Accounting for Earnings Announcements in the Pricing of Equity Options
aa r X i v : . [ q -f i n . P R ] A p r Accounting for Earnings Announcements in the Pricing ofEquity Options
Tim Leung ∗ Marco Santoli † September 4, 2018
Abstract
We study an option pricing framework that accounts for the price impact of an earnings announce-ment (EA), and analyze the behavior of the implied volatility surface prior to the event. On theannouncement date, we incorporate a random jump to the stock price to represent the shock dueto earnings. We consider different distributions of the scheduled earnings jump as well as differentunderlying stock price dynamics before and after the EA date. Our main contributions include ana-lytical option pricing formulas when the underlying stock price follows the Kou model along with adouble-exponential or Gaussian EA jump on the announcement date. Furthermore, we derive analyticbounds and asymptotics for the pre-EA implied volatility under various models. The calibration re-sults demonstrate adequate fit of the entire implied volatility surface prior to an announcement. Wealso compare the risk-neutral distribution of the EA jump to its historical distribution. Finally, wediscuss the valuation and exercise strategy of pre-EA American options, and illustrate an analyticalapproximation and numerical results.
Keywords: earnings announcement, equity options, pre-earnings announcement implied volatility
JEL Classification:
G12, G13, G14
Contents ∗ Industrial Engineering & Operations Research Department, Columbia University, New York, NY 10027, USA.E-mail: leung@ ieor.columbia.edu. Corresponding author. † Industrial Engineering & Operations Research Department, Columbia University, New York, NY 10027, USA.E-mail: [email protected]. Calibration and Parameter Estimators 15
Appendix 22
Public companies routinely release summaries of their operations and performance, including incomestatements, balance sheets, and other reports. Such events are commonly referred to as earnings an-nouncements . In the market session following a scheduled announcement , empirical studies (see, forexample, Patell and Wolfson (1984)) suggest that the opening stock price can move drastically. For in-stance, Dubinsky and Johannes (2006) study a sample of stocks and report that the variance of stockprice returns on the earnings announcement date is over five times greater than those on other dates. Inaddition to the immediate price impact, earnings releases may also affect the drift of the stock price overa longer horizon. This empirical observation is commonly referred to as the post-earnings-announcementdrift; see Chordia and Shivakumar (2006) and references therein.Since many public companies also have options written on their stock prices, this motivates us toinvestigate the problem of pricing equity options prior to an earnings announcement (EA). As optionsare intrinsically forward-looking contracts, their prices should account for the uncertain stock price impactof an upcoming earnings release. A natural question is how to extract some information on such an impactfrom observed options prices, especially a few days before the announcement. Since traders often quoteor study the implied volatility (IV) for each option, in practice it is important to better understand thebehavior of the pre-earnings announcement implied volatility (PEAIV). One main feature of the PEAIV is that it tends to rise, often rather drastically, up until the announcementdate. In Figure 1, we show the price and IV of the front month (i.e. with the nearest maturity date) at-the-money (ATM) call option written on IBM. Following an earnings announcement on July 17, 2013, theoption expires on the Friday in the same week. As we can see on the right panel, the IV increases rapidlyfrom 30% to over 70% as time approaches the earnings announcement, and then drops significantly toclose to 20% after the earnings report is released. At the same time, the ATM call price stays well abovezero even though the time to maturity is very short (Figure 1 (left)). Moreover, we observe that the IVreaches a maximum just before an earnings event.Let us briefly discuss the intuitions behind these observations. First, the market price of an optionexpiring after earnings announcement reflects the possibility of a jump in the stock price on the announce-ment date. Since the Black-Scholes formula, which is used to convert option prices to IVs, assumes alog-normal model for the underlying price without jumps, the only way to account for the deterministi-cally timed yet random jump in the stock price is to apply a higher volatility parameter. Furthermore,since the volatility parameter in the Black-Scholes formula is coupled with time-to-maturity, an evenhigher volatility parameter is needed as time progresses. As a result, even though the stock price may Earnings reports are commonly released after the market closes. Therefore, the price impact of earnings are first reflectedat the beginning of the next market session.
Date O p t i on P r i c e Date I V Figure 1:
The price (left) and implied volatility (right) of the front month ATM option on IBM with expirationdate July 18, 2013. The earnings announcement date is July 17, 2013.
Date I V
100 150 200 0 200 400 60000.20.40.60.8 MaturityStrike I m p li ed V o l a t ili t y Figure 2:
The time series (left) of the IV of the front month ATM call on IBM. The red dots mark the value ofthe IV on EA dates. The IV surface (right) of IBM at the market close on July 15, 2013, two days before an EA. .2 Objectives and Related Studies These market observations call for a pricing framework that explicitly incorporates the jump to berealized on the earnings announcement date. In this paper, we analyze an option pricing frameworkthat accounts for the price impact of an earnings announcement, with emphasis on the behavior ofthe implied volatility prior to the event. Specifically, we introduce a random-sized jump scheduled onthe earnings announcement date to represent the shock to the company stock price. The introductionof a jump related to earnings is consistent with prior empirical studies (Maheu and McCurdy (2004);Piazzesi (2005); Dubinsky and Johannes (2006); Lee and Mykland (2008)). We first apply this idea tothe Black and Scholes (1973) (BS) model, where we obtain formulas for the pre-EA European optionprice, its IV and various Greeks. In this model, we find that the IV increases with a specific pattern astime approaches the announcement date. We further incorporate additional random-time jumps in thestock price by extending the Kou (2002) double-exponential (DE) jump-diffusion model to incorporate anEA jump. Specifically, we assume that the EA jump size for the log stock price is either a DE or Gaussianrandom variable. In both cases, analytic option pricing formulas are derived and used to generate IVs.To better understand the behavior of the PEAIV, we derive analytic bounds and asymptotics forextreme strikes for the IV under a general class of stock price models with an EA jump. In the bounds,we observe the explicit dependence on EA jump parameters and a common time-dependence of thePEAIV across these models. In addition, we calibrate a number of models, namely, the extended Hestonand Kou models with an EA jump, to market option prices. Our results demonstrate a more adequatefit of the entire PEAIV surface, as compared to these models without the EA jump. Furthermore, weextract the risk-neutral distribution of the EA jump and compare it to its historical distribution. Thediscrepancy between the two distributions can be interpreted as a risk premium for the jump risk due toearnings. We conclude the paper analyzing the pricing of American options before and after an EA date.For implementation, we discuss a numerical method, as well as an analytical approximation based on thework of Barone-Adesi and Whaley (1987).There is a wealth of literature in empirical finance and accounting that examines how earnings an-nouncements affect stock and option prices. Most of the studies focus on the informational content,market reactions and price patterns associated with earnings releases. Systematic reviews of the empiri-cal studies can be found in Dubinsky and Johannes (2006), Barth et al. (2011), Rogers et al. (2009) andBillings and Jennings (2011). For the price impact of news on futures, we refer to Chatrah et al. (2009)and references therein. Below, we summarize a number of related studies that focus on option pricesprior to an earnings announcement.The model by Patell and Wolfson (1981) (PW) is among the first empirical studies on the prices of op-tions in the presence of an earnings announcement. They build on the pricing models of Black and Scholes(1973) and Merton (1973), and assume the instantaneous volatility as a deterministic piece-wise constantfunction of time, with two volatility levels over two time windows around the earnings release in orderto reflect the uncertainty surrounding the earnings impact. Since their seminal work, other papers haveadopted the same model or its variations and performed empirical tests (see Donders and Vorst (1996),Isakov and Perignon (2001), Barth et al. (2011), among others). In particular, Barth et al. (2011) extendPW’s model by allowing the deterministic piece-wise constant volatility to take different values before,during, and after the announcement date. In all these empirical studies, there are no jumps incorporatedin the stock price at anytime. Moreover, without stochastic volatility or jumps in the stock price, the IVwill not exhibit any skew, though this is commonly observed in the options market.In this paper, we adopt a different approach by incorporating a deterministic-time random jump inthe stock price dynamics. We consider different distributions of the scheduled earnings jump as well asdifferent underlying stock price dynamics before and after the EA date. Our study generalizes that ofDubinsky and Johannes (2006), which focuses on the BS model with a Gaussian EA jump and provide a4ormula for the time-deterministic implied volatility function. Our main contributions include analyticaloption pricing formulas when the underlying stock price admits doubly exponential jumps over time(the Kou model), along with a double-exponential or Gaussian EA jump on the announcement date (seeTheorem 3.1). Furthermore, we derive analytic bounds and asymptotics for the pre-EA implied volatilityunder various models (see Propositions 4.1 and 4.2). In addition, we study the pricing of pre-earningsAmerican options under the Kou model and discuss an analytic approximation to the option price byextending the method of Barone-Adesi and Whaley (1987).Our analysis draws motivation from empirical studies and market observations to introduce moresophisticated models that help reproduce the entire implied volatility surface accurately and tractably.For calibration, we employ recent options data from the whole observed IV surface, instead of ATM onlyoptions used in Dubinsky and Johannes (2006). Using earnings data from 1994 to 2013, we also comparethe risk-neutral distribution of the EA jump to its historical distribution. The framework and methodsintroduced herein can be readily generalized to other option pricing models. They can also be useful formonitoring the PEAIV, and estimating the historical and risk-neutral distributions of the EA jump andthe associated risk premium.The rest of the paper is structured as follows. In Section 2, we extend the Black-Scholes model byincorporating an EA jump in the stock price, and derive the corresponding implied volatility functionand Greeks. In Section 3, we discuss the extensions of other models, including the Kou and Hestonmodels. In Section 4, we study the bounds and asymptotics for the IV surface under various models.In Section 5, we discuss our calibration results from the extended BS, Kou, and Heston models, andcompare the risk-neutral and historical distributions of EA jumps. In Section 6, we study the valuationof American options prior to an earnings announcement. The Appendix contains the proofs and detailsfor our analytical results.
