 Theoretical Economics Letters, 2012, 2, 400-407 http://dx.doi.org/10.4236/tel.2012.24074 Published Online October 2012 (http://www.SciRP.org/journal/tel) Joint Characteristic Function of Stock Log-Price 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 closed-form expression for the joint conditional characteristic function of a stock’s log-price 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 American-style options. The discussed approach for European-style derivatives improves on the option formula of Bates. The suggested approach for American-style derivatives, based on a compound-option technique, offers an alternative solution to exist- ing finite-difference methods Keywords: Bates Model; Stochastic Volatility; Jump-Diffusion; Characteristic Function; Option Pricing 1. Introduction Stochastic volatility and jump-diffusion are standard tools of modeling asset price dynamics in finance research (see Aït-Sahalia and Jacod, 2011 [1]). Popularity of stochastic volatility models, such as the continuous-time 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 time-series, which may feature discontinuities (for a discussion on jumps in asset data, see Aït-Sahalia and Jacod, 2009 [5]). In this paper, I analytically derive and provide exam- ples for the use of a closed-form expression for the joint conditional characteristic function of a stock’s log-price 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. 72-73). 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 jump-diffusion dynamics with stochastic vola- tility, the values of derivative securities such as Euro- pean-style 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 state-price vector. Bates (1996) [4] solved for the marginal conditional characteristic function of the logprice and derived a formula for the value of a European-style call option that involves two separate inversions. In contrast, the problem of finding the joint conditional characteristic function of the log-price 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 closed-form 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 European-style 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 European-style 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 American-style options by proposing an extension of the Geske-Johnson compound-option 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 American-style options using finite-difference 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 log-price. Section 4 shows that the joint conditional characteristic function is a martingale and uses this result to derive a partial dif- ferential-integral equation for the function. Section 5 solves this equation analytically to obtain a closed-form 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 non-decreasing 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 risk-free 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 “risk-neutral” 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 mean-reverting 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 real-valued a.s., t v vS and are assumed to satisfy a restriction 22 (see Chernov and Ghysels, 2000 [12]). 3. Dynamics of Log-Price Let t denote the stock’s log-price, 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 closed-form ex- pression for the joint characteristic function of some future, date- log-price and squared volatility given their present, date-t 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 differential-integral 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. Closed-Form 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 complex-valued 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 closed-formolu- 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 non-stochastic stock volatility3. Case (1). Suppose that ial circumces of 0 . Let 1 A and 12 ,B be complex-vnd 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 closed-form, 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 state-price 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 European-style derivative securities. Let ,,, Ett PXSv denotee date-value of a Euro- th with a strike t price pean-style 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 date-t value of a correspond- ing European-style call option; its date-T 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 state-price 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 log-price 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 closed-form 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 colex-valued 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 European-style 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 non-Black-Scholes stock ics. In comparison to the first example in this section, which ut using the Gauss-Kronr s et al., ore e C integration, adra bly, o . Fo qu resp Equa ons thod, for exam m ropean-style 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 American-style options by building on the methodologiproach developed by Zhylyevskyy (2010) [10]. The approach extends the Geske-Johnson compound-option 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]). American-style 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 American-style 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 European-style counterparts, except the American- style ones are allowed to be exercised at any uropean-style ones may be exercised y on thexpiration date). Th bi expi onl but omplicates the problem of de- termining the alue ofn American-style deivative - curity; thus, closed-formutions are generally not available (Epps, 2000 [19]). Bermudan-style os may be viewed as an intermediate case betwee =1 ,, ntt n DsvTt be a sequence of Bermudan-style 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 European-style option, which may be ex- ercised only once, on the expiration date, with =1n and 1=tT. 2 D is the value of a Bermudan-style option that may be exercised on two dates, 1=2ttT (i.e., half-way to expiration) and 2=tT. 34 , ,DD are defined similarly. The limit of the sequence, D , corresponds to the value of an American-style option, which features a continuum of possible exercise dates before expiration. Let the exercise value of the Bermudan-style 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. Bermudan-style 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 Bermudan-styl 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 American-style derivative securities under the Bates m dynamics using a finite-difference-type 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 American-style option price ,, tt DsvTt by applying a Richardson extrapolation (e.g., see Zhylyevskyy, odel the log-price asquarer the jum-dif- 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 closed-form 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 American-style 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 finite-difference techniques, particularly in the case of American-style options. [1] Y. Aït-Sahalia 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. 1007-1050. [2] S. L. Heston, “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Cur- rency Options,” Review of Financial Studies, Vol. 6, No. 2, 1993, pp. 327-343. 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. 637-654. 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. 69-107. doi:10.1093/rfs/9.1.69 [5] Y. Aït-Sahalia and J. Jacod, “Testing for Jumps in a Dis- cretely Observed Process,” Annals of Statistics, Vol. 37, No. 1, 2009, pp. 184-222. doi:10.1214/07-AOS568 [6] D. Duffie, J. Pan and K. Singleton, “Transform Analysis and Asset Pricing for Affine Jump-Diffusions,” Econo- metrica, Vol. 68, No. 6, 2000, pp. 1343-1376. doi:10.1111/1468-0262.00164 [7] G. Bakshi andand Derivative-Se- curity Valuationics, Vol. D. Madan, “Spanning ,” Journal of Financial Econom 55, No. 2, 2000, pp. 205-238. doi:10.1016/S0304-405X(99)00050-1 [8] R. Geske and H. E. Johnson, “The American Put Option Valued Analytically,” Journal of Finance, Vol. 39, No. 5, 1984, pp. 1511-1524. [9] C. Chiarella, B. Kang, G. H. Meyer and A. Ziogas, “The Evaluation of American Option Prices under Stochastic Volatility and Jump-Diffusion 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. 1-24. doi:10.1007/s11147-009-9041-6 [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. 381-408. doi:10.1016/0022-0531(79)90043-7 [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. 407-458. doi:10.1016/S0304-405X(00)00046-5 [13] P. Protter, “Stochastic Integration and Differential Equa- tions: A New Approach,” Springer-V 1990. 4Chiarella et al. propose a method of lines, in which a partial differen- tial-integral equation is replaced with a system of simpler differential equations to be solved using a stabilized finite-difference 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 European-Style 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. 16-20. 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. 141-183. 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. 519-529. doi:10.1017/S0266466600004746 [19] T. W. Epps, “Pricing Derivative Securities,” World Sci- entific, River Edge, 2000.
|
