Numerical Solutions to an integro-differential parabolic problem arising in the pricing of financial options in a Levy market

Transcription

1 September 2, 2 7:6 Quantitative Finance methodsqfstyle Quantitative Finance, Vol., No., Sep 2, 5 Numerical Solutions to an integro-differential parabolic problem arising in the pricing of financial options in a Levy market Ionuţ Florescu, Maria Cristina Mariani and Granville Sewell 4 Dept. of Mathematical Sciences, Stevens Institute of Technology Department of Mathematical Sciences, The University of Texas at El Paso (September 2, 2) We study the numerical solutions for an integro-differential parabolic problem modeling a process with jumps and stochastic volatility in Financial Mathematics. We present two general algorithms to calculate numerical solutions. The algorithms are implemented in PDE2D, a general purpose, partial differential equation solver. Keywords: integro-differential equation, numerical methods, option valuation. Introduction In Financial Mathematics the old problem of finding the price of derivatives (options, futures, etc.) leads to the study of Partial Differential Equations. The standard type of equations obtained are of parabolic type. In recent years, the complexity of the models used has increased and in turn this lead to more and more complicated equations for the derivative prices. Of particular interest in a type of differential equations containing an integral term. These equations aptly named Partial Integro-Differential Equations (PIDE) are difficult to solve and numerical methods specially constructed for them are not easy to find. In Florescu and Mariani (2) we study these type of problems and we prove the existence of the solution under general hypotheses about the integral term. In the present work we are extending the work on PIDE by providing a completely novel algorithm which is suggested in the proof of existence of the solution. Additionally, we present another algorithm which is a more classical finite element scheme coupled with a discretization of the integral term. In our numerical applications the two schemes are convergent to the same solution. The work is structured as follows. In Sections and 2 we Corresponding author. mcmariani@utep.edu Quantitative Finance ISSN print/issn online c 2 Taylor & Francis DOI:.8/469768YYxxxxxxxx

2 September 2, 2 7:6 Quantitative Finance methodsqfstyle 2 I. Florescu, M.C. Mariani and G. Sewell introduce the problem, as well as previous results and definitions; in Section 3 we describe the two algorithms that we will use to find numerical solutions; in Section 4 we explain how do we find numerical solutions by using PDE2D, a general purpose partial differential equation solver which has been used to solve many mathematical finance applications. In Sections 4 and 5 we present and discuss the results obtained. 2. Problem Motivation and general PIDE results In financial mathematics, usually the Black-Scholes model (Black and Scholes 973), or variants of the Black-Scholes model (Duffie 2, Hull 28, Ikeda 989, Jarrow 22, Merton 992) have been used for pricing derivatives on the equity. By applying the fundamental theorem of asset pricing (Harrison and Pliska 98, Delbaen and Schachermayer 994) one obtains different types of backward parabolic partial differential equations. In all these models, an important quantity is the volatility which is a measure of the fluctuation (risk) in the asset prices, and corresponds to the diffusion coefficient in the Black-Scholes equation. 2.. Stochastic Volatility models In the standard Black-Scholes model, a basic assumption is that the volatility is constant. It was soon discovered that this assumption does not allow matching an entire option chain (option values for different strike values). Several models were proposed, allowing the volatility to be modeled as a stochastic variable the so called stochastic volatility models (Hull and White 987, Wiggins 987, Scott 987, Chesney and Scott 989, Stein and Stein 99, Hagan et al. 22, Heston 993). To exemplify, in Heston (993) model the underlying security S follows, ds t = µs t dt + σ t S t dz t, where Z = {Z t } t is a standard Brownian motion. Unlike the classical model, the variance v(t) = σ 2 (t) also follows a stochastic process given by dv t = κ(θ v(t)) dt + γ v t dw t, where W = {W t } t is another standard Brownian motion. The correlation coefficient between W and Z is denoted by ρ: Cov(dZ t, dw t ) = ρ dt This leads to a generalized PDE in two state variables and one temporal variable: 2 vs2 2 U S 2 + ργvs 2 U v S + 2 vγ2 2 U U + rs v2 S + [κ(θ v) λv] U v ru + U t =. We should mention that any stochastic volatility model (no matter how simple) has the capability of matching an entire option chain for a fixed maturity (Fouque et al. 2, Florescu and Viens 28).