The effect of a scheduled earnings announcement on the stock price is modeled by a deterministic-timerandom-size price jump in the price dynamics. Naturally, this modification can be applied to virtuallyany model, and we start with an extension of the Black and Scholes (1973) model.Let ( W t ) t ≥ be a standard Brownian motion defined on the risk-neutral probability space (Ω , F , Q ).Throughout, we assume a constant risk-free rate r ≥
0. Let T e be the EA date, and we call the r.v. Z e the EA jump, which is the jump size of the log stock price immediately after the earnings announcement.We assume Z e to be independent of W . Under the risk-neutral measure, the company stock price ( S t ) t ≥ evolves according to dS t S t − = rdt + σdW t + d (cid:0) { t ≥ T e } (cid:0) e Z e − (cid:1)(cid:1) , (2.1)where 1 {·} denotes the indicator function, r is the positive constant interest rate, and σ represents theconstant volatility. The martingale condition S = E (cid:8) e − rt S t (cid:9) , ≤ t ≤ T, where E {·} denotes theexpectation under Q , implies that E (cid:8) e Z e (cid:9) = 1.In this section, we assume that Z e is normally distributed. This yields closed form expressions forthe price and the IV surface. The martingale condition E (cid:8) e Z e (cid:9) = 1 implies that Z e ∼ N (cid:16) − σ e , σ e (cid:17) , sothe EA jump is parametrized by σ e only. We notice that, for T ≥ T e ,log (cid:18) S T S t (cid:19) ∼ N (cid:18)(cid:18) r − σ − σ e T − t ) (cid:19) ( T − t ) , σ ( T − t ) + σ e (cid:19) . (2.2)5herefore, the price of a European call with strike K and maturity T is given by C ( t, S t ) = C BS T − t, S t ; r σ + σ e T − t , K, r ! , ≤ t < T e , (2.3)where C BS ( τ, S ; σ, K, r ) represents the usual BS formula with time to maturity τ and spot price S . The price formula (2.3) allows us to express the implied volatility (IV) as the deterministic function I ( t ; K, T ) = (q σ + σ e T − t ≤ t < T e ,σ T e ≤ t < T. (2.4)As we can see, although the IV surface, for a fixed time t , is flat across strikes, it has a decreasing termstructure in T . In addition, the IV for any particular option increases in time as we approach the earningsannouncement. Alternatively, the option price formula (2.3) can be obtained as if the stock has no jumpand follows the dynamics dS t S t = rdt + I ( t ; K, T ) dW t . This model is used in Patell and Wolfson (1981), who assume I ( t ; K, T ) to be a deterministic, piece-wiseconstant function to reflect the impact of earnings on the implied volatility.In Figure 3 (left), we compare the IV function in (2.4) for fixed (
K, T ), against the IV time series ofthe front month ATM IBM option with expiration date 7/18/2013. The parameters of the model havebeen chosen by a least-square regression. As we can see, the observed ATM IV and the model IV increasein a similar fashion. In the same figure (right), we also compare the IV function in (2.4) for fixed ( t, K ),against the term structure of the IV of the ATM IBM call option on 7/17/2013—just before the earningsannouncement and one day before the expiration date. Again, the parameters of the model are obtainedvia an additional least-square regression and they differ from the one obtained from the IV time seriesanalysis. The model term structure approximates the observed one even if the IV surface presents noskew and a term structure which allows only for the functional form (2.4). In order to understand the sensitivity of options prices approaching the earnings announcement, we deriveand analyze the Greeks based on the price function (2.3).For a call or put with maturity T after the EA date T e , the Delta and Gamma are the simply theBlack-Scholes Delta and Gamma functions, but with the volatility parameter set at higher value I ( t ), for t < T e . On the other hand, we notice that there are two parameters related to the volatility of the stockprice. Consequently, in addition to the standard Black-Scholes Vega, we introduce the EA-Vega V e ≡ ∂C∂σ e = 1 √ T − t q σ ( T − t ) σ e + 1 V BS ( T − t, S ; I ( t ) , K, r ) , where V BS ( τ, S ; σ, K, r ) = Sφ ( d ) √ τ represents the usual Black-Scholes Vega function with spot price S and time-to-maturity τ . The IVs of the ATM call and put are observed to be very close but not identical. Each point of the time series representsthe average of the two IVs. Date I V IVModel
Maturity I V IVModel
Figure 3:
Left: The IV function (2.4) (solid line) prior to an EA, with σ = 0 . , σ e = 0 . σ = 0 . , σ e = 0 . For the Theta of a call, we obtainΘ ≡ ∂C∂t = Θ BS ( T − t, S ; I ( t ) , K, r ) + 12 I ( t ) (cid:18) σ e T − t (cid:19) V BS ( T − t, S ; I ( t ) , K, r ) , ≤ t < T. (2.5)where Θ BS ( τ, S ; σ, K, r ) = − S σ √ τ φ ( d ) − rKe − rτ Φ ( d ) is the Black-Scholes Theta function. First, wenote that Θ BS ( T − t, S ; I ( t ) , K, r ) ≤ Θ ≤ . The left part of the inequality is a direct consequence of(2.5) while the right part is a consequence of the fact that the option price is decreasing in time. On theother hand, it is not true in general that Θ ≤ Θ BS ( S, K, r, T − t, σ ) . This means that the option maylose value less rapidly over time than an option priced with a lower volatility. To illustrate this point, wesuppose r = 0, and the Black-Scholes PDE implies that ∂C∂t = − σ S ∂ C∂S = − σ S Γ BS ( T − t, S ; I ( t ) , K, . On the other hand, we also have Θ BS ( T − t, S ; σ, K,
0) = − σ S Γ BS ( T − t, S ; σ, K, . Therefore, ∂C∂t − Θ BS ( T − t, S ; σ, K,
0) = − σ S (Γ BS ( T − t, S ; I ( t ) , K, − Γ BS ( T − t, S ; σ, K, . For ATM options, we have Γ BS ( T − t, S ; σ, S,
0) = √ πσS e − σ T − t )2 , which is decreasing in σ . This impliesthat ∂C∂t > Θ BS ( T − t, S ; σ, S,
0) for σ e >
0, implying a less rapid time decay in the option value. In fact,the same is true for other puts and calls whose Γ BS is decreasing in σ .In Figure 4 (left), we illustrate Θ as a function of the spot price for a call with 5 days to matu-rity. For comparison, we plot two additional benchmarks based on the BS Theta with different volatilityparameter values I (0) and σ , respectively. As expected from (2.5), the time-decay of the call price isless severe than Θ BS ( T, S ; I (0) , K, r ). In addition, for spot prices near the strike K , we observe thatΘ > Θ BS ( T, S ; σ, S, r ) > Θ BS ( T, S ; I (0) , S, r ). Figure 4 (right) shows how Θ changes in time for an ATM( S = K = 100) call, up to one day before earnings. Again, we notice the same dominance of Theta’s,7ut their differences increase as time approaches the EA date. Interestingly, Θ, which accounts for theEA jump, appears to be significantly more constant as compared to the other two BS Theta’s.
80 90 100 110 120−50−40−30−20−100 Spot Price Θ ΘΘ BS (I) Θ BS ( σ ) 0.005 0.01 0.015 0.02−150−100−500 Time to Maturity Θ ΘΘ BS (I) Θ BS ( σ ) Figure 4:
The shape of Θ and Θ BS changes w.r.t. the spot price (with T = ) and time to maturity (with S = K = 100). In both panels, we see that Θ for ATM options is less negative than the ordinary BS Theta Θ BS .Common parameters: r = 0 . , σ = 0 . , σ e = 0 . Although the extended BS model (2.1) is capable of showing an increasing IV approaching an earningsannouncement, the IV has no skew and its term structure only admits the particular two-parameterfunctional form q σ + σ e T − t . On the other hand, in addition to the scheduled jump, the stock pricemay also experience randomly timed jumps which cannot be adequately captured by diffusion. This hasmotivated many models that incorporate jumps of various distributions into the stock price dynamics,with notable examples in the Merton (1976), Kou (2002), Variance Gamma (Madan et al. (1998)), CGMY(Carr et al. (2002))) models. In this section, we present an analytic formula for pricing European optionsunder an extension of the jump-diffusion model of Kou (2002), and also discuss the pricing of optionsprior to an earnings announcement using a transform method. We consider an extension of the Kou model and derive an analytic formula for the price of a Europeanoption with an EA jump. Under this model, the risk-neutral terminal stock price followslog (cid:18) S T S (cid:19) = − σ T + σW T − mκT + N T X i =1 J i + 1 { t ≥ T e } (cid:0) Z e − log (cid:0) E (cid:8) e Z e (cid:9)(cid:1)(cid:1) , (3.1)where each jump J i ∼ DE ( p, λ , λ ) is double-exponentially distributed with the p.d.f. f J i ( x ) =1 { x ≥ } pλ e − λ x + 1 { x ≤ } (1 − p ) λ e λ x . The number of randomly timed jumps is modeled by N T ∼ Poi( κT ). To ensure that the martingale condition holds, we set m = p λ λ − +(1 − p ) λ λ +1 −
1. We shall con-sider separately two distributions for the EA log jump size Z e : (i) double-exponential Z e ∼ DE ( u, η , η ),and (ii) Gaussian Z e ∼ N (cid:0) , σ e (cid:1) . 8et C ( S ) = E (cid:8) e − rT ( S T − K ) + | S = S (cid:9) be the price of the European call option at time t = 0 whenthe spot price is S . We now present the price formula when the EA jump Z e is a double exponentialrandom variable. Proposition 3.1.
Suppose the terminal stock price follows (3.1), with Z e ∼ DE ( u, η , η ) and η = λ , η = λ . Then, the European call option price is given by C ( S ) = e − α S Υ (cid:16) S, K, r + σ , T, σ, ˆ κ, ˆ p, ˆ λ , ˆ λ , ˆ u, ˆ w, ˆ η , ˆ η (cid:17) − Ke − rT Υ (
S, K, r, T, σ, κ, p, λ , λ , u, w, η , η ) , (3.2) where the function Υ is given in Appendix A.1, and the constants are ˆ η = η − , ˆ η = η + 1 , ˆ λ = λ − , ˆ λ = λ + 1 , ˆ u = u η η − , w = 1 − u, ˆ w = w η η + 1 , ˆ κ = ( m + 1) κ, ˆ p = λ λ − pm + 1 , α = log (cid:18) u η η − w η η + 1 (cid:19) . When the jump size parameters of the EA and randomly timed jumps are the same, i.e. η = λ or η = λ , an analytic pricing formula can also be derived but omitted here. In practice, the parameters λ , λ are usually of an order of magnitude greater than η and η , as we will observe from our calibrationin Section 5.Alternatively, if the EA jump is normally distributed, i.e. Z e ∼ N (cid:0) , σ e (cid:1) , then one can directly adaptthe result from Kou (2002) to account for the EA jump. Specifically, the EA jump parameter σ e can beincorporated in the volatility coefficient of the Brownian motion W T :log (cid:18) S T S (cid:19) d = − σ + σ e T − mκ ! T + r σ + σ e T W T + N T X i =1 J i . From this, we see that the resulting analytic formula is in fact identical to the original formula withoutthe EA jump, except with σ replaced by q σ + σ e T .The analytic formula (3.2) allows for the fast computation of the price, and simultaneously its delta,∆ = e − α Υ ( · ), where Υ ( · ) is the first term on RHS of (3.2). In Table 3.1, we apply (3.2) to calculateoption prices and we report the corresponding implied volatilities. As we can see, the IV decreases astime to maturity increases, which is typical of the IV before an earnings announcement. In addition, theIV skew becomes flatter as maturity increases. In Figure 5, we plot the IV of the ATM option when theunderlying follows (3.1) with either a Gaussian or DE EA jump. For comparison, we have chosen thejumps parameters so that the variances of the Gaussian and DE EA jumps coincide. We notice that theIVs have similar dependence on time. Comparing Figures 3 and 5, it is natural to wonder whether theIV increases in time in a similar fashion, not only in the extended Black-Scholes and Kou, but also inother models. This motivates us to explore the properties of the IV under different models in Section 4.9 \ T C IV C IV C IV C IV
90 11.031 0.799 11.380 0.425 12.348 0.300 16.050 0.23992.5 8.958 0.767 9.400 0.415 10.529 0.297 14.485 0.23995 7.048 0.738 7.598 0.407 8.871 0.295 13.027 0.23997.5 5.357 0.714 6.007 0.401 7.384 0.294 11.675 0.239100 3.945 0.699 4.651 0.397 6.075 0.293 10.428 0.239102.5 2.849 0.696 3.536 0.396 4.942 0.292 9.284 0.239105 2.047 0.704 2.651 0.396 3.979 0.292 8.240 0.238107.5 1.475 0.719 1.968 0.399 3.173 0.292 7.292 0.238110 1.069 0.738 1.455 0.403 2.509 0.293 6.434 0.238
Table 1:
Option prices and IVs under model (3.1). Prices are computed via formula (3.2). Parameters: r = 2%, S = 100, κ = 10, p = 0 . λ = 60, λ = 50, u = 0 . η = 15, η = 12, σ = 20%. I V Gaussian EA JumpDE EA Jump
Figure 5:
The time-series of the ATM option IV under the extended Kou model (3.1) when the EA and expirationdates are on the 10th and 11th days respectively. Parameters: T e = 14 days , T = 15 days , r = 2%, S = 100, κ = 10, p = 0 . λ = 50, λ = 50, u = 0 . η = 25, η = 25, σ = 20%. In general, let us write the terminal stock price as S T = S e X T + Z e . If X T and Z e are independent,and they both admit analytic characteristic functions, then we obtain the characteristic formula of thelog-price Ψ ( ω ) := E (cid:26) e i ω log (cid:16) STS (cid:17) (cid:27) = e ˆ ψ ( ω )+ ψ e ( ω ) , where ˆ ψ ( ω ) := log (cid:16) E n e i ω log ( X T ) o(cid:17) , and ψ e ( ω ) := log (cid:16) E n e i ω log ( Z e ) o(cid:17) . It is then possible to make use of available methods to price vanilla as well as exotic options. Forexample, the methods of Carr and Madan (1999), Duffie and Singleton (2000), Lee (2004), and Raible(2000), among others, can be used to price European options. Alternatively, the methods developed inJackson et al. (2008) or Lord et al. (2008) can be adapted to price both European and American optionsin models incorporating the jump Z e . 10n the following sections, we will also consider an extension of the Heston model with dynamics dS t S t = rdt + σ t dW t + d (cid:0) { t ≥ T e } (cid:0) e Z e − (cid:1)(cid:1) ,dσ t = ν (cid:0) ϑ − σ t (cid:1) dt + ζσ t d ˜ W t , (3.3)where W and ˜ W are standard Brownian motions, with E { dW t d ˜ W t } = ρdt, and Z e is independent of both W and ˜ W . Write the option price as C (0 , S ) = ˆ ∞−∞ (cid:16) e log( S + x ) − K (cid:17) + f ( x ) dx = ( e · − K ) + ∗ f ( − · ) (log( S )) , where f denotes the p.d.f. of log( S T S ) and ∗ denotes the convolution operator. Denoting with F ( g ( x )) ≡ ´ R e − iωx g ( x ) dx the Fourier operator acting on the function g , we can then write the price of the Europeancall as C (0 , S ) = e γ log( S ) F − (cid:8) F (cid:8) e − γx ( Se x − K ) + (cid:9) Ψ ( ω − iγ ) (cid:9) . (3.4)where the introduction of the dampening factor e − γx is necessary because the payoff is not integrable.The same observations have been made in Lord et al. (2008), where (3.4) is implemented as part of anew pricing algorithm. Jackson et al. (2008) also derive the same formula by analyzing the associatedpricing PIDE in Fourier space. One can then apply a Fast Fourier transform (FFT) algorithm to priceaccording to (3.4). In addition, these methods can be adapted to price American options, as we will doin Section 6. The market observations in Figures 1-3 motivate us to analyze some characteristics of the pre-earningsannouncement implied volatility. In Section 4.1, we provide upper and lower bounds for the IV under aclass of models. In Section 4.2, we study some IV asymptotics with focus on small and large strikes.
