of Proposition 7.2. To this end, we make a secondaryrepresentation I ∗ = q I ∗ − q e K ( u ) I ∗ , where I ∗ = (cid:90) ∞−∞ f ( − φ ) (cid:90) ∞ ln A ( u,v u ) e i ( φ − i ) x u d x u d φ I ∗ = (cid:90) ∞−∞ f ( − φ ) (cid:90) ∞ ln A ( u,v u ) e iφx u d x u d φ. Evaluating the integral with respect to x u , I ∗ can be written as I ∗ = lim b →∞ (cid:90) ∞−∞ f ( − φ ) (cid:34) e i ( φ − i ) b − e i ( φ − i ) ln A ( u,v u ) i ( φ − i ) (cid:35) d φ = lim b →∞ (cid:90) ∞−∞ f ( φ − i ) (cid:34) e − iφ ln A ( u,v u ) − e − iφb iφ (cid:35) d φ = (cid:90) ∞ f ( φ − i ) e − iφ ln A ( u,v u ) − f ( − φ − i ) e iφ ln A ( u,v u ) iφ d φ − lim b →∞ (cid:90) ∞ f ( φ − i ) e − iφb − f ( − φ − i ) e iφb iφ d φ. Note that f ( φ − i ) ≡ f (cid:16) u − t, ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( u − t ) e Υ ,m − Υ ,n , y ; φ − i, v u (cid:17) = ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( u − t ) e Υ ,m − Υ ,n × f (cid:16) u − t, ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( u − t ) e Υ ,m − Υ ,n , v ; φ, v u (cid:17) , where f is defined in the proposition. For ease of notation, let f ( φ ) be shorthandfor f ( u − t, ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( u − t ) e Υ ,m − Υ ,n , v ; φ, v u ). With this representation, I ∗ can40e written as I ∗ = ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( u − t ) e Υ ,m − Υ ,n × (cid:34) (cid:90) ∞ f ( φ ) e − iφ ln A ( u,v u ) − f ( − φ ) e iφ ln A ( u,v u ) iφ d φ − lim b →∞ (cid:90) ∞ f ( φ ) e − iφb − f ( − φ ) e iφb iφ d φ (cid:35) = ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( u − t ) e Υ ,m − Υ ,n × (cid:34) π + (cid:90) ∞ f ( φ ) e − iφ ln A ( u,v u ) − f ( − φ ) e iφ ln A ( u,v u ) iφ d φ (cid:35) , where the limit was evaluated using Theorem 3 of Shephard (1991).Similar calculations apply in simplifying I ∗ . Integration with respect to x u yields I ∗ = lim b →∞ (cid:90) ∞−∞ f ( − φ ) (cid:34) e iφb − e iφ ln A ( u,v u ) iφ (cid:35) d φ = (cid:90) ∞ f ( φ ) e − iφ ln A ( u,v u ) − f ( − φ ) e iφ ln A ( u,v u ) iφ d φ − lim b →∞ (cid:90) ∞ f ( φ ) e − iφb − f ( − φ ) e iφb iφ d φ. By Theorem 3 of Shephard (1991), the limit reduces to − π , giving us I ∗ = π + (cid:90) ∞ f ( φ ) e − iφ ln A ( u,v u ) − f ( − φ ) e iφ ln A ( u,v u ) iφ d φ. From our calculations, we now find that I ∗ is given by I ∗ = q ˜ se − (˜ λ ˜ κ +˜ λ ˜ κ − )( u − t ) e Υ ,m − Υ ,n × (cid:32) π + (cid:90) ∞ f ( φ ) e − iφ ln A ( u,v u ) − f ( − φ ) e iφ ln A ( u,v u ) iφ d φ (cid:33) − q e ( q − q ) u (cid:32) π + (cid:90) ∞ f ( φ ) e − iφ ln A ( u,v u ) − f ( − φ ) e iφ ln A ( u,v u ) iφ d φ (cid:33) . Substituting this expression in place of the innermost double integral in equation (E1),multiplying by e − q u / (2 π ), and expressing the resulting expression in terms of P A and P A (as defined in the proposition) yields equation (62) for the diffusion component ofthe early exercise premium. Evaluating the jump components: