Theoretical Economics Letters, 2012, 2, 400407 http://dx.doi.org/10.4236/tel.2012.24074 Published Online October 2012 (http://www.SciRP.org/journal/tel) Joint Characteristic Function of Stock LogPrice and Squared Volatility in the Bates Model and Its Asset Pricing Applications Oleksandr Zhylyevskyy Department of Economics, Iowa State University, Ames, USA Email: oz9a@iastate.edu Received July 17, 2012; revised August 17, 2012; accepted September 17, 2012 ABSTRACT The model of Bates specifies a rich, flexible structure of stock dynamics suitable for applications in finance and eco nomics, including valuation of derivative securities. This paper analytically derives a closedform expression for the joint conditional characteristic function of a stock’s logprice and squared volatility under the model dynamics. The use of the function, based on inverting it, is illustrated on examples of pricing European, Bermudan, and Americanstyle options. The discussed approach for Europeanstyle derivatives improves on the option formula of Bates. The suggested approach for Americanstyle derivatives, based on a compoundoption technique, offers an alternative solution to exist ing finitedifference methods Keywords: Bates Model; Stochastic Volatility; JumpDiffusion; Characteristic Function; Option Pricing 1. Introduction Stochastic volatility and jumpdiffusion are standard tools of modeling asset price dynamics in finance research (see AïtSahalia and Jacod, 2011 [1]). Popularity of stochastic volatility models, such as the continuoustime model of Heston (1993) [2], is partly due to their ability to account for several aspects of stock price data that are not cap tured by analytically simpler geometric Brownian motion dynamics. For example, these models can help to account for an empirically relevant “leverage effect,” which re fers to an increase in the volatility of a stock when its price declines, and a decrease in the volatility when the price rises. They also can help to partly correct for defi ciencies of the famous Black and Scholes (1973) [3] op tion pricing formula (e.g., the implied volatility “smile”). The model of Bates (1996) [4] extends the Heston model by incorporating jumps in stock dynamics. Allowing for jumps enables a more realistic representation of stock price timeseries, which may feature discontinuities (for a discussion on jumps in asset data, see AïtSahalia and Jacod, 2009 [5]). In this paper, I analytically derive and provide exam ples for the use of a closedform expression for the joint conditional characteristic function of a stock’s logprice and squared volatility under the dynamics of the Bates model. The model offers a rich distributional structure of stock returns. For instance, a skewed distribution can arise due to a correlation between shocks to the stock price and shocks to the volatility or due to nonzero aver age jumps. Excess kurtosis can arise from variable vola tility or from a jump component. Also, the model can help to distinguish between two alternative explanations for skewness and excess kurtosis: stochastic volatility implies a positive relationship between the length of the holding period and the magnitude of skewness and kur tosis, whereas jumps imply a negative relationship (Bates, 1996 [4], pp. 7273). The flexibility of the model makes it particularly attractive for the task of valuation of de rivative securities. As such, it is useful in applied research and practice. Under jumpdiffusion dynamics with stochastic vola tility, the values of derivative securities such as Euro peanstyle options are typically impossible to express in simple form. Instead, they may be computed numeri cally by applying the transform methods of Duffie et al. (2000) [6] and Bakshi and Madan (2000) [7], which require inverting a conditional characteristic function of an underlying stateprice vector. Bates (1996) [4] solved for the marginal conditional characteristic function of the logprice and derived a formula for the value of a Europeanstyle call option that involves two separate inversions. In contrast, the problem of finding the joint conditional characteristic function of the logprice and squared volatility was not posed, and to the best of my knowledge, a solution for this function is not available in C opyright © 2012 SciRes. TEL