For our analysis of the IV bounds of European options, we consider a general framework where theterminal stock price is written in the form S T = S t e X t,T + Z e , (4.1)with S t the stock price at time t < T e . The martingale condition on S implies that ( X t,T ) ≤ t ≤ T satisfies E (cid:8) e X t,T (cid:9) = e r ( T − t ) . The r.v. Z e is a continuous mixture of Gaussian r.v.’s with p.d.f. f Z e ( y ) = ´ R + φ (cid:16) y ; − ˆ σ , ˆ σ (cid:17) G ( d ˆ σ ), where φ ( · ; a, b ) represents the p.d.f. of a Gaussian r.v. with mean a andvariance b , and G ( · ) is a measure over the space R + with G [0 , ∞ ) = 1.Note that we have not specified the distribution of X t,T , therefore the base model can be very general.We have the following lower bound for the IV volatility function. Proposition 4.1.
Suppose the terminal stock price S T follows (4.1). Then, the implied volatility I ( t ; K, T ) admits the lower bound I ( t ; K, T ) ≥ ˆ σ min √ T − t , t < T e , (4.2)11 here ˆ σ min := inf { ˆ σ ∈ R + : G [0 , ˆ σ ] > } . In addition, if X t,T is also distributed as a Gaussian mixture, f X t,T ( y ) = ´ R + φ (cid:16) y ; r ( T − t ) − ˜ σ , ˜ σ (cid:17) H ( d ˜ σ ) , then the lower bound improves to I ( t ; K, T ) ≥ s ˜ σ min + ˆ σ min T − t , t < T e , where ˜ σ min := inf { ˜ σ ∈ R + : H [0 , ˜ σ ] > } . We note that ˆ σ min ≥
0, so the bound (4.2) is nontrivial only if ˆ σ min >
0. This means that the measure G has zero weight on Gaussian r.v.’s with variance smaller than ˆ σ min >
0. Such a condition is satisfied,for example, by any finite mixture of Gaussian r.v.In addition, we obtain an upper bound for the implied volatility.
Proposition 4.2.
Suppose the terminal stock price S T follows (4.1) and assume that both X T and Z e are distributed as continuous Gaussian mixtures, f Z e ( y ) = ˆ R + φ (cid:18) y ; − ˆ σ , ˆ σ (cid:19) G ( d ˆ σ ) , f X t,T ( y ) = ˆ R + φ (cid:18) y ; r ( T − t ) − ˜ σ , ˜ σ (cid:19) H ( d ˜ σ ) . (4.3) Then, the following upper bound for the implied volatility of a European ATM-forward call option, K = e r ( T − t ) S t , holds: I (cid:16) t ; e r ( T − t ) S t , T (cid:17) ≤ s ˆ R + ˜ σ T − t H ( d ˜ σ ) + ˆ R + ˆ σ T − t G ( d ˆ σ ) , t < T e . (4.4)Notable examples of models that satisfy conditions (4.3) include the extended Merton and Heston(when ρ = 0) models. However, we remark that Gaussian mixtures can also be used to approximateother distributions. Moreover, our bounds can serve as analytical benchmarks for the IV under differentmodels. As an example, we derive the explicit expressions for the bounds (4.4) under the Heston model.According to Propositions 4.1 and 4.2, the IV bounds under different models exhibit similar behaviorsas time approaches the earnings announcement. Comparing the bounds to the IV function I in (2.4), itis not too surprising that the simple extended BS model was able to fit the observed ATM IV over time(see Figure 3). Example 4.3.
Assume that S follows the Heston dynamics (3.3). In the case ρ = 0 it is known that,conditioned on the path of ( σ u ) t ≤ u ≤ T , ´ Tt σ u dW u ∼ N (cid:0) , ˜ σ (cid:1) , where ˜ σ ≡ ´ Tt σ u du . From this we observethat X t,T ≡ (cid:16) r − ˜ σ T − t ) (cid:17) ( T − t )+ ´ Tt σ u dW u satisfies the second part of (4.3) . In turn, direct computationyields that ´ R + ˜ σ T − t H ( d ˜ σ ) = ϑ + σ t − ϑν ( T − t ) (cid:0) − e − ν ( T − t ) (cid:1) . Therefore, for example, in the case of a GaussianEA jump, the term ´ R + ˆ σ T − t G ( d ˆ σ ) is equal to σ e T − t and the bound (4.4) reads I (cid:16) t ; e r ( T − t ) S t , T (cid:17) ≤ s ϑ + σ t − ϑν ( T − t ) (cid:0) − e − ν ( T − t ) (cid:1) + σ e T − t . (4.5) If we instead assume that the jump is distributed as a symmetric double-exponential, Z e ∼ DE (cid:0) , η, η (cid:1) ,then we use the fact that Z e d = q η ǫZ, where ǫ ∼ Exp (1) and Z ∼ N (0 , are independent. In fact, inthat case, G ( x ) = η xe − η x , ´ R + ˆ σ T − t G ( d ˆ σ ) = η ( T − t ) , and the bound (4.4) reads I (cid:16) t ; e r ( T − t ) S t , T (cid:17) ≤ s ϑ + σ t − ϑν ( T − t ) (cid:0) − e − ν ( T − t ) (cid:1) + 2 η ( T − t ) . (4.6)12n Figure 6 we plot the explicit bounds (4.5) and (4.6) from Example 4.3. As we can see, the upperbound is relatively close to the model IV curve and shares a very similar time dependence. In particular,when the EA jump is Gaussian (Figure 6, left), the bound is almost indistinguishable from the modelIV. To see this, as t → T , the lower and upper bounds ((4.2) and (4.5)) share a common leading term σ e T − t . In practical cases as in this example, the term ´ R + ˜ σ T − t H ( d ˜ σ ) ≈ σ t is typically at least one order ofmagnitude smaller than σ e / ( T − t ) since T − t is very small and σ e and σ t are of the same order. Also,in Figure 6, we observe that the model IV with a non-zero ρ still admits a similar time behavior and arevery close to the bounds (4.5) and (4.6) for the case ρ = 0. As a curious note, the coefficient η of theleading term in (4.6) as t → T , is exactly the variance of Z e when Z e ∼ DE ( , η, η ), the same way thatthe coefficient σ e of the leading term in (4.5) is the variance of the Gaussian Z e . Therefore, an interestingquestion is whether the rate of change of the IV is approximately proportional to the standard deviationof the EA jump, at least as time approaches the EA date (see also Figure 5).To conclude, we recall that Propositions 4.1 and 4.2 also give us information about the term struc-ture of the IV. Indeed, for models whose dynamics are time-homogeneous, we observe the relationship: ∂I ( t ; K,T ) ∂t = − ∂I ( t ; K,T ) ∂T . This implies that one should expect a decreasing term structure for options priorto the EA date. I V Model IV, ρ = 0Model IV, ρ =-0.54Upper Bound IV
0 1 2 3 4 5 6 7 8 9 100.30.350.40.450.50.550.6 Time (days) I V Model IV, ρ = 0Model IV, ρ =-0.54Upper Bound IV Figure 6:
The model IV plotted against the upper bounds in (4.5) and (4.6) under the extended Heston modelwith a Gaussian EA jump (left) and DE EA jump (right) respectively. Parameters: ν = 4 . , ϑ = 0 . , σ =1 . , ρ ∈ { , − . } , ζ = 0 . , σ e = . , u = 0 . , η = η = 40. We now analyze the asymptotics of the IV surface for small and large strikes for the extended Hestonand Kou model. Our asymptotics follow from an adaptation of the results of Benaim and Friz (2008)(see also Benaim et al. (2012)). We state the results here and provide the proofs in Appendix A.4.
Proposition 4.4.
Let S satisfy the extended Heston dynamics (3.3) where Z e is either:Case 1: normally distributed, Z e ∼ N (cid:16) , σ e (cid:17) ; orCase 2: double-exponentially distributed, Z e ∼ DE ( u, η , η ) . hen for any fixed t < T e , the implied volatility I ( t ; K, T ) satisfies I ( t ; K, T ) ( T − t )log (cid:16) KS t (cid:17) ∼ ξ ( q ∗ ) , as K → , (4.7) I ( t ; K, T ) ( T − t )log (cid:16) KS t (cid:17) ∼ ξ ( r ∗ − , as K → ∞ , (4.8) where ξ ( x ) is defined by ξ ( x ) ≡ − (cid:16) √ x + x − x (cid:17) and q ∗ = ( p − case , min { p − , η } case , , r ∗ = ( p + case , min { p + , η } case , and p ± is the smallest positive solution to, respectively, ν ∓ ρζp ± + q ( ν ∓ ρζp ± ) + ζ ( ± p ± − ( p ± ) ) coth (cid:18) ( T − t )2 q ( ν ∓ ρζp ± ) + ζ ( ± p − ( p ± ) ) (cid:19) = 0 . Proposition 4.5.
Let S follow the extended Kou dynamics (3.1) where Z e is either:Case 1: normally distributed, Z e ∼ N (cid:16) , σ e (cid:17) ; orCase 2: double-exponentially distributed, Z e ∼ DE ( u, η , η ) .Then for any fixed t < T e , the implied volatility I ( t ; K, T ) satisfies I ( t ; K, T ) ( T − t )log (cid:16) KS t (cid:17) ∼ ξ ( q ∗ ) , as K → , (4.9) I ( t ; K, T ) ( T − t )log (cid:16) KS t (cid:17) ∼ ξ ( r ∗ − , as K → ∞ , (4.10) where ξ ( x ) ≡ − (cid:16) √ x + x − x (cid:17) and q ∗ = ( λ case , min { λ , η } case , , r ∗ = ( λ case , min { λ , η } case . First, we observe that if the EA jump is Gaussian, then it has no role in the IV asymptotics in strikes(see (4.7) and (4.9)). Hence, in either the Heston or Kou model, the large/small strikes asymptotics withand without the EA jump are in fact identical. On the other hand, if the tails of the EA jump are fatterthan those of the base model, then the IV asymptotics are determined by the EA jump parameters. Insuch cases, the asymptotics are observed for less extreme strikes with short maturities, which is when theEA jump variance dominates. For longer maturities, however, the asymptotics hold for more extremestrikes. An intuitive explanation is that, as time-to-maturity increases, the EA jump variance is relativelylow and the tails behavior is manifest only for extreme values.In Figure 7, we show the IV asymptotics under the extended Kou (left) and Heston (right) modelscompared to the IV obtained by inverting the BS formula on option prices calculated via Fourier transform(see Section 3.2). We plot the asymptotic volatility function I ( t ; K, T ) = c + p log ( K/S t ) ξ ( ω ) for fixed( t, T ), where c is a constant chosen so that the asymptotics and the model IVs coincide at the mostextreme strikes considered. In each particular case, ω is a constant set according to (4.9)-(4.10) (for the14xtended Kou model) and (4.7)-(4.8) (for the extended Heston model). In Figure 7 (left) the double-exponential EA jump tails dominates those of the daily jumps, as this generally holds in practice. InFigure 7 (right), the EA jump is Gaussian and thus does not affect the IV asymptotics from the basemodel.
60 80 100 120 140 160 1800.40.50.60.70.80.911.11.21.3 Strike I V Model IVAsymptotics 60 80 100 120 140 160 1800.10.150.20.250.30.350.40.45 Strike I V Model IVAsymptotics
Figure 7: (Left) The IV obtained from numerical option prices against the asymptotics (4.9)-(4.10) under theextended Kou model with a log-DE EA jump. Parameters: S = 100 , r = 0 . , T = , κ = 300 , p =0 . , λ = λ = 100 , u = 0 . , η = 30 , η = 25. (Right) The IV obtained from numerical option pricesagainst the asymptotics (4.7)-(4.8) under the extended Heston model with a log-normal EA jump. Parameters: S = 100 , r = 0 . , T = 1 , ν = 2 . , ϑ = 0 . , σ = 0 . , ρ = − . , ζ = 0 . , σ e = 0 . . In this section, we perform calibrations of the extended BS, Kou and Heston models to the observedmarket prices of options near an earnings announcement. This allows us to evaluate whether the modelextensions improve the accuracy of the calibration compared to that of the base models. Calibrationresults can also be used to infer information about the distribution of the EA jump and, in one simpleexample, we compare the estimators obtained through calibration on IBM options data to estimates givenby its historical distribution.