3 September 2, 2 7:6 Quantitative Finance methodsqfstyle Numerical Solutions for PIDE s Jump models, Levy processes At about the same time as the stochastic volatility models were developed researchers argued that the bad fitting to real data was caused by the path continuity of the price process. Thus, the resulting model may have difficulties fitting financial data exhibiting large fluctuations. The necessity of taking into account large market movements and a great amount of information arriving suddenly (i.e., a jump) led researchers to propose models with jumps. Merton (992), Andersen and Andreasen (2) model the stock as a jump-diffusion model (a geometric Brownian motion plus a compound Poisson process multiplying the stock process). Under this modeling assumption one may derive differential equations for option prices but these equations will contain an integral term coming from the compensator of the Poisson process. This has led to the study of partial integro-differential equations (PIDE). As an example, in the above cited work, the derivative value F (S, t) solves the following PIDE under appropriate boundary conditions: 2 σ2 S 2 F SS + (r λk)sf S + F t rf + λe{f (SY, t) F (S, t)} =. (2.) Here r denotes the riskless rate, λ the jump intensity, and k = E(Y ), where E is the expectation operator and the random variable Y measures the percentage change in the stock price if the jump occurs The jump-diffusion component is a particular case of a Lévy process and indeed stock evolution was soon modeled using these more general processes (e.g., Barndorff-Nielsen et al. (998), Madan et al. (998), Cont and Tankov (23), Geman (22)). Similar with the jump-diffusion case, when using any of these more general models, the resulting partial differential equation for derivative prices contains an integral term coming from the associated Lévy measure. Practical applications confirm that a Lévy like stochastic process appears to be the best fit when modeling high frequency data (see Mariani et al. (29) and its references) Combining the stochastic volatility and the Levy component Each of the two proposed models are doing something different. One is allowing for varying fluctuations and the other cope well with large amount of information driving the price suddenly up or down. It is only natural to attempt and combine the two modeling assumptions. As we shall see this complicates the resulting equations for the stock price but they do remain of the same integro differential type. As a simple example we consider a jump-diffusion process with volatility replaced by a geometric Brownian motion: ds t = S t (µdt + σ t dz t + Y (g)dn t ) dσ t = σ t (αdt + βdw t ), where Z and W are two standard Brownian motion with correlation coefficient ρ, N t is a Poisson process with intensity λ, and Y (g) is the jump amplitude random variable with density g. The jump part may be written in a perhaps more traditional way as a compound Poisson process: Nt i= Y i. To obtain the price of a derivative in the model above we follow Merton (992) and we obtain

4 September 2, 2 7:6 Quantitative Finance methodsqfstyle 4 I. Florescu, M.C. Mariani and G. Sewell the following PIDE, F t + 2 σ2 S 2 2 F S σ2 β 2 2 F σ 2 + ρσ2 βs 2 F S σ + λ R + (r λk)s F S 2 ρσ2 β F σ [F (SY, σ, t) F (S, σ, t)]g(y )dy rf =. (2.2) Here, r denotes the riskless rate, and k = E (Y ), where E is the expectation operator, g is the density of the Y random variable. We mention that the quantity Y measures the percentage change in the stock price if the jump occurs, for more discussion see (Merton 992, Section 9.2). We note that the derivative price still solves a PIDE and this is a general characteristic of these type of processes A general existence result The previous discussion motivates to consider more general integro-differential parabolic problems. In recent work Florescu and Mariani (2) the authors prove the existence of solutions to a general partial integro-differential equation in an unbounded smooth domain (2.3) using the method of upper and lower solutions. Since we shall use some of the parts of the proof in the algorithm we state the problem and the essential results here as well. Let Ω R d an unbounded smooth domain, L a second order elliptic operator in non divergence form, i.e., Lu := d a ij (x, t)u xix j + i,j= d b i (x, t)u xi + c(x, t)u, with coefficients of L in the Hölder Space C δ,δ/2 ( Ω [, T ] ) and satisfying the following conditions: λ v 2 i= d a ij (x, t)v i v j Λ v 2 ( λ Λ) i,j= b i (x, t) C; c(x, t). These are classical conditions to insure the strict ellipticity of the operator. The problem we are solving is: Lu u t = G(t, u) in Ω (, T ) u(x, ) = u (x) on Ω {} u(x, t) = h(x, t) on Ω (, T ) (2.3) The operator G is a completely continuous integral operator as the ones defined in (2.) and (2.2), coming from the jump distribution. More precisely, we assume that G(t, u) = Ω g(x, t, u)dx, where g is any continuous function. The proof however is not constrained to integral operators, indeed any operator with the same properties will work. In this general model, the case in which g is increasing with respect to u and all jumps are positive corresponds to the evolution of a call option near a crash.

5 September 2, 2 7:6 Quantitative Finance methodsqfstyle Numerical Solutions for PIDE s 5 The existence result in Florescu and Mariani (2) reads as follows: Theorem 2. : Let L and G be the operators defined above. Assume that either: G is nonincreasing with respect to u, or there exist some continuous, and increasing one dimensional function f such that G(t, u) f(u) is nonincreasing with respect to u Furthermore, assume there exist α and β a lower and an upper solution of the problem with α β in Ω (, T ). Then, the problem (2.3) admits a solution u such that α u β in Ω (, T ). Remark : The assumptions above say that the result is applicable in particular for any G that is dominated by a polynomial. Remark 2 : We remind the interested reader that a smooth function u is called an upper (lower) solution of problem (2.3) if Lu u t ( ) G(t, u) in Ω (, T ) u(x, ) ( ) u (x) on Ω {} u(x, t) ( ) h(x, t) on Ω (, T ) The new algorithm we propose is inspired by the proof of the theorem and we outline here the main steps in the proof. We construct a series of regular PDE discretizations (for n {, 2,..., K}) for the problem (2.3), Lu n+ u n+ t = G(t, u n ) in U (, T ) u n+ (x, ) = u (x) in U {} u n+ (x, t) = φ U (x, t) in U (, T ) Note that the boundary conditions are identical with the original problem 2.3, but by using the known solution from the previous step in the G term the main equation loses its integral term and becomes a regular PDE. (2.4) Under the hypotheses given each of the systems (2.4) have a unique solution u n+ W 2, p (V ) where W 2, p (V ) = {v L p : v xi, v xi x j L p, v t L p } (see e.g., Krylov (996), Lieberman (996)) We prove that α u n β, n using the special properties of L and G We show that {u n } is a Cauchy sequence in Wp 2, (V ) using that G is a completely continuous operator, the previous step and the Lebesgue s dominated convergence theorem We conclude that u n u in the Wp 2, -norm, and then u is a strong solution of the problem. 3. Algorithms for numerical solutions We propose two algorithms to find the solution of the problem (2.3). The first algorithm is new and is inspired by the constructive proof of the theorem. It can be considered that this algorithm handles the integral part of the PIDE using an implicit method

6 September 2, 2 7:6 Quantitative Finance methodsqfstyle 6 I. Florescu, M.C. Mariani and G. Sewell because, after convergence, the u appearing in the integral term is evaluated at the current value of t. The second algorithm is a more standard explicit approximation, where the u appearing in the integral is evaluated at the previous time step. We mention Cont and Voltchkova (25) where the authors provide a similar approach for a more specific PIDE in only one spatial dimension and using a finite difference scheme (we are using finite elements). We have no proof of convergence for this algorithm and indeed we observed that for stability the time step had to be less than a threshhold value, which decreases as λ increases. This is not surprising, since while both algorithms use implicit methods for the partial differential equations, the second algorithm handles the integral part of the PIDE using an explicit scheme. The purpose of presenting this algorithm is twofold. One, we want to compare the solution provided by this algorithm with the solution provided by the other algorithm in hopes of showing that under reasonable parameter values and discretizations the two solutions are close. Two, this algorithm is much faster than the other algorithm (in our tests 5 times faster) and thus there is interest in applying it in the financial applications where computer time is very important. 3.. Algorithm The new algorithm we propose comes from the proof of theorem 2.. We first find α() lower and β() upper solutions of the problem and we ensure that the operators L and G verify the hypotheses of the theorem. We start with u = α the lower solution. For every n we solve the system (2.4). The PDE in this system cannot be solved analytically and we implement a finite element scheme for it. We take a three dimensional grid in t [, T ], S [, S max ], σ [, σ max ]. The upper bounds are suitably chosen large numbers. This grid is kept the same for all iterations. We approximate the integrals in G(t, u n ) using a midpoint rule, with the required values of u n (S, σ, t) interpolated from the values saved at all the points on the grid, using quadratic interpolation in S and σ and linear interpolation in t. Since the boundary condition is usually at T we solve the resulting partial differential equation backwards in time using a PDE solver (we will use PDE2D in our experiments). The result is u n+ calculated at all grid points and to be used in the next iteration. The u n sequence converges to the solution of the main system (2.3). Thus the algorithm stops when the maximum difference between two consecutive iterations at all points on the grid is small Algorithm 2 Algorithm 2 may be viewed as an explicit numerical scheme. It does not use the discretization (2.4), instead it works directly with the given system (2.3). It works backward in time from t = T. Specifically: Starts with the final condition at T. When a PDE solver (we will use PDE2D) is used to solve (2.3) backwards from t i+ to t i, G(t, u(t)) is replaced by G(t i+, u(t i+ )), the integral is approximated using a midpoint rule, and the required values of u at t i+ are approximated using quadratic interpolation to the values saved at the S, σ grid points. When t = we stop. The solution approximation is obtained for all grid points.

7 September 2, 2 7:6 Quantitative Finance methodsqfstyle Numerical Solutions for PIDE s 7 4. Numerical solutions using PDE2D In order to test the two algorithms we consider the PIDE resulting from the model combining stochastic volatility and jumps (system (2.2)). In this system we use the following parameter values. Parameters characterizing the stock: initial value S =, riskfree rate r =.5, volatility of volatility β =.4, correlation ρ =.3. The drift parameters µ and α do not enter in the PIDE. Parameters characterizing the jump component: jump frequency λ = 5 (expected five jumps per year) and the expected jump percentage k =.5, which can be calculated from the jump distribution. We are using a Gaussian mixture model for g(y ) =.75 g (Y ) +.25 g 2 (Y ), where g (Y ) and g 2 (Y ) are Normal densities with means.7 and.3 respectively, and variances equal to.. The values are chosen to insure that a jump is noticeable and separable from the stochastic variability of the model. Fan and Wang (27) show that in the presence of both jumps and stochastic volatility it is hard to separate the small sized jumps from the large stock variability. Both algorithms are solved for the value of an European Call in this model. Specifically, the option maturity was T =, and the strike price K =. The boundary conditions characterizing this option type were: F (S, σ, T ) = max(s K, ), F (, σ, t) =, and F S (S max, σ, t) =. We had no boundary conditions at σ = or σ = σ max. The algorithms were implemented using PDE2D, a general purpose partial differential equation solver (Sewell 25, 2) available from Visual Numerics, Inc. ( which has been used to solve many mathematical finance applications (Topper 25). PDE2D solves linear or nonlinear, steady-state, time-dependent and eigenvalue problems, in D intervals, general 2D regions (with curved boundaries), and a wide range of simple 3D regions. It has a sophisticated GUI interface which makes it extremely easy to use. It can solve D, 2D or 3D problems similar to the problems presented (2.),(2.2), using a collocation finite element method with cubic Hermite basis functions to discretize the spatial derivatives and adaptive finite difference methods to discretize the time derivatives. Without the integral term, a problem such as (2.2) is a very straightforward PDE2D application, and can be solved with little user effort. The integral term in (2.2), however, required special efforts. The two algorithms were implemented using PDE2D, and we present below the specific details of each implementation. For both algorithms we use S max = 4, σ max =. A equally spaced grid was constructed on [, T ] [, S max ] [, σ max ]. A simple modification of the example 2.3 in Florescu and Mariani (2) shows that the operators L and G would satisfy the hypotheses of the theorem if σ min >, and that upper and lower solutions exist with the lower solution α. Algorithm implementation. The iteration (2.4) is repeatedly solved backward in time from a final condition using PDE2D. The first iteration uses u = α =, so that no integral term appears, and the solution was saved for each point on the grid. Each iteration thereafter, (2.4) is solved with the F in the integral term approximated by interpolating the solution saved on the previous iteration, using quadratic interpolation in S and σ, and linear interpolation in time. For our examples, g(y ) was negligible outside of < Y < 2, so the integral limits in (2.4) were taken to be [, 2], and a midpoint rule was used to numerically approximate the integral. Since Y can be as large as 2, however, values of F are needed at points SY beyond S max, where they have not been calculated. For these values, a value of F is extrapolated using the boundary condition F S =, that is: F (SY, σ, t) F (S max, σ, t) + SY S max