O. ZHYLYEVSKYY 401 existing finance studies. This paper aims to fill in the gap by deriving a closedform expression for the function, which is an analytically challenging task. In addition, I provide two practically relevant examples illustrating the use of the function. The first example revisits the prob lem of the valuation of Europeanstyle options. I show that the marginal characteristic function is a special case of the joint characteristic function and then apply results from prior research to obtain formulas for Europeanstyle put and call options that require a single inversion; this approach is more efficient than the solution suggested by Bates involving two inversions. The second example addresses the problem of valuation of Bermudan and Americanstyle options by proposing an extension of the GeskeJohnson compoundoption technique (Geske and Johnson, 1984 [8]). In this case, knowledge of the joint (rather than marginal) characteristic function is indis pensable. The proposed approach provides an alternative to pricing Americanstyle options using finitedifference methods (e.g., Chiarella et al., 2008 [9]), which can pose practical challenges when dealing with stochastic volatil ity (for a review, see Zhylyevskyy, 2010 [10]). The em pirical relevance of the example is due to a large num ber of single name equity and commodity futures op tions traded on organized exchanges being American style. The remainder of the paper is organized as follows. Section 2 sets up the Bates model and outlines the as sumptions and notation. Section 3 derives a stochastic differential equation for the stock’s logprice. Section 4 shows that the joint conditional characteristic function is a martingale and uses this result to derive a partial dif ferentialintegral equation for the function. Section 5 solves this equation analytically to obtain a closedform expression for the function. Section 6 provides examples for the use of the function when pricing derivative secu rities. Section 7 concludes. 2. The Bates Model I first outline the assumptions and introduce the notation. The financial market is assumed to admit no arbitrage opportunities. Thus, there is an equivalent martingale pro bability measure (see Harrison and Kreps, 1979 [11]), denoted here as 1. Random variables and stochastic processes are defined on a probability space with as the probability measure. An expected value taken with respect to is denoted by . To rigorously analyze stochastic processes, I work with a filtered probability space 0t, where is the set of outcomes, indexes time, is a filtration (i.e., a nondecreasing sequence of P t P P ,, t []E tt f ,P elds 0 i ), and fiel d . Stochastic processes are assumed to be adapted to tt 0 tt One of the assets traded in the financial market is a riskless bond fund with a share worth . 0 =rt t Me 0r on date , where 0 is an initial value and is a riskfree interest rate, which is assumed to be constant over time. In contexts involving asset pricing (e.g., valuation of derivative securities), such riskless fund is often used as a numéraire asset, with prices of other assets being discounted by t t>0M . Also, is often referred to as the “riskneutral” probability measure. P I focus on a stock process 0 tt, where t denotes the price of the stock on date . The stock is allowed to pay dividends continuously at a rate S tS 0 =0 , which is assumed to be constant over time ( in the case of no dividend). Since the stock process in the Bates model incorporates a jump component, which results in discon tinuities in the stock price, it is helpful to introduce the notion of a “left limit” of a stochastic process. In particular, the left limit of 0 tt S on date t is defined as 1 mn t =li SS t n , where is a positive integer. If there is a jump on date , then . nttt In the Bates model, the dynamics of t under P are described by a system of two stochastic differential equa tions: SS S 1 =, t tt dSdtv dWUdN tt S r (1) 2 =. ttt dv t vdtv dW (2) Equation (1) shows that the instantaneous net return on the stock, tt dS S , is a sum of three distinct com ponents: 1) a deterministic drift term ; 2) a stochastic diffusion term rd t 1tt 1 W vdW , and (3) a stochastic jump term t UdN . A process 0 tt underlying the stochastic diffusion term is a standard Brownian motion. A process N0 tt underlying the stochastic jump term is a Poisson process with intensity 0 , so that = t EN t . The processes 10 tt and are independent of each other. A value of in dicates that the stock price has undergone t jumps as of date . The magnitudes of such jumps are governed by independent and identically distributed (i.i.d.) ran dom variables such that W 0 tt N >0 t N N t 12 ,,UU 22 ln1ln1/ 2,,UN (3) where is a generic random variable having the same distribution as 12 , and U ,,UU>1 =1 t and are the distribution parameters. In Equation (1), is the random percentage jump of the stock price given a jump occurring at (i.e., given ). The Bates model reduces to the Heston stochastic volatility model (Heston, 1993 [2]) if (1) 20 U tdN =0 , or (2) =0 and , since these cases effectively eliminate jumps from the stock dynamics. 2=0 Equation (2) describes a meanreverting square root 1P need not be unique, as the financial market may be incomplete. Copyright © 2012 SciRes. TEL
O. ZHYLYEVSKYY 402 process for the squared volatility2. This equation is bor rowed directly from the Heston model. A process 0 tt is a standard Brownian motion possibly cor related with 10 {, so that 2 W } tt W12 ,= t dWW dt , with 1 . The process 2tt W 12 ,...UU 0, process 0 tt, and the random variables are mutually independent. Constants N , 0 , 0 , and 0 are parameters. In order for to be almost surely (a.s.) positive so that t and t are realvalued a.s., t v vS and are assumed to satisfy a restriction 22 (see Chernov and Ghysels, 2000 [12]). 3. Dynamics of LogPrice Let t denote the stock’s logprice, t =ln t S. The dy namics of t under are derived using a generalized Itô formula for semimartingales, which allows me to properly account for possible discontinuities in the time path of the stock price. See Theorem 32 of Protter (1990 [13], p. 71) for details on the formula. Before applying the Itô formula, observe that Equation (1) implies that t and t are, in general, not equal to each other because of the presence of the jump term; more speci fically, tt Thus, P = t SU SS SS. t dN =1 tt t SS UdN , and therefore, lnln= ln1. tt t SS UdN Also, note that by the properties of the Poisson process, is effectively either 0 or 1. Therefore, t dN lnln =ln1=ln1. tt t SS UdN UdN t Hence, the generalized Itô formula applied to the function ln t S indicates that the dynamics of t under are described by a stochastic differential equation P 2 2 111 =2 lnln, tt tt tt ttttt dsdSv Sdt SS SSSSS which is straightforward to simplify as: 1 =2ln1. 12 ,,UU fi e l d ttttt dsrvdtvdWUdN h (4) 4. Martingale Property and Dynamics of Joint Characteristic Function My main interest lies in deriving a closedform ex pression for the joint characteristic function of some future, date logprice and squared volatility given their present, datet values, where . Consider an arbitrary date such that , and note that since , . Equations (2) and (4), the properties of the Poisson process and standard Brownian motion, and the assumption of i.i.d. random variables imply that the information contained in the T h <tT Th <th <tt h relevant for conditioning the joint dis tribution of T and T on h comprises the values of v h and h, and the time remaining at until , vhT 0 Th . Thus, let 12 ,;,, hh vT h denote the joint conditional characteristic function of T , T v given , hh v, evaluated at real arguments 1 and 2 . By definition of the characteristic function, 12 is v TT h e 12 is v TT t e , hh sv 12 ,; 12 ,; t ,T ,T =E =E   . h t Likewise, , hh sv . Since , the law of iterated expectations im plies that h 12 12 ,; 12 = =;,, v TTt v TT hh T te Es 12 , is is e ,, tt =E EE sv a t h .s., ht vT which shows that 12 ,; :0 , tt ttT svt ,T is a martingale. The martingale property of implies that 12 ,;s,= s tt t v ,T t0 a..Ed In what follows, it is helpful to denote by the dura tion of the time interval between and T, t=Tt . Also, observe that Equation (2) implies that has a continuous time path; therefore, v. In comparison, Equation (4) implies that t v = tt v t may have discontinuities in its path, with 1=ln tt t sUdN . The goal is to find a solution for as a function with continuous second order partial derivatives. Apply ing the generalized Itô formula, 12 12 12 12 , , ; , ; 1tt dt tW ,; ds, tt , ;,, ,, 2 ,, t t stt t vtt v s sv rv sv = v dvd 2 t dW 12 12 ;,, ;,, , tt sst t vvt t t t t v sv 2 t t vdt 12 , , , ; vd 0.5 0.5 sv vv , vdt d dt ,Nv sv 12 12 , ,ln1 ,;,,, tt t tt sv ; t sU 2In applications, the unobserved value of is often treated as an ad t v ditional parameter to estimate. Copyright © 2012 SciRes. TEL
O. ZHYLYEVSKYY 403 and y where symbolic terms of the form de note partial derivatives / and 2/ y , re spectively. By the properties of the Poisson process and mutual in dependence of and Ut dN , 12 12 12 12 ,; ln1,, ,;,, =,;ln1,, ,;,,. ttt ttt tt tt EsUdNv sv EsUv sv dt t Then, applying the relationship =0 t Ed ownian motion, it is and the properties of the standard Br straightforward to show that the function must satisfy the following partial differentialintegral etion: qua 2 12 22 ,;n 12 2 l1,, ,;,,, 0= tv t tt tt v v Uv sv where the arguments of the partial derivatives and the term are omitted to shorten the notation. Note that in the special case of ss tvv tsvt t vvv Es (5) r dt =0 , 12 =exp TT is v . 5. ClosedForm Solution for Joint Characteristic Function Solving for the joint characteristic function he eq solution com in closed form using Equation (5) presents a substantial ana lytical challenge. My approach to address this problem is to first propose a general form of a solution to tua tion, and then analytically derive all of the  ponents. Suppose that is of the form: 1212 12 11 ,;,, =exp;,;, tt t t vpq is v (6) where 12 ;,p and 12 ;,q are complex valued functions of tytically, ano be sd olved for anal 1 is complexvalued and constant with respect to t , t v, and . The expression for i shortly. expression for s provided Differentiating the in Equation (6): 12 12 1212 1 ,;,, =,;,, ;,,, tt tt t sv sv pqv 12 12 1 1212 12 ; ,;,, =,;,,, ,;,, =,;,,;,, st t tt vtt tt sv sv i svsv q 2 1212 1 ,;,, =,;,,[], ,;,, tt tt svsv i sv sv 2 12 12 12 =,;,, [;,], ss vvttt tq and ,;,,sv 12 1 2 =, ;,, , tt sv iq wher 12 1 ;, svt t =pdpd and =qdqd . ln 1U e Next, recall from Equation (3) that is a normal random variable. By assumption,pen dent of the information contained inusing Equation (6), it is inde t. Thus, 12 12 12 1 12 12 1 12 2 111 ,;ln1,, ,;,, =,;,,expln1 ,;,, =,;,,expln11 =,;,, expln121, ttt tt tt t tt tt tt EsUv sv EsviU sv sv EiU sv ii w res ress here the last equality follows from the properties of the characteristic function of a normal random variable (see Chung, 2001 [14], p. 156). Given this ult, it is con venient to exp as follows: 2 11 11 =exp ln121ii . (7) Then, the integral term in Equation (5) is 12 12 12 1 ,;ln1,, ,;,, =,;,,. tt tt tt EsUv sv sv t By plugging in the obtained expressions into Equation (5) and simplifying it (e.g., note that is differenced out), I get 12 1 2 2 121 12 ; ,2 ;,2;,.qiq 12 112 2 12 1 0= ;,;, ;,2 t pir qv qiq is equation mpar ticular value of , the functions Since thust hold irrespective of a t v() and ()q need to solve the ng system of two tions: followi differential equa 12 112 ;, =;,pir q , Copyright © 2012 SciRes. TEL
O. ZHYLYEVSKYY 404 2 12 11112 2 2 ;, =22;, qiiq 12 ;,2.q Observe that the relationship 121 2 12 12 1 ,;,,0 exp = exp0;, 0;, TTTT TT svi sv p qvi s im i plies that the system has initial conditions 12 0;,=0p and 122 0; ,=q . The system is similar, although not identical, to a system of differential equations analyzed by Zhylyevskyy (2010) [10] in the case of the Heston model. By appro priately modifying the prior analysis, a closedformolu tion for and s p q 0 , in the case of the Bates model studied here, can be split into three cases. In Case ( beloter 1) w, parame , which implies that the square voIn comparison, Case (2) and ns for and d latility process is stochastic. Case (3) provide solutio p q when =0 , that is, under the spec stan nonstochastic stock volatility3. Case (1). Suppose that ial circumces of 0 . Let 1 A and 12 ,B be complexvnd ctant with respect alued aons to , and defined as follows: 222 11 =1 2Ai 2 2 , 1 2 112 12 2 112 ,= . iAi BiAi Then, 12 1 2 ;, = 1 2ln, 1 pr i B ABe (8) A 12 2 ;, = ,qiA 1 1 A A Be (9) 1 1Be where 1 AA and 12 ,BB at =0 as defined above. Ca se thse (2). Suppo but 0 . Then, 12 1 2 211 2 11 2 ;, = 21 2 1 , 2 pri eii ei 22 12112 11 1 ;, =2. 2 qeii i ) (11 Case (3). Suppose that (10) =0 and =0 . Then, 12 1 2 11 2 4 ;, = 4, pri ii (12 ) 2 ;, .qi 12 1 12 =2i (13) Together, the expression for in Equation (6), the expression for in Equati), and th for on (7e solution and q (1 f p given byons (8 respectively (alternatively, E10) and (11) or Eq (12)3), resp depending on the val Equati quations ( ectively, ) and (9), uations particular and ues o and , as shown above), joint characteristic function provide a closedform, analytical expression for the of T and T v, conditional on t and 6. Applications of Joint Characteristic The derived joint characteristic function may be em ployed in asset pricing applications. To illustr provide two examples related to implementing the trans form methods of Duffie et al. (2000) [6] and Bakshi and Madan (2000) [7] to price derivative securities dynamics of the Bates model. These method in ve expression for er t v. Function ate its use, I  under the s require verting a conditional characteristic function of an un derlying stateprice ctor. Thus, knowledge of a closed form the characteristic function, such as the one obtained in this paper, is essential for their imple mentation. In the first example, I considthe problem of deter mining the values of Europeanstyle derivative securities. Let ,,, Ett PXSv denotee datevalue of a Euro th with a strike t price peanstyle put option and time to expiration , gi exerc ven the (current) ing stock’s ion is allised on date d its date un T derly , an price t S and squared volatility t v. This put opt owed to be T value is ,,,0=max0, TT T PXSv XS. Likewise, let E ,,, Ett CXSv be the datet value of a correspond ing Europeanstyle call option; its dateT value is ,,,0=max0, ETT T CXSvSX. The dynamics of t S and t v, under the equivalent martingale probability measure, are described by Equations (1) and (2). Thus, there are two state variables (comprising the stateprice vector), namely, t S and t v, or equivalently, (the log price) t and t v. The analysis of Zhylyevskyy (2012) [15] adapted to the case of the Bates model indicates that the valuation of the options 3Observe from Equation (2) that the value of =0 eliminates the diffusion component from the dynamics of . t v E P and requires knowledge of E C Copyright © 2012 SciRes. TEL
O. ZHYLYEVSKYY 405 the marginal conditional characteristic function of the date T logprice T given t and t v. Let this fun ction ;,, tt sv be denoted as , where is a real number. By definition, ;,, =. is T tt t sv Ee A closedform expression for ;,,sv tt is easily obtained aof the epression for 12 ,;,, tt sv s a special case x , which was derived earlier. Namely, 0 = =,0;,,. is t is v TT t tt Ee sv ;, tt sv,=T Ee Then, applying Equation (10) of Zhylyevskyy (2012) [15], the value of E P is 0 ,,, =1 ;,,d, 2π Ett rtt PXSv X eX Rsv 2 i 12 e i where Re In tur denotes the real part of a colexvalued number.n, can be calculated ng a put ca A practical implementation of ese form for mp usi ulas E C ll parity relationship for Europeanstyle options (Mer ton, 1973 [16]): = ,,,. E r tt t PXSve Se X ,,, E tt C XSv th E P n to cal and would require nrical integratio E C culate the term ume dRe od qu 1 [17]). Not t approach t n of B posed here re ereas the cor , which is straightforward, ture me (Pres 200a these forulas pr a mfficienprice Eu riv r instance, for proires a singnu whonding fo Bates (see Bates, 1996 [4], tion (15) on p. 77) quires two separate integrations. In the second example, I cider the pro of pric uda cal ap ique (Geske and Johon, 1984 [8]) to a case of nonBlackScholes stock ics. In comparison to the first example in this section, which ut using the GaussKronr s et al., ore e C integration, adra bly, o . Fo qu resp Equa ons thod, for exam m ropeanstyle the fo le rm blem ns dynam ple ovide de rmula