In our calibration procedure, we consider a set of N vanilla calls and puts with observed market prices,ˆ C i , i = 1 , ..., N . These options have different contractual features, such as strike, maturity, and optiontype. For a given model, the set of model parameters is denoted by Θ. In turn, the model price of option i is denoted by C i (Θ). To calibrate a given model, we minimize the sum of squared errors (see e.g. Dennis(1977); Andersen and Andreasen (2000); Bates (1996); Cont and Tankov (2002)):min Θ N X i =1 (cid:16) C i (Θ) − ˆ C i (cid:17) , (5.1)We use the best-bid and best-ask mid-point prices. Furthermore, we adapt a trust-region-reflectivegradient-descent algorithm (Coleman and Li (1994); Coleman and Li (1996)), starting from different ini-tial points to guarantee a better exploration of the parameters space. While our numerical tests show15he adopted method results in an effective calibration, we remark that there are many alternative, pos-sibly more advanced, calibration procedures available (see e.g. Cont and Tankov (2002) and referencestherein).The majority of equity options in the US are of the American type. While in Section 6 we willdiscuss the pricing of American options, the methods are generally too computationally-intensive to bepractically used in conjunction with a gradient-descent method for calibration, especially as the number ofoptions and parameters increase. Related studies typically circumvent this issue by simply assuming theAmerican options are European-style (see e.g. Dubinsky and Johannes (2006); Broadie et al. (2009)). Incontrast, our procedure begins by inverting the market prices of American puts and calls via a relativelyfast American option pricer under the Black-Scholes model. This gives us the observed IVs. In turn,we apply the Black-Scholes European put/call pricing formula, with the volatility parameter being theobserved IV, and derive the associated European put or call price. We then use the resulting prices asinputs to calibrate against the option prices generated from a model. In all our experiments, we obtainoption price data, available up to August 2013, from the OptionMetrics Ivy database.We now present an example using the extended BS, Kou and Heston models with different distributionsfor the EA jump. Our objective is to illustrate the calibrated IV surfaces under these models and comparethem with the empirical IV surface. Recall from Figure 2 the empirical implied volatility surface of IBMon July 15, 2013. That is observed 2 days prior to the earnings announcement by IBM on July 17, 2013after market. The closest options expiration date was Friday July 19, 2013. As noted earlier, the frontmonth IVs are significantly higher than those for options with longer maturities. In Figure 8 we showthe associated calibrated IV surfaces for the 3 base models and their extensions, resulting in a total of 9calibrated models. Table 2 summarizes the calibrated parameters. In the original Black-Scholes model,the implied volatility surface is flat and takes a high value of 28.11%. As we incorporate the EA jump,under both Gaussian and DE distributions, the calibrated values of the stock price volatility σ are lower.More importantly, in every case, the base model is unable to generate the characteristic shape of the IVsurface before the EA. Between the extended Heston and Kou models, the IV surface generated from theKou model tends to flatten more rapidly as maturity lengthens. Overall, the Heston model seems to beable to reproduce the IV surface more accurately, and the incorporation of a Gaussian EA jump seemsto reproduce the IV surface better than with a DE EA jump in this example. We will further comparethe two EA jump distributions in Section 5.3. Black-Scholes σ σ e u η η Base 28.11% — — — —Gaussian jump 27.69% 7.11% — — —DE jump 21.42% — 38.28% 23.03 6.3Heston ν ϑ ζ ρ σ σ e u η η Base 3.70 0.05 0.90 -0.51 0.05 — — — —Gaussian jump 4.04 0.05 1.01 -0.55 0.03 4.73% — — —DE jump 3.10 0.05 0.84 -0.54 0.03 — 42.06% 34.77 24.95Kou σ κ p λ λ σ e u η η Base 2.80% 232.3 46.85% 275.9 82.1 — — — —Gaussian jump 0.03% 193.6 51.17% 998.9 70.0 3.61% — — —DE jump 0.39% 85.1 79.46% 990.3 30.0 — 98.73% 32.9 2.0
Table 2:
Summary of calibrated parameters based on the observed IV surface in Figure 2. The correspondingcalibrated IV surfaces are displayed in Figure 8
00 200 300 100 200 300 400 500−1−0.500.511.5 I m p li ed V o l a t ili t y Strike Maturity 100 200 300 100 200 300 400 5000.40.50.60.70.8 MaturityStrike I m p li ed V o l a t ili t y
100 200 300 200 400 60000.511.52 MaturityStrike I m p li ed V o l a t ili t y
100 200 300 100 200 300 400 5000.10.20.30.40.5 MaturityStrike I m p li ed V o l a t ili t y
100 200 300 100 200 300 400 5000.10.20.30.40.50.6 MaturityStrike I m p li ed V o l a t ili t y
100 200 300 200 400 60000.20.40.60.8 MaturityStrike I m p li ed V o l a t ili t y
100 200 300 100 200 300 400 5000.180.20.220.240.260.280.3 MaturityStrike I m p li ed V o l a t ili t y
100 150 200 0 200 400 6000.10.20.30.40.5 MaturityStrike I m p li ed V o l a t ili t y
100 200 300 200 400 60000.20.40.60.81 MaturityStrike I m p li ed V o l a t ili t y Figure 8:
Calibrated surfaces for the Black-Scholes, Heston, and Kou models (1st, 2nd, 3rd rows respectively)without EA jump (left column), and with Gaussian (middle) and DE (right) EA jumps. The calibrated modelparameters are listed in Table 2.
In the extended BS model (2.1), it is possible to derive analytical estimators for the models parameters( σ, σ e ), as discussed in Dubinsky and Johannes (2006). We apply these estimators to compare with theestimators obtained by calibrating other models (see Table 3). First, we consider the extended BS modelwhere the EA jump Z e is normally distributed. To calibrate this model, it suffices to use any pair ofoptions of different maturities. Let σ IV ( T ) , σ IV ( T ) represent the implied volatilities of two optionswith maturities T and T , respectively. Then, applying (2.4), the model parameters ( σ, σ e ) can beestimated by σ T S = s ( T − t ) σ IV ( T ) − ( T − t ) σ IV ( T ) T − T , σ T Se = s σ IV ( T ) − σ IV ( T ) T − t − T − t . (5.2)where the superscript T S indicates the relevance of the IV term structure to these estimators. In par-ticular, we call σ T Se the term structure estimator of the jump volatility under the risk-neutral measure Q . We stress that a set of two options with identical maturity would not allow us to estimate σ and σ e separately, but only the aggregate value σ + σ e T − t .Alternatively, one can utilize option prices at different times for parameter estimation. In fact, giventhe implied volatilities σ IV,t and σ IV,t at times t and t , with t < t < T e , we apply (2.4) to get the17ollowing estimators: σ ts = s ( T − t ) σ IV,t ( T ) − ( T − t ) σ IV,t ( T ) t − t , σ tse = vuut σ IV,t ( T ) − σ IV,t ( T ) T − t − T − t . (5.3)They are called the time-series estimators (see also Dubinsky and Johannes (2006)).We observe from (5.2) that one must require that σ IV ( T ) > σ IV ( T ) in order to obtain well-definedestimators. Similarly for σ ts and σ tse in (5.3), their definitions suggest that σ IV,t > σ IV,t must hold.In our empirical tests, we find that, before an earnings announcement, σ IV ( T ) > σ IV ( T ) alwaysholds, but the condition σ IV,t > σ IV,t is sometimes violated. Similar observations are also discussedin Dubinsky and Johannes (2006), who have also conducted a comprehensive empirical test using ATMoptions.We emphasize that these analytical estimators are based on a specific extension of the BS model. Sincemarket prices are not necessarily generated by this model, the analytical estimators and the calibratedparameters may not coincide. Moreover, they also depend on the choice of options whose IVs are inputsto the estimator formulas. On the other hand, the main advantage of these analytical estimators is thatthey can be computed instantly, and they are also used in practice (see e.g. Mehra et al. (2014)) andrelated studies. With the choice of a pricing model, our calibration procedure extracts the implied distribution of theEA jump. One useful application is to compare the risk-neutral and historical distributions of the EAjump. Their discrepancy will shed some light on the risk premium associated with the EA jump. As anexample, let us consider the empirical EA jumps of the IBM stock starting from 1994. We assume thatboth risk-neutral and historical distributions are Gaussian, which is amenable for comparison since weonly need to estimate a single parameter, i.e. the EA jump volatility. In Table 3, we report the estimateof EA jump volatility obtained by calibrating the Heston model extended with a Gaussian EA jump, σ Qe ,and the empirical EA jump volatility, σ Pe based on data from 1994 up to the given date. For comparison,we also list the EA jump volatility estimators according to (5.2). As we can see, for each given date,the ratio σ Pe /σ Qe is very close to 1. This suggests that under the extended Heston model, the EA jumpdistributions are very similar under both historical and risk-neutral measures. On the other hand, theratio σ Pe /σ T Se is smaller and less than 1 in this example, suggesting that the extended BS model wouldimply a higher EA jump volatility than the empirical one. In summary, the volatility σ Qe calibrated fromthe extended Heston model is smaller than the term structure EA jump volatility estimator σ T Se whichis based on the extended BS model without stochastic volatility.
Date σ Pe σ Qe σ Pe /σ Qe σ TSe σ Pe /σ TSe
Table 3:
The implied and historical EA jump volatilities for IBM. For each date in the table, the historical EAjump volatility σ Pe is estimated using price data from 1994 up to that date. The implied volatility σ Qe is calibratedfrom the extended Heston model. American Options
While index options are typically of European style, most US equity options are American-style. Ingeneral, the American option pricing problem does not admit closed-form formulas, so we discuss anumerical method for computing the option price and exercise boundary. In addition, we apply theanalytic results from the European case to approximate the American option price before an earningsannouncement.