merical ula due to ative securities under the Bates model dynamics than the original solutioates E re ing Bermn and Americanstyle options by building on the methodologiproach developed by Zhylyevskyy (2010) [10]. The approach extends the GeskeJohnson compoundoption techn ilizes only a special case ;,, =,0;,, tt tt sv sv of the joint characteristic function , this second example requires knowledge of the value of 12 ,;,, tt sv for any combination of real numbers 1 and 2 , including cases of 20 . Let ,;,, TTtt fsvsv be the joint probability density func T tion of and T v, conditional on t and t v. The density function is an inverse Fourier transform of the characteristic function (see Shephard, 1991 [18]; Chung, 2001 [14]): 12 121 2 2 ,;,, 1 =,;,,dd. 2π TTtt is v TT tt fsvsv esv In practice, numerical values of can be ef ficiently computed using values of by applying a fast Fourier transform algorithmel smoothing (see Press ., 2001 [17]; Zh, 20[10]). Americanstyle put and call tions are similar to their t t time before ration E lity of an early exerciseantially c v ar se sol ption n their Euro pean and Americanstyle Bermudan style option is allowed to be fore expiration, only on a selected number ofmined dates. To clarify the iea, let s with ke ylyevskyy op ha e possi counterparts. A exercised be predeter tion rn et al (the e subst d 10 Europeanstyle counterparts, except the American style ones are allowed to be exercised at any uropeanstyle ones may be exercised y on thexpiration date). Th bi expi onl but omplicates the problem of de termining the alue ofn Americanstyle deivative  curity; thus, closedformutions are generally not available (Epps, 2000 [19]). Bermudanstyle os may be viewed as an intermediate case betwee =1 ,, ntt n DsvTt be a sequence of Bermudanstyle op, where n D is the value of an option that may be exercised on dates = j jT t tt n for =1, ,jn. In the sequence, 1 D represents the value of a Europeanstyle option, which may be ex ercised only once, on the expiration date, with =1n and 1=tT. 2 D is the value of a Bermudanstyle option that may be exercised on two dates, 1=2ttT (i.e., halfway to expiration) and 2=tT. 34 , ,DD are defined similarly. The limit of the sequence, D , corresponds to the value of an Americanstyle option, which features a continuum of possible exercise dates before expiration. Let the exercise value of the Bermudanstyle option n D on its first potential exte be de as ercise da 1>tt noted 1 11 ,, tt vTt. For example, 1 st =max 0, e Copyright © 2012 SciRes. TEL
O. ZHYLYEVSKYY 406 in the case of a put option and 1 =max 0, st eX in the case of a call option. Bermudanstyle derivative se curities obey a recursive relationship: 1 11 1 11 11 1 ,, = max,,,,, =max ntt rtt ttnttt rtt DsvTt e EsvTtDsvTt e where and 11 1 0 1 ,,,,, ,;,,, n tt s vTtDsvTt fsvsvt tdvds 00D 1 ,; , , tt svs v tt 12 1 ,;,, tt can be com putedrting by inve vt t relationship prov ny Bermudanstyl approximate the price of a tforwar 2,, tt DsvT t 3,, tt DsvTt, and t 2 , as discussed es a way to comice of ae derivative se curity, to n American style onevalue of . In pr d to coute and , and ifm hen 010 [10])is methodological approach is an alternative to pricing Americanstyle derivative securities under the Bates m dynamics using a finitedifferencetype scheme proposed by Chiarella et al. (2008) [9]4. 7. Conclusion This paper contributes to the literature by solving in closed form for the joint conditional characteristic function of nd d volatility undep ates model. The ving a syste REFERENCES earlier. In theory, the pute the pr as well as ,, tt vT t putationally feasible, id by choosing a sufficiently large actice, it should be straigh n mp co use these . Th 1 Ds computed values to approximate the Americanstyle option price ,, tt DsvTt by applying a Richardson extrapolation (e.g., see Zhylyevskyy, odel the logprice asquarer the jumdif fusion dynamics of the B model features a flexible distributional structure of asset returns. As such, it has a number of applications in finance and economics, including the problem of valuation of derivative se curities. Obtaining a closedform expression for the joint characteristic function