We assume that the stock price evolves according to the extended Kou model with an EA jump definedin (3.1). The value of the American option is defined by A ( t, S ) = sup t ≤ τ ≤ T E n e − r ( τ − t ) ( K − S τ ) + | S t = S o , t ≤ T, (6.1)where τ is a stopping time w.r.t. the filtration generated by S . By the dynamic programming principle,the option price can be written as (see, e.g. (Øksendal, 2003, Chap. 10)) A ( t, S ) = sup t ≤ τ ≤ T e E n e − r ( τ − t ) (cid:0) { τ 0; (6.3) D ( t, S ) ≥ ( K − S ) + , ≤ t < T e , S ≥ ,rD ( t, S ) − ∂D∂t ( t, S ) − L D ( t, S ) ≥ , ≤ t < T e , S ≥ , (cid:0) D ( t, S ) − ( K − S ) + (cid:1) (cid:0) rD ( t, S ) − ∂D∂t ( t, S ) − L D ( t, S ) (cid:1) = 0 , ≤ t < T e , S ≥ ,D ( T e , S ) = ´ R e D ( T e , Se z ) g ( z ) dz , S ≥ . (6.4)We have denoted by L the infinitesimal generator of S under model (3.1): L V ( S ) ≡ σ S ∂ V∂S + rS ∂V∂S + κ ˆ ∞−∞ ( V ( Se y ) − V ( S )) f J ( y ) dy, (6.5)and f J is double exponential p.d.f.It is worth noting that the integration at time T e in (6.4) must be approximated with a sum because˜ D is not in closed form. The numerical computation of the integration may introduce computationalerrors, but it also adds to the computational burden since the sum is O (cid:0) n (cid:1) , where n is the number of19iscretized stock price values. We remark that it is possible, for example, to reduce the complexity ofthe integration to O ( n log ( n )) using an FFT algorithm. It is also worth noticing that when T e = T , thecomplexity reduces to O ( n ) if closed form formulas are available for the European option (e.g. if the EAjump is Gaussian or double-exponential). On the other hand, the complexity also reduces to O ( n ) whenthe announcement is imminent, i.e. T e = 0 + . This motivates us to look for a closed-form approximationto the American option price based on these scenarios, as discussed in Sect. 6.2. It is useful to comparethe American option prices under the same model but with different earnings announcement dates. Proposition 6.1. Let A ( t, S ; u ) denote the American option price as in (6.1) with T e = u . Then, wehave A ( t, S ; l ) ≤ A ( t, S ; u ) , l ≥ u > t. (6.6)As a result, the American option price is monotonically decreasing as the EA date T e approachesmaturity. Consequently, A ( t, S ; t ) and A ( t, S ; T ) become the upper and lower bounds, respectively, forthe American option price A ( t, S ; u ), t ≤ u ≤ T . As an interesting comparison, the European optionprice is completely independent of the exact EA date as long as it is at or prior to maturity. In Figure9 (right & top) we show the time-value of an American put, with strike, maturity, and spot price fixed,over different announcement times T e . Consistent with Proposition 6.1, the time-value of the Americanput is indeed monotonically decreasing in T e .To solve problems (6.3)-(6.4) we use the Fourier transform based method presented in Jackson et al.(2008). Unless T e = T or T e = 0 + , we solve backward in time for ˜ D in (6.3), perform the numericalintegration and feed it as a terminal condition for problem (6.4) which is solved backward in time as well.Figure 9 (left) shows the exercise boundary for different values of T e under the Kou model with a DEEA jump, along other common parameters. Naturally, the scheduled announcement introduces a discon-tinuity in the exercising boundary. We mark the EA dates with three crosses. As expected, after thelargest date T e the three boundaries coincide. Interestingly, the exercise boundary is decreasing rapidly intime near T e , which means that the option holder is more likely to wait until the earnings announcement,rather than exercising immediately. This can also be seen in terms of the option’s time-value. In Figure9 (right & bottom), we illustrate that the time-value of an American put, with both strike and spot pricefixed, is increasing as time approaches the EA date. One major method proposed by Barone-Adesi and Whaley (1987) to analytically approximate the priceof an American option is to express the American option price in terms of an European option price plusa correction term. The correction term is determined as the solution of an approximation of the Black-Scholes equation, with the addition of elementary boundary condition. Kou and Wang (2004) presentan analytical approximation to the American option price under the Kou model. Here, we adapt theBarone-Adesi approximation and apply it to our extended Kou model.When T e = T the approximation is virtually identical to the original one. Let P E ( t, S ) denote theEuropean put price (as in Proposition 3.1). The approximate price for the American put is similar tothat of Kou and Wang (2004) and is given by˜ A ( t, S ) = ( P E ( t, S ) + γ S − β + γ S − β , if S > α ( t ) , ( K − S ) + , if S ≤ α ( t ) , (6.7) The time-value of an American put is defined as A − ( K − S ) + where A is the put price, S is the spot price and K isthe strike. With spot price S fixed, inequality (6.6) also holds for the corresponding time-values. S po t P r i c e Time T i m e − V a l ue e T i m e − V a l ue Figure 9: Left: The American put exercising boundary when T e assumes different values, T e = 1 , , T e changes and the time-valueof a put (bottom) with strike $100 and spot $87.33 when T e = 2 months and time t changes. Parameters: r = 0 . , T = 4 months , σ = 0 . , κ = 252 , p = 0 . , λ = 300 , λ = 300 , u = 0 . , η = 30 , η = 30 with the positive constants γ ≡ α β β − β (cid:16) β K − (1 + β ) ( α + P E ( t, α )) + Ke − r ( T − t ) q ( α ( t )) (cid:17) , (6.8) γ ≡ α β β − β (cid:16) β K − (1 + β ) ( α + P E ( t, α )) + Ke − r ( T − t ) q ( α ( t )) (cid:17) , (6.9)where q ( s ) ≡ Q { S T ≤ K | S t = s } , and β , , 0 < β < λ < β < ∞ , are the two positive solutions to theequation r − e − r ( T − t ) = β (cid:18) mκ − σ − r (cid:19) + σ β κ (cid:18) p λ λ + β + (1 − p ) λ λ − β − (cid:19) . (6.10)Also, α ( t ) ∈ [0 , K ] is the solution to the equation c K − c ( α ( t ) + P E ( t, α ( t ))) = ( c − c ) Ke − r ( T − t ) q ( α ( t )) , (6.11)with c = β β (1 + λ ) and c = λ (1 + β ) (1 + β ). The analytical expression for P E ( t, S ) and q ( s )can be directly obtained from Proposition 3.1. In contrast to the formula given in Kou and Wang (2004)(see equation (7) there), the calculation of P E ( s, t ), q ( α ( t )), γ , and α accounts for the EA jump r.v. Z e (see Appendix A.6).When t ≤ T e < T , one could apply the same method used to derive the approximation above, writingthe American option as ˜ A ( t, S ) = E n e − r ( T e − t ) ˜ A ( T e , S T e ) | S t = S o + ǫ ( t, S ). The functional form of ǫ would be identical, ǫ ( t, S ) = γ S − β + γ S − β , because it is derived from the same PIDE (see AppendixA.6). The constants β and β are indeed solutions to equation (6.10), after adjusting the time parametersaccordingly. In the case when the EA jump is imminent ( T e = t + ), we need to evaluate the expectation E n ˜ A (cid:0) t + , Se Z e (cid:1)o , where S is the stock price at time t . From (6.7), this amounts to computing ˆ + ∞−∞ (cid:16)(cid:16) P E ( t + , Se z ) + γ e − β z S − β + γ e − β z S − β (cid:17) { Se z >α } + ( K − Se z ) 1 { Se z ≤ α } (cid:17) f Z e ( z ) dz (6.12)21 \ T e T − 2D 1 . 5M 3D T T − 2D 1 . 5M 3D T BA L BA H EU FST FST FST BA L BA H EU80 0.10 0.10 0.10 0.10 0.11 0.10 0.01 0.01 0.01 0.01 0.01 0.0185 0.37 0.37 0.37 0.37 0.37 0.37 0.05 0.05 0.05 0.05 0.05 0.0590 1.02 1.02 1.02 1.02 1.02 1.02 0.22 0.22 0.23 0.22 0.23 0.2295 2.30 2.31 2.31 2.30 2.31 2.29 0.84 0.86 0.87 0.84 0.87 0.84100 4.40 4.42 4.43 4.40 4.42 4.38 2.54 2.60 2.63 2.54 2.62 2.53105 7.35 7.39 7.40 7.36 7.38 7.32 5.68 5.81 5.91 5.70 5.91 5.66110 11.07 11.11 11.14 11.06 11.11 10.98 10.01 10.10 10.28 10.03 10.30 9.87115 15.36 15.40 15.45 15.35 15.42 15.20 15.00 15.00 15.08 15.00 15.10 14.57120 20.06 20.07 20.14 20.04 20.13 19.77 20.00 20.00 20.01 20.00 20.04 19.45 Table 4: American put prices under the extended Kou model with a DE EA jump. The column “FST” shows theprices calculated by a Fourier transform method for three different values of T e with expiration date T = 3 monthsfixed. The extended Barone-Adesi approximations (6.7) and (6.12) are given under columns “ BA L ” and “ BA U ”.Column “EU” shows the corresponding European put price (see (3.2)). The first 6 columns are calculated withmodel parameters: S = 100 , r = 0 . , σ = 0 . , κ = 252 , p = 0 . , λ = 300 , λ = 300 , u = 0 . , η = 30 , η = 30. Thelast 6 columns with S = 100 , r = 0 . , σ = 0 . , κ = 200 , p = 0 . , λ = 350 , λ = 350 , u = 0 . , η = 25 , η = 25. where α , γ , , and β , are determined by (6.8)-(6.11). Notice that at time t there is no exercisingboundary since ˜ A ( t, S ) ≥ ( K − S ) + due to Jensen’s inequality. In the extended Kou model with double-exponential EA jump, semi-closed formulas can be obtained in a similar fashion as in Proposition 3.1.In Table 4, we present numerical results for the prices of American put options with different strikesand maturities under the extended Kou model. We compare the price computed from the Fourier trans-form method to that analytical approximations (6.7) and (6.12). For different choices of model parameters,strikes and maturities, the approximations perform very well when T e ∼ t or T e ∼ T . For the second setof parameters (last 6 columns), the EA jump has a greater impact on the options prices as the tails of thejump are fatter, and the volatility of the other part of the dynamics is lower. In this case, the differencebetween the “true” prices for different values of T e increases, and so does the difference between the twoapproximations which might then not be suitable to approximate the options values for in-the-moneyoptions when T e is not close to t or T . A Appendix In this Appendix, we provide a number of detailed proofs and formulas. A.1 Details for Formula (3.2) In this section we write the expression of the function Υ in (3.2) explicitly. For ease of notation, we referto the parameters given as input to Υ as a vector Θ ≡ ( θ , ..., θ ). In addition, let ˜Θ be a permutationof the vector Θ where only the 8th and 9th components (the parameters for the randomly timed jumps)are switched. The function Υ is given byΥ(Θ) = ∞ X n =0 ( θ θ ) n e − θ θ n ! Z n (Θ) , (A.1)where Z n (Θ) = θ n X l =0 ( P n,l T ,n ( k, Θ) + Q n,l T ,n ( k, Θ)) + θ n X l =0 ( P n,l T ,n ( k, Θ) + Q n,l T ,n ( k, Θ)) , = log (cid:18) θ θ (cid:19) − (cid:18) θ − θ − mκ (cid:19) θ − α, m = p λ λ − q λ λ + 1 − ,T ,n +1 ( s, Θ) = θ θ − θ T ,n ( s, Θ) − θ e ( θ θ θ √ π (cid:16) θ p θ θ (cid:17) n I n (cid:18) s ; − θ , − θ √ θ , − θ θ p θ (cid:19) ,T ,n +1 ( s, Θ) = θ θ + θ T ,n ( s, Θ) + θ e ( θ θ θ √ π (cid:16) θ p θ θ (cid:17) n I n (cid:18) s ; θ , θ √ θ , − θ θ p θ (cid:19) ,T ,n +1 ( s, Θ) = 1 − T ,n +1 (cid:16) − s, ˜Θ (cid:17) , T ,n +1 ( s, Θ) = 1 − T ,n +1 (cid:16) − s, ˜Θ (cid:17) ,T , ( s, Θ) = T , ( s, Θ) = θ e ( θ θ ) θ / √ π I (cid:18) s ; − θ , − θ √ θ , − θ θ p θ (cid:19) ,T , ( s, Θ) = 1 θ − θ e ( θ θ θ √ π I (cid:18) s ; − θ , − θ √ θ , − θ θ p θ (cid:19) − e ( θ θ θ √ π I (cid:18) s ; − θ , − θ √ θ , − θ θ p θ (cid:19) ,T , ( s, Θ) = 1 θ − θ e ( θ θ θ √ π I (cid:18) s ; − θ , − θ √ θ , − θ θ p θ (cid:19) − e ( θ θ θ √ π I (cid:18) s ; − θ , − θ √ θ , − θ θ p θ (cid:19) ,P n,m = n − X i = m (cid:18) n − m − i − m (cid:19)(cid:18) ni (cid:19) (cid:18) θ θ + θ (cid:19) i − m (cid:18) θ θ + θ (cid:19) n − i θ i (1 − θ ) n − i ,Q n,m = n − X i = m (cid:18) n − m − i − m (cid:19)(cid:18) ni (cid:19) (cid:18) θ θ + θ (cid:19) n − i (cid:18) θ θ + θ (cid:19) i − m θ n − i (1 − θ ) i ,P n,n = θ n , Q n,n = (1 − θ ) n , P , = 1 , Q , = 0 ,Hh n ( x ) = 1 n ( Hh n − ( x ) − xHh n − ( x )) , Hh ( x ) = ˆ − x −∞ e − t / dt, I n ( k ; α, β, δ ) = − e αk α Hh n ( βk − δ )+ βα I n − ,I − ( k ; α, β, δ ) = √ πβ e αδβ + α β Φ (cid:16) − βk + δ + αβ (cid:17) if β > α = 0 , − Φ (cid:16) βk − δ − αβ (cid:17) if β < α < . As suggested by these definitions, the implementation of the formula involves successive computation offunctions, followed by a summation in (A.1) that must be truncated. In order to control the error, wenotice that Z n ≤ 2, and the error boundΥ (Θ) − M X n =0 ( θ θ ) n e − θ θ n ! Z n (Θ) ≤ ∞ X n = M +1 ( θ θ ) n e − θ θ n ! = 2 − M X n =0 ( θ θ ) n e − θ θ n ! ≡ ǫ ( θ θ , M ) . This can be verified directly from the proof of Proposition 3.1 presented in A.2 M th term, the upper bound for the truncationerror is given by e − α Sǫ (( m + 1) κT, M ) + e − rT Kǫ ( κT, M ) . This can be computed quickly and used to limit the error to the a priori desired decimal place. Forexample, take S = K = 100, T = 1 and an error tolerance of 0 . 01. Then, with κ ≈ m, α < . M = 143. A.2 Proof of Propositions 3.1 In order to price the European call option, we need to evaluate the terms Q { S T > K } and E (cid:8) e S T { S T >K } (cid:9) .We first state some useful facts (see Kou (2002) for proofs). Lemma A.1. Define two i.i.d. exponential r.v.’s, namely, J + i ∼ Exp ( λ ) , J − i ∼ Exp ( λ ) . For every n ≥ , we have n X i =1 J i d = (P mi =1 J + i , w.p. P n,m , P mi =1 J − i , w.p. Q n,m , where P n,m = n − X i = m (cid:18) n − m − i − m (cid:19)(cid:18) ni (cid:19) (cid:18) λ λ + λ (cid:19) i − m (cid:18) λ λ + λ (cid:19) n − i p i q n − i ,Q n,m = n − X i = m (cid:18) n − m − i − m (cid:19)(cid:18) ni (cid:19) (cid:18) λ λ + λ (cid:19) n − i (cid:18) λ λ + λ (cid:19) i − m p n − i q i ,P n,n = p n , Q n,n = q n , P , = 1 , Q , = 0 , q = 1 − p. Next, for every n ≥ 0, we define the functions Hh n ( x ) = ˆ ∞ x Hh n − ( y ) dy = 1 n ! ˆ ∞ x ( t − x ) n e − t dt,Hh − ( x ) = e − x / , Hh ( x ) = ˆ ∞ x e − t dt = √ π Φ ( − x ) , where Φ ( x ) denotes the standard normal c.d.f. In addition, for n ≥ 0, define the integral I n ( k, α, β, δ ) = ˆ ∞ k e αx Hh n ( βx − δ ) dx. Lemma A.2. ( i ) If β > and α = 0 , then for all n ≥ − , we have I n ( k, α, β, δ ) = − e αk α n X i =0 (cid:18) βα (cid:19) n − i Hh i ( βk − δ ) + (cid:18) βα (cid:19) n +1 √ πβ e αδβ + α β Φ (cid:18) − βk + δ + αβ (cid:19) . The Hh functions can be calculated using either of the following facts: Hh n = 2 − n √ πe − x F (cid:16) n +12 , , x (cid:17) √ (cid:0) n (cid:1) − x F (cid:16) n + 1 , , x (cid:17) Γ (cid:0) n +12 (cid:1) ,nHh n ( x ) = Hh n − ( x ) − xHh n − ( x ) , n ≥ , where F denotes the confluent hypergeometric function and Γ is the gamma function. ii ) If β < and α < , then for all n ≥ − , we have I n ( k, α, β, δ ) = − e αk α n X i =0 (cid:18) βα (cid:19) n − i Hh i ( βk − δ ) − (cid:18) βα (cid:19) n +1 √ πβ e αδβ + α β Φ (cid:18) βk − δ − αβ (cid:19) . In particular, I − ( k ; α, β, δ ) = √ πβ e αδβ + α β Φ (cid:16) − βk + δ + αβ (cid:17) if β > , α = 0 , − Φ (cid:16) βk − δ − αβ (cid:17) if β < , α < . In addition, under either assumption (i) or (ii) over the parameters α and β given above, the functions( I n ) satisfy the recursive relation I n ( k ; α, β, δ ) = − e αk α Hh n ( βk − δ ) + βα I n − ( k ; α, β, δ ) . In our extended Kou model, we need to understand the distribution of the sum of double exponentiallydistributed randomly timed and EA jumps. Hence, we consider the associated p.d.f.’s. Lemma A.3. Let Z + e , Z − e be i.i.d. exponential r.v.’s, Z + e , ∼ Exp ( η ) , Z − e , ∼ Exp ( η ) . In addition, let J + i , J − i , Z + e , Z − e be independent. Then, we have f P n J + i + Z + e ( t ) = λ − η ! n (cid:16) A n ( t ) e − λ t + B n ( t ) e − η t (cid:17) , t > , λ = η , (A.2) f − P n J − i + Z + e ( t ) = λ + η ! n C n ( t ) e λ min (0 ,t ) − η max(0 ,t ) , t ∈ R , λ = η , (A.3) where f Y denotes the p.d.f. of Y , and B n ( t ) = − η − ( n − , A n ( t ) = 1 + n − X i η − ( i − (cid:18) λ − η (cid:19) n − i λ n − i ( t ) n − i ( n − i )! , t > ,C n ( t ) = C n − ( t ) λ + λ n − (max (0 , t ) − t ) n − ( n − , t ∈ R . In addition, they satisfy the following recursive relations: f P n +11 J + i + Z + e ( t ) = λ ( λ − η ) (cid:18) f P n J + i + Z + e ( t ) − λ n ( t ) n n ! η e − λ t (cid:19) , (A.4) f − P n +11 J − i + Z + e ( t ) = λ λ + η (cid:18) f − P n J − i + Z + e ( t ) + λ n (max (0 , t ) − t ) n n ! η e λ t e − ( λ + η ) max(0 ,t ) (cid:19) . (A.5) Proof. We first notice that the positive r.v., J + i + Z + e , has the p.d.f. ˆ t λ e − λ ( t − x ) η e − η x dx = 1 λ − η (cid:16) e − λ t − e − η t (cid:17) ≡ λ − η ! (cid:16) A ( t ) e − λ t + B ( t ) e − η t (cid:17) , t > , λ = η . n ≥ f P n J + i + Z + e ( t ) = (cid:18) λ − η (cid:19) n (cid:0) A n ( t ) e − λ t + B n ( t ) e − η t (cid:1) , we obtain f P n J + i + Z + e ( t ) = ˆ t λ n ( t − x ) n − ( n − e − λ ( t − x ) η e − η x dx = λ n ( t ) n n ! η e − λ t + ( λ − η ) λ f P n +11 J + i + Z + e ( t ) , = ⇒ f P n +11 J + i + Z + e ( t ) = λ ( λ − η ) (cid:18) f P n J + i + Z + e ( t ) − λ n ( t ) n n ! η e − λ t (cid:19) = λ − η ! n +1 (cid:16) A n +1 ( t ) e − λ t + B n +1 ( t ) e − η t (cid:17) , where the coefficients satisfy B n +1 = 1 η B n , A n +1 = A n η − C n +1 , C n +1 = (cid:18) λ − η (cid:19) n λ n ( t ) n n ! ,B n = − η − ( n − , A n = 1 + n − X i η − ( i − (cid:18) λ − η (cid:19) n − i λ n − i ( t ) n − i ( n − i )! , and this yields (A.2). Next, we note that the real-valued r.v. − J − i + Z + e has the p.d.f. ˆ ∞ max(0 ,t ) λ e λ ( t − x ) η e − η x dx = e λ min (0 ,t ) − η max(0 ,t )1 λ + η , t ∈ R , λ = η . For a fixed n ≥ 1, assume that f − P n J − i + Z + e ( t ) = (cid:18) λ + η (cid:19) n C n ( t ) e λ min (0 ,t ) − η max(0 ,t ) , then we get f − P n J − i + Z + e ( t ) = ˆ ∞ max(0 ,t ) λ n ( − t + x ) n − ( n − e λ ( t − x ) η e − η x dx = − λ n (max (0 , t ) − t ) n n ! η e λ t e − ( λ + η ) max(0 ,t ) + ( λ + η ) λ f − P n +11 J − i + Z + e ( t ) , = ⇒ f − P n +11 J − i + Z + e ( t ) = λ λ + η (cid:18) f − P n J − i + Z + e ( t ) + λ n (max (0 , t ) − t ) n n ! η e λ t e − ( λ + η ) max(0 ,t ) (cid:19) ≡ λ + η ! n +1 C n +1 ( t ) e λ min (0 ,t ) − η max(0 ,t ) . Matching terms yields C n +1 ( t ) = C n ( t ) λ + λ n ,t ) − t ) n n ! . We now calculate the distribution of the sum of a normal r.v. and double-exponentials. Proposition A.4. Let W ∼ N (cid:0) , σ (cid:1) . Then, we have the p.d.f.’s: f W + P n +1 i =1 J + i + Z + e ( t ) = (cid:18) λ λ − η (cid:19) n η e ( ση √ π e − η t Hh (cid:18) − tσ + ση (cid:19) + (A.6) − n X i =1 (cid:18) λ λ − η (cid:19) n − i +1 η e ( σλ √ π (cid:0) σ i λ i (cid:1) e − λ t Hh i (cid:18) − tσ + σλ (cid:19) , t > , λ = η , W − P n +1 i =1 J − i + Z + e ( t ) = (cid:18) λ λ + η (cid:19) n η e ( ση √ π e − η t Hh (cid:18) − tσ + ση (cid:19) + (A.7)+ n X i =1 (cid:18) λ λ + η (cid:19) n − i +1 η e ( σλ √ π (cid:0) σ i λ i (cid:1) e λ t Hh i (cid:18) tσ + σλ (cid:19) , t ∈ R , λ = η . Moreover, they admit the recursive relations: f W + P n +1 i =1 J + i + Z + e ( t ) = λ λ − η f W + P ni =1 J + i + Z + e ( t ) − η e ( σλ √ π ( σ n λ n ) e − λ t Hh n (cid:18) − tσ + σλ (cid:19) , (A.8) f W − P n +1 i =1 J − i + Z + e ( t ) = λ λ + η f W − P n +1 i =1 J − i + Z + e ( t ) + η e ( σλ √ π ( σ n λ n ) e λ t Hh n (cid:18) tσ + σλ (cid:19) . (A.9) Proof. We start with the p.d.f. for W + Z + e : f W + Z + e ( t ) = η e ( ση √ π e − η t Hh (cid:18) − tσ + ση (cid:19) . Using (A.4), we also write f W + P n +1 i =1 J + i + Z + e ( t ) = ˆ t −∞ f P n +11 J + i + Z + e ( t − x ) e − x σ √ πσ dx = λ λ − η f W + P ni =1 J + i + Z + e ( t ) − η e ( σλ √ π ( σ n λ n ) e − λ t Hh n (cid:18) − tσ + σλ (cid:19) , which leads directly to f W + P n +1 i =1 J + i + Z + e ( t ) = (cid:18) λ λ − η (cid:19) n η e ( ση √ π e − η t Hh (cid:18) − tσ + ση (cid:19) + − n X i =1 (cid:18) λ λ − η (cid:19) n − i +1 η e ( σλ √ π (cid:0) σ i λ i (cid:1) e − λ t Hh i (cid:18) − tσ + σλ (cid:19) . To prove (A.7), we apply (A.5) to get the recursive expression f W − P n +1 i =1 J − i + Z + e ( t ) = ˆ R f − P n +1 i =1 J − i + Z + e ( t − x ) e − x σ √ πσ dx = λ λ + η f W − P n +1 i =1 J − i + Z + e ( t ) + η e ( σλ √ π ( σ n λ n ) e λ t Hh n (cid:18) tσ + σλ (cid:19) , f W − P n +1 i =1 J − i + Z + e ( t ) = (cid:18) λ λ + η (cid:19) n η e ( ση √ π e − η t Hh (cid:18) − tσ + ση (cid:19) ++ n X i =1 (cid:18) λ λ + η (cid:19) n − i +1 η e ( σλ √ π (cid:0) σ i λ i (cid:1) e λ t Hh i (cid:18) tσ + σλ (cid:19) . We can now calculate the tail probabilities that will allow us to price the call option. Proposition A.5. Let F X ( z ) ≡ Q { X ≤ z } . Then, F W + P n +1 i =1 J + i + Z + e ( z ) = (cid:18) λ λ − η (cid:19) n η e ( ση √ π I (cid:18) z ; − η , − σ , − η σ (cid:19) + (A.10) − n X i =1 (cid:18) λ λ − η (cid:19) n − i +1 η e ( σλ √ π (cid:0) σ i λ i (cid:1) I i (cid:18) z ; − λ , − σ , − λ σ (cid:19) , z > , λ = η ,F W − P n +1 i =1 J − i + Z + e ( z ) = (cid:18) λ λ + η (cid:19) n η e ( ση √ π I (cid:18) z, − η , − σ , − ση (cid:19) + (A.11)+ n X i =1 (cid:18) λ λ + η (cid:19) n − i +1 η e ( σλ √ π (cid:0) σ i λ i (cid:1) I n (cid:18) z, λ , σ , − σλ (cid:19) , z ∈ R , λ = η , In addition, these c.d.f.’s admit the following recursive relations: F W + P n +1 i =1 J + i + Z + e ( z ) = λ ( λ − η ) F W + P ni =1 J + i + Z + e ( z ) − η e ( σλ √ π ( σ n λ n ) I n (cid:18) z ; − λ , − σ , − λ σ (cid:19) , (A.12) F W − P n +1 i =1 J − i + Z + e ( z ) = λ λ + η F W − P n +1 i =1 J − i + Z + e ( z ) + η e ( σλ √ π ( σ n λ n ) I n (cid:18) z, λ , σ , − σλ (cid:19) . (A.13) Proof. The above expressions can all be derived by integrating the corresponding densities given in (A.6),(A.7), (A.8), (A.9) and taking into account the definitions of the Hh and I functions.We remark that although we did not provide the tail probability formulas F σW T + P mi =1 ˆ J + i − ˆ Z − e and F σW T − P mi =1 ˆ J − i − ˆ Z − e we point out that they can be derived by symmetry. For example, we have F σW T + P mi =1 ˆ J + i − ˆ Z − e ( s ) = 1 − F σW T − P mi =1 ˆ J + i + ˆ Z − e ( − s ) . In fact, these c.d.f.’s appear in the pricing formula as the functions T i,j ’s, defined in Section A.1. We nowgive the formula for the call option price. Recall the call price can be written as C ( S ) = S E (cid:26) e ( − σ − αT − κζ ) T + σW T + P NTi =1 J i + Z e { S T >K } | S = S (cid:27) − e − rT K Q { S T > K | S = S } . (A.14)28sing the results above, we can write Q { S T > K | S = S } (A.15)= ∞ X n =1 Q { N T = n } Q ( σW T + n X i J i + Z e > log (cid:18) KS (cid:19) − (cid:18) r − σ − αT − κζ (cid:19) T ) = ∞ X n =1 κ n e − κ n ! (cid:20) u n X m =1 P n,m Q ( σW T + m X i J + i + Z + e > k ) + Q n,m Q ( σW T − m X i J − i + Z + e > k )! ++ w n X m =1 P n,m Q ( σW T + m X i J + i − Z − e > k ) + Q n,m Q ( σW T − m X i J − i − Z − e > k )! (cid:21) = ∞ X n =1 κ n e − κ n ! " u n X m =1 ( P n,m T ,n ( k, Θ) + Q n,m T ,n ( k, Θ)) + w n X m =1 ( P n,m T ,n ( k, Θ) + Q n,m T ,n ( k, Θ)) , where k ≡ log (cid:0) KS (cid:1) − (cid:16) r − σ − κζ (cid:17) T − α .It remains to compute E n e σW T + P NTi =1 J i + Z e { S T >K } | S = S o (A.16)= ∞ X n =1 ( κT ) n e − κT n ! (cid:20) u n X m =1 P n,m E n e σW T + P mi =1 J + i + Z + e { σW T + P mi =1 J + i + Z + e >k } o + (A.17)+ u n X k =1 Q n,k E n e σW T − P mi =1 J − i + Z + e { σW T − P mi =1 J − i + Z + e >k } o + (A.18)+ w n X k =1 P n,k E n e σW T + P mi =1 J + i − Z − e { σW T + P mi =1 J + i − Z − e >k } o + (A.19)+ w n X k =1 Q n,k E n e σW T − P mi =1 J − i − Z − e { σW T − P mi =1 J − i − Z − e } o (cid:21) . (A.20)In order to compute the expectation in (A.17), we apply the p.d.f. from Prop. A.4 to write E n e σW T + P mi =1 J + i + Z + e { σW T + P mi =1 J + i + Z + e >k } o == ˆ ∞ k ˆ R e t − x e − ( t − x )22 σ T √ πσ T ˆ R e x − y f P mi =1 J + i ( x − y ) e y f Z + e ( y ) dydxdt = ˆ ∞ k ˆ R e σ T e − ( t − x − σ T ) σ T √ πσ T ˆ R (cid:18) λ λ − (cid:19) m f P mi =1 ˆ J + i ( x − y ) η η − f ˆ Z + e ( y ) dydxdt = e σ T (cid:18) λ λ − (cid:19) m η η − Q σW T + m X i =1 ˆ J + i + ˆ Z + e > k − σ T ! = e σ T (cid:18) λ λ − (cid:19) m η η − F σW T + P mi =1 ˆ J + i + ˆ Z + e (cid:0) k − σ T (cid:1) , (A.21)where ˆ J + i ∼ Exp ( λ − 1) and ˆ Z + e ∼ Exp ( η − E (cid:26) e ( − σ − αT − κζ ) T + σW T + P Ni =1 J i + Z e { S T >K } | S = S (cid:27) = e − α ∞ X n =1 (ˆ κT ) n e − ˆ κT n ! ˆ u n X k =1 (cid:16) ˆ P n,k T ,n ( k, Θ) + ˆ Q n,k T ,n ( k, Θ) (cid:17) + ˆ w n X k =1 (cid:16) ˆ P n,k T ,n (cid:16) ˆ k, Θ (cid:17) + ˆ Q n,k T ,n (cid:16) ˆ k, Θ (cid:17)(cid:17)! , where ˆ P n,k , ˆ Q n,k are calculated as P n,k , Q n,k but with the parameters ˆ η , , ˆ λ , in place of η , and λ , . In addition, the Poisson intensity parameter has also been transformed to ˆ κ ≡ ( m + 1) κ , with m = p λ λ − + q λ λ +1 − 1. Finally, substituting the expressions for (A.15) and (A.16) into (A.14) concludesthe proof. A.3 Proof of Propositions 4.1 and 4.2 The first part of Proposition 4.1 follows from Jensen’s inequality, namely, C ( t, S ) ≥ E n e − rτ (cid:0) Se rτ + Z e − K (cid:1) + o = ˆ R + C BS (cid:18) τ, S ; ˆ σ √ τ , K, r (cid:19) G ( d ˆ σ ) ≥ C BS (cid:18) τ, S ; ˆ σ min √ τ , K, r (cid:19) , (A.22)where τ ≡ T − t . In (A.22), the equality follows from the tower property of the conditional expectationand the last inequality follows from the monotonicity of C BS w.r.t. the volatility parameter σ . Thesecond part of the proposition is proved in a similar fashion.To prove Proposition 4.2, we first observe that C ( t, S ) = ˆ R + × R + C BS τ, S ; r ˜ σ + ˆ σ τ , K, r ! H ( d ˜ σ ) G ( d ˆ σ ) . For an ATM-forward option, i.e. K = e rτ S , we notice that the Black-Scholes price C BS is concave in itsvolatility parameter σ . Consequently, by Jensen’s inequality, we obtain the upper bound C ( t, S ) ≤ C BS τ, S ; s ˆ R + ˜ σ τ H ( d ˜ σ ) + ˆ R + ˆ σ τ G ( d ˆ σ ) , K, r ! . A.4 Proof of Propositions 4.4 and 4.5 Propositions 4.4 and 4.5 are applications of the more general results developed in Benaim and Friz (2008).Denote by M ( ω ) ≡ E (cid:8) e ωX (cid:9) the m.g.f. of the r.v. X , and with F its c.d.f. As mentioned in Section 4,if r ∗ ≡ inf { ω s.t. M ( ω ) < ∞} is finite, than it follows that lim sup x →∞ − log(1 − F ( x )) x = r ∗ . In turn, if F is well-behaved, the lim sup can be replaced by a lim and the tail asymptotics − log(1 − F ( x )) x ∼ r ∗ x , alongwith the condition r ∗ > I ( t ; K, T ) ( T − t )log (cid:16) KS t (cid:17) ∼ ξ ( p ∗ ) , as K → ∞ ;see Theorems 9 and 10 of Benaim and Friz (2008) for this result and the associated technical conditions.A symmetric argument holds for the negative tails of F and of the implied volatility. Therefore, given that30he log stock price under the extended Kou and Heston models admits a m.g.f, it remains to prove that − log(1 − F ( x )) x ∼ r ∗ x . Theorem 7 and Theorem 8 in Benaim and Friz (2008) provide sufficient conditionsfor models admitting a m.g.f. to ensure that such condition holds.The m.g.f. M of X ≡ log (cid:16) S T S t (cid:17) under the extended Heston model (3.3), given σ t = σ , satisfies log M ( ω ) = C + ωD − νϑζ − ge − d ( T − t ) − g ! + d ( T − t ) ! + ν − ρζω − dζ − e − d ( T − t ) − ge − d ( T − t ) σ + ψ e ( ω ) , with g = ν − ρζω − dν − ρζω + d , d = q ( ν − ρζω ) + ζ ( ω − ω ) , ψ e ( ω ) = log (cid:18) u η η − ω + w η η + ω (cid:19) , where C and D are constants. Clearly, we have r ∗ = min { p, η } , where p is the smallest positive solutionof 1 − ge − d ( T − t ) | ω = p = 0 . In turn, the last equation is equivalent to ν − ρζp + q ( ν − ρζp ) + ζ ( p − p )coth (cid:18) ( T − t )2 q ( ν − ρζp ) + ζ ( p − p ) (cid:19) = 0 . Notice that, when r ∗ = p , the dominating term for ω → p is ν − ρζωi − dζ − e − d ( T − t ) − ge − d ( T − t ) σ . Using l’Hopital’srule, it follows that p − ω − ge − d ( T − t ) → constant, as ω → p − . This means that ν − ρζω − dζ − e − d ( T − t ) − ge − d ( T − t ) σ t is regularly varying of index 1 as a function of p − ω and CriterionII of Theorem 8 in Benaim and Friz (2008) is satisfied. When r ∗ = η , we have M ( η − z ) ∼ uη z − as z → + , and thus Criterion I of Theorem 7 in Benaim and Friz (2008) is satisfied. A similar argument holds forthe negative tail.Now consider the m.g.f. of X ≡ log (cid:16) S T S t (cid:17) under the extended Kou model:log M ( ω ) = µω + σ ω κ (cid:18) p λ λ − ω + (1 − p ) λ λ + ω − (cid:19) + log (cid:18) u η η − ω + (1 − u ) η η + ω (cid:19) , where µ is a constant. This m.g.f. satisfies the tail asymptotics condition with r ∗ = min { λ , η } , and ( log M ( λ − z ) ∼ κpλ z − as z → + if λ ≤ η ,M ( η − z ) ∼ uη z − as z → + if λ > η . Hence, M satisfies either Criterion I or II (depending on whether λ > η or not) of Theorems 7 and 8in Benaim and Friz (2008). A similar argument holds for the negative tail. A.5 Proof of Proposition 6.1 Let A ( t, S ; u ) be the American put price when the earnings announcement is scheduled at time T e = u, t < u ≤ T . Our goal is to show that A ( t, S ; u ) ≥ A ( t, S ; l ), for t < u ≤ l . W.l.o.g., let t = 0 and write A ( S ; l ) ≡ A (0 , S ; l ). Let X s = log( S s /S ) − { s ≥ T e } Z e be the log stock price excluding the EA jump, anddenote by ( F us ) ≤ s ≤ T (resp. ( F ls ) ≤ s ≤ T ) the filtration generated by S with T e = u (resp. T e = l ). Notice For the derivation of the Heston m.g.f., see, e.g. del Ba˜no Rollin et al. (2009). F ls = F us for s < u or s ≥ l , and F ls ⊂ F us , for u ≤ s < l . Therefore, the sets of stopping times w.r.tto F u and F l , denoted respectively by T u and T l , satisfy T l ⊂ T u . Thus, for any candidate stoppingtime τ ∈ T l , we have E n e − rτ (cid:0) K − Se X τ +1 { τ ≥ u } Z e (cid:1) + | F ll − o (A.23)= E n { τ Here we give a sketch of the derivation of the approximation formula (6.7). The arguments are adaptedfrom Barone-Adesi and Whaley (1987) and Kou and Wang (2004). We start by writing A ( t, S ) = P E ( t, S ) + ǫ ( t, S ) , where P E is the European put price with an EA jump, and ǫ is a correction term. Note that P E iscomputed using the results in Proposition 3.1. In the continuation region, ǫ must satisfy the same PIDEas that of P E and A , namely rǫ ( t, S ) − ∂ t ǫ ( t, S ) − L ǫ ( t, S ) = 0 , (A.27)where the operator L is defined in (6.5). The idea of the approximation in Barone-Adesi and Whaley(1987) is to remove the term ǫ t in (A.27). This involves letting ǫ ( t, S ) ≡ g ( z, S ) z , with z ≡ − e − r ( T − t ) ,substituting in the above PIDE, and ignoring the term (1 − z ) g z . This results in the OIDE rz ǫ ( t, S ) − L ǫ ( t, S ) = 0 . (A.28)While (A.28) holds in the continuation region, in the exercise reason we have ǫ ( t, S ) = K − S − P E ( t, S ).Following Kou and Wang (2004), we consider the ansatz ǫ ( t, S ) = ( γ ( t ) S − β + γ ( t ) S − β , S > α ( t ) ,K − S − P E ( t, S ) , S ≤ α ( t ) , where α ( t ) is the boundary at time t . One can directly verify that the ansatz solves the OIDE (A.28) if β , are two positive solutions to 6.10, and if Kλ − α ( t )1 + λ − ˆ −∞ P E ( t, αe y ) e λ y dy = γ ( t ) α − β λ − β + γ ( t ) α − β λ − β , (A.29)In turn, we impose the continuous-fit and smooth-pasting conditions. The first condition yields (6.11)while the second condition, altogether with (A.29), yields the constants γ and γ in (6.8) and (6.9).32 eferences Andersen, L. and Andreasen, J. (2000). Jump diffusion models: volatility smile fitting and numerical methods forpricing. Review of Derivatives Research , 4:231–262.Barone-Adesi, G. and Whaley, R. (1987). Efficient analytic approximation of American option values. The Journalof Finance , 17(2):301–320.Barth, M., Johnson, T., and So, C. (2011). Dynamics of earnings announcement news: Evidence from option prices. Working paper .Bates, D. (1996). Jumps and stochastic volatility: The exchange rate processes implicit in Deutschemark options. Review of Financial Studies , 9(1):69–107.Benaim, S. and Friz, P. (2008). Smile asymptotics II: models with known moment generating functions. Journalof Applied Probability , 45 (1):1–291.Benaim, S., Friz, P., and Lee, R. (2012). On the Black-Scholes implied volatility at extreme strikes. In Cont, R.,editor, Frontiers in Quantitative Finance: Volatility and Credit Risk Modeling , pages 19–45. Wiley & Sons.Bensoussan, A. and Lions, J. (1984). Impulse control and quasivariational inequalities. Gaunthier-Villars.Billings, M. and Jennings, R. (2011). The option markets anticipation of information content in earnings announce-ments. Review of Accounting Studies Conference Version , 16:587–619.Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy ,81:637–654.Broadie, M., Chernov, M., and Johannes, M. (2009). Understanding index option returns. Review of FinancialStudies , 22(11):4493–4529.Carr, P., Geman, H., Madan, D., and Yor, M. (2002). The fine structure of asset returns: An empirical investigation. Journal of Business , 75(2):305–332.Carr, P. and Madan, D. (1999). Option pricing and the fast Fourier transform. Journal of Computational Finance ,2:61–73.Chatrah, A., Christie-David, R., and Lee, K. (2009). How potent are news reversals?: Evidence from the futuresmarkets. Journal of Futures Market , 29:42–73.Chordia, T. and Shivakumar, L. (2006). Earnings and price momentum. Journal of Financial Economics , 80(3):627–656.Coleman, T. and Li, Y. (1994). On the convergence of reflective Newton methods for large-scale nonlinear mini-mization subject to bounds. Mathematical Programming , 67, 2:189–224.Coleman, T. and Li, Y. (1996). An interior, trust region approach for nonlinear minimization subject to bounds. SIAM Journal on Optimization , 6:418–445.Cont, R. and Tankov, P. (2002). Calibration of jump-diffusion option-pricing models: a robust non-parametricapproach. Working paper .Cont, R. and Voltchkova, E. (2005). A finite difference scheme for option pricing in jump diffusions and exponentialL´evy models. SIAM Journal on Numerical Analysis , 43:1596–1626.del Ba˜no Rollin, S., Ferreiro-Castilla, A., and Utzet, F. (2009). A new look at the Heston characteristic function.Preprint.Dennis, J. (1977). Nonlinear least-squares and equations. In Jacobs, D., editor, State of the Art in NumericalAnalysis , pages 269–312. Academic Press.Donders, M. and Vorst, T. (1996). The impact of firm specific news on IVs. Journal of Banking and Finance ,20:1447–1461.Dubinsky, A. and Johannes, M. (2006). Fundamental uncertainty, earning announcements and equity options.Working paper. uffie, D., J. P. and Singleton, K. (2000). Transform analysis and option pricing for affine jump-diffusions. Econo-metrica , 68:1343–1376.Isakov, D. and Perignon, C. (2001). Evolution of market uncertainty around earnings announcements. Journal ofBanking and Finance , 25:1769–1788.Jackson, K., Jaimungal, S., and Surkov, V. (2008). Fourier space time-stepping for option pricing with L´evy models. Journal of Computational Finance , 12(2):1–29.Kou, S. (2002). A jump-diffusion model for option pricing. Management Science , 48:1086–1101.Kou, S. G. and Wang, H. (2004). Option pricing under a double exponential jump diffusion model. ManagementScience , 50(9):1178–1192.Lee, R. (2004). Option pricing by transform methods: extensions, unification, and error control. Journal ofComputational Finance , 7:51–86.Lee, S. and Mykland, P. (2008). Jumps in financial markets: A new nonparametric test and jump dynamics. Reviewof Financial Studies , 21(6):2535–2563.Lord, R., Fang, F., Bervoets, F., and Oosterlee, C. W. (2008). A fast and accurate FFT-based method for pricingearly-exercise options under L´evy processes. SIAM Journal of Scientific Computing , 30(4):1678–1705.Madan, D., Carr, P., and Chang, E. (1998). The variance gamma process and option pricing. European FinanceReview , 2(8):79–105.Maheu, J. and McCurdy, T. (2004). News arrival, jump dynamics and volatility components for individual stockreturns. Journal of Finance , 59:755–793.Mehra, A., Kolanovic, M., and Kaplan, B. (2014). Earnings and option volatility monitor. Technical report, J.P.Morgan.Merton, R. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science ,4:141–183.Merton, R. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Eco-nomics , 3:125–144.Øksendal, B. (2003). Stochastic Differential Equations: An Introduction with Applications . Springer.Patell, J. and Wolfson, M. (1981). The ex ante and ex post price effect of quarterly earnings announcementsreflected in option and stock prices. Journal of Accounting Research , 19:434–458.Patell, J. and Wolfson, M. (1984). The intraday speed of adjustment of stock prices to earnings and dividendannouncements. Journal of Financial Economics , 13:223–252.Piazzesi, M. (2005). Bond yields and the Federal Reserve. Journal of Political Economy , 113:311–344.Raible, S. (2000). L´evy processes in finance: Theory, numerics, and empirical facts . PhD thesis, Univ. Freiburg.Rogers, J., Skinner, D., and Van Buskirk, A. (2009). Earnings guidance and market uncertainty. Journal ofAccounting and Economics , 48:90–109. References Andersen, L. and Andreasen, J. (2000). Jump diffusion models: volatility smile fitting and numerical methods forpricing. Review of Derivatives Research , 4:231–262.Barone-Adesi, G. and Whaley, R. (1987). Efficient analytic approximation of American option values. The Journalof Finance , 17(2):301–320.Barth, M., Johnson, T., and So, C. (2011). Dynamics of earnings announcement news: Evidence from option prices. Working paper . ates, D. (1996). Jumps and stochastic volatility: The exchange rate processes implicit in Deutschemark options. Review of Financial Studies , 9(1):69–107.Benaim, S. and Friz, P. (2008). Smile asymptotics II: models with known moment generating functions. Journalof Applied Probability , 45 (1):1–291.Benaim, S., Friz, P., and Lee, R. (2012). On the Black-Scholes implied volatility at extreme strikes. In Cont, R.,editor, Frontiers in Quantitative Finance: Volatility and Credit Risk Modeling , pages 19–45. Wiley & Sons.Bensoussan, A. and Lions, J. (1984). Impulse control and quasivariational inequalities. Gaunthier-Villars.Billings, M. and Jennings, R. (2011). The option markets anticipation of information content in earnings announce-ments. Review of Accounting Studies Conference Version , 16:587–619.Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy ,81:637–654.Broadie, M., Chernov, M., and Johannes, M. (2009). Understanding index option returns. Review of FinancialStudies , 22(11):4493–4529.Carr, P., Geman, H., Madan, D., and Yor, M. (2002). The fine structure of asset returns: An empirical investigation. Journal of Business , 75(2):305–332.Carr, P. and Madan, D. (1999). Option pricing and the fast Fourier transform. Journal of Computational Finance ,2:61–73.Chatrah, A., Christie-David, R., and Lee, K. (2009). How potent are news reversals?: Evidence from the futuresmarkets. Journal of Futures Market , 29:42–73.Chordia, T. and Shivakumar, L. (2006). Earnings and price momentum. Journal of Financial Economics , 80(3):627–656.Coleman, T. and Li, Y. (1994). On the convergence of reflective Newton methods for large-scale nonlinear mini-mization subject to bounds. Mathematical Programming , 67, 2:189–224.Coleman, T. and Li, Y. (1996). An interior, trust region approach for nonlinear minimization subject to bounds. SIAM Journal on Optimization , 6:418–445.Cont, R. and Tankov, P. (2002). Calibration of jump-diffusion option-pricing models: a robust non-parametricapproach. Working paper .Cont, R. and Voltchkova, E. (2005). A finite difference scheme for option pricing in jump diffusions and exponentialL´evy models. SIAM Journal on Numerical Analysis , 43:1596–1626.del Ba˜no Rollin, S., Ferreiro-Castilla, A., and Utzet, F. (2009). A new look at the Heston characteristic function.Preprint.Dennis, J. (1977). Nonlinear least-squares and equations. In Jacobs, D., editor, State of the Art in NumericalAnalysis , pages 269–312. Academic Press.Donders, M. and Vorst, T. (1996). The impact of firm specific news on IVs. Journal of Banking and Finance ,20:1447–1461.Dubinsky, A. and Johannes, M. (2006). Fundamental uncertainty, earning announcements and equity options.Working paper.Duffie, D., J. P. and Singleton, K. (2000). Transform analysis and option pricing for affine jump-diffusions. Econo-metrica , 68:1343–1376.Isakov, D. and Perignon, C. (2001). Evolution of market uncertainty around earnings announcements. Journal ofBanking and Finance , 25:1769–1788.Jackson, K., Jaimungal, S., and Surkov, V. (2008). Fourier space time-stepping for option pricing with L´evy models. Journal of Computational Finance , 12(2):1–29.Kou, S. (2002). A jump-diffusion model for option pricing. Management Science , 48:1086–1101. ou, S. G. and Wang, H. (2004). Option pricing under a double exponential jump diffusion model. ManagementScience , 50(9):1178–1192.Lee, R. (2004). Option pricing by transform methods: extensions, unification, and error control. Journal ofComputational Finance , 7:51–86.Lee, S. and Mykland, P. (2008). Jumps in financial markets: A new nonparametric test and jump dynamics. Reviewof Financial Studies , 21(6):2535–2563.Lord, R., Fang, F., Bervoets, F., and Oosterlee, C. W. (2008). A fast and accurate FFT-based method for pricingearly-exercise options under L´evy processes. SIAM Journal of Scientific Computing , 30(4):1678–1705.Madan, D., Carr, P., and Chang, E. (1998). The variance gamma process and option pricing. European FinanceReview , 2(8):79–105.Maheu, J. and McCurdy, T. (2004). News arrival, jump dynamics and volatility components for individual stockreturns. Journal of Finance , 59:755–793.Mehra, A., Kolanovic, M., and Kaplan, B. (2014). Earnings and option volatility monitor. Technical report, J.P.Morgan.Merton, R. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science ,4:141–183.Merton, R. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Eco-nomics , 3:125–144.Øksendal, B. (2003). Stochastic Differential Equations: An Introduction with Applications . Springer.Patell, J. and Wolfson, M. (1981). The ex ante and ex post price effect of quarterly earnings announcementsreflected in option and stock prices. Journal of Accounting Research , 19:434–458.Patell, J. and Wolfson, M. (1984). The intraday speed of adjustment of stock prices to earnings and dividendannouncements. Journal of Financial Economics , 13:223–252.Piazzesi, M. (2005). Bond yields and the Federal Reserve. Journal of Political Economy , 113:311–344.Raible, S. (2000). L´evy processes in finance: Theory, numerics, and empirical facts . PhD thesis, Univ. Freiburg.Rogers, J., Skinner, D., and Van Buskirk, A. (2009). Earnings guidance and market uncertainty. Journal ofAccounting and Economics , 48:90–109., 48:90–109.