is an analytically demanding task, which involves applying a generalized Itô formula for semimartingales and solm of differential equations, among other steps. The use of the derived function is illustrated on empirically relevant examples of pricing European, Bermudan, and Americanstyle options. The proposed methodological approach is based on inverting the characteristic function, and may be em ployed in practice as an alternative to pricing derivative securities using finitedifference techniques, particularly in the case of Americanstyle options. [1] Y. AïtSahalia and J. Jacod, “Analyzing the Spectrum of Asset Returns: Jump and Volatility Components in High Frequency Data,” Journal of Economic Literature, Vol. 50, No. 4, 2012, pp. 10071050. [2] S. L. Heston, “A ClosedForm Solution for Options with Stochastic Volatility with Applications to Bond and Cur rency Options,” Review of Financial Studies, Vol. 6, No. 2, 1993, pp. 327343. doi:10.1093/rfs/6.2.327 [3] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637654. doi:10.1086/260062 [4] D. S. Bates, “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options,” Re view of Financial Studies, Vol. 9, No. 1, 1996, pp. 69107. doi:10.1093/rfs/9.1.69 [5] Y. AïtSahalia and J. Jacod, “Testing for Jumps in a Dis cretely Observed Process,” Annals of Statistics, Vol. 37, No. 1, 2009, pp. 184222. doi:10.1214/07AOS568 [6] D. Duffie, J. Pan and K. Singleton, “Transform Analysis and Asset Pricing for Affine JumpDiffusions,” Econo metrica, Vol. 68, No. 6, 2000, pp. 13431376. doi:10.1111/14680262.00164 [7] G. Bakshi andand DerivativeSe curity Valuationics, Vol. D. Madan, “Spanning ,” Journal of Financial Econom 55, No. 2, 2000, pp. 205238. doi:10.1016/S0304405X(99)000501 [8] R. Geske and H. E. Johnson, “The American Put Option Valued Analytically,” Journal of Finance, Vol. 39, No. 5, 1984, pp. 15111524. [9] C. Chiarella, B. Kang, G. H. Meyer and A. Ziogas, “The Evaluation of American Option Prices under Stochastic Volatility and JumpDiffusion Dynamics Using the Me thod of Lines,” Quantitative Finance Research Centre, University of Technology, Sydney, 2008. O. Zhylyevskyy, “A Fast Fourier Transform Tec[10] hnique for Pricing American Options under Stochastic Volatil ity,” Review of Derivatives Research, Vol. 13, No. 1, 2010, pp. 124. doi:10.1007/s1114700990416 [11] J. M. Harrison and D. M. Kreps, “Martingales and Arbi trage in Multiperiod Securities Markets,” Journal of Eco nomic Theory, Vol. 20, No. 3, 1979, pp. 381408. doi:10.1016/00220531(79)900437 [12] M. Chernov and E. Ghysels, “A Study towards a Unified Approach to the Joint Estimation of Objective and Risk Neutral Measures for the Purpose of Options V Journal of Financial Economicaluation,” s, Vol. 56, No. 3, 2000, pp. 407458. doi:10.1016/S0304405X(00)000465 [13] P. Protter, “Stochastic Integration and Differential Equa tions: A New Approach,” SpringerV 1990. 4Chiarella et al. propose a method of lines, in which a partial differen tialintegral equation is replaced with a system of simpler differential equations to be solved using a stabilized finitedifference scheme. The integral component of the equation is approximated using an Hermite Gauss uadrature. erlag, New York, [14] K. L. Chung, “A Course in Probability Theory,” 3rd Edi Copyright © 2012 SciRes. TEL
O. ZHYLYEVSKYY Copyright © 2012 SciRes. TEL 407 ficient Pricing of EuropeanStyle tion, Academic Press, San Diego, 2001. [15] O. Zhylyevskyy, “Ef Options under Heston’s Stochastic Volatility Model,” Theo retical Economics Letters, Vol. 2, No. 1, 2012, pp. 1620. doi:10.4236/tel.2012.21003 [16] R. C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science, Vol. 4, No. 1, 1973, pp. 141183. doi:10.2307/3003143 [17] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, “Numerical Recipes in Fortran 77: The Art of Scientific Computing,” 2nd Edition, Cambridge Univer sity Press, Cambridge, 2001. for the Theory,” [18] N. G. Shephard, “From Characteristic Function to Distri bution Function: A Simple Framework Econometric Theory, Vol. 7, No. 4, 1991, pp. 519529. doi:10.1017/S0266466600004746 [19] T. W. Epps, “Pricing Derivative Securities,” World Sci entific, River Edge, 2000.