8 September 2, 2 7:6 Quantitative Finance methodsqfstyle 8 I. Florescu, M.C. Mariani and G. Sewell σ 5 S 5 Figure. F(S,σ,) for the integral term equal to (Algorithm first iteration) Algorithm 2 implementation. This time (2.2) is only solved once using PDE2D, and at each time point t i+ = (i + ) t ( t = T/), PDE2D saves the solution on the S σ grid. Then when it integrates backward from t i+ to t i, the terms F (SY, σ, t) and F (S, σ, t) appearing in the integral are approximated by F (SY, σ, t i+ ) and F (S, σ, t i+ ), and these values are obtained using quadratic interpolation to the solution saved at the grid points at t = t i+. The numerical approximation to the integral is the same as in Algorithm. 4.. Results The solution is a function of three variables, so it is impossible to view on a regular 3D plot. For this reason we only plot the solutions when t = since usually this is the most relevant value. Figure shows the solution from Algorithm after the first iteration. This is the same as the homogeneous problem with λ =, that is, without the integral term, corresponding to the stochastic volatility model where the jump diffusion term does not exist. This figure is provided for comparison to view the effect of the integral term in the PIDE.

9 September 2, 2 7:6 Quantitative Finance methodsqfstyle Numerical Solutions for PIDE s σ 2 S 3 4 Figure 2. The final solution F(S,σ,) after convergence of Algorithm Algorithm showed significant oscillations near σ = for the first few iterations, but the oscillations damped after further iteration, and (2.4) converged after 3 iterations, using our stopping criterion of maximum difference < 3 5. The oscillations are believed to be due to the lack of a boundary condition at σ =. Figure 2 shows F (S, σ, ) after convergence, and Figure 3 shows the same solution in the vicinity of the origin. Algorithm 2 produced a solution which agreed with that produced by Algorithm closely (which is why we do not plot it), the maximum difference between the two solutions on the three dimensional grid was about.2. Since Algorithm 2 does only one iteration (only solves (2.2) once), it is expected to be the more efficient algorithm, and indeed it required about 5 times less computer time for this example. To give more details about the order of magnitude of the difference between the two solutions we calculate and plot in Figure 4 the Root Mean Squared Error defined as RMSE(t) = n (F (S, σ, t) F 2 (S, σ, t)) 2, and the Mean Absolute Deviation defined as MAD(t) = n F (S, σ, t) F 2 (S, σ, t). F and F 2 are the solutions obtained using Algorithm respectively Algorithm 2 and the sum runs over all the grid points. At t = we do not plot since the two solutions coincide with the boundary value. Finally, since at t = we know the S variable but the σ is unobservable we present in Figure 5 the option price evolution depending on the realized volatility value σ.

10 September 2, 2 7:6 Quantitative Finance methodsqfstyle I. Florescu, M.C. Mariani and G. Sewell σ 5 S 5 Figure 3. The final solution F(S,σ,) a closeup at origin 4.2. A discussion of the influence of the integral term The approach we present in the current article deals specifically with partial differential equations containing an integral term. As we may observe in any of the equations presented, λ, the parameter which governs the frequency of the jumps, ends multiplying the integral term in the equation. Thus we chose to present the results when λ increases in value and thus the integral term becomes dominant. In Figure 6 we present the graphs of the solution at t =, obtained when λ changes in value. We see in Figure 6(a) the result of the regular PDE with no integral term, and as λ increases the integral term becomes dominant. Note that when λ = (Figure 6(d)) the solution is almost F = S, which is to be expected since F = S is the steady-state solution when λ is infinite (and satisfies the boundary conditions).

11 September 2, 2 7:6 Quantitative Finance methodsqfstyle Numerical Solutions for PIDE s Root mean squared error Mean absolute deviation times Figure 4. Two measures of difference between the two solutions 5. Conclusions In this article we present a completely novel algorithm proven to converge to solve partial differential equations with an integral term. We compare the new algorithm with another more explicit algorithm that is not proven to converge. In our numerical example we found the difference between the two solutions to be minimal. This opens the way for numerical schemes to approximate more general PIDE produced by Lévy models of the stock price. The algorithms presented should work for these problems although a more detailed study of convergence for Algorithm 2 should be performed. In our numerical experiments we discovered that increasing λ produced instability which was fixed by using a smaller time step. This is very similar with

12 September 2, 2 7:6 Quantitative Finance methodsqfstyle 2 REFERENCES option Figure 5. The call option value depending on the volatility value at time t = vol the traditional Explicit Finite Difference scheme and a relation between parameters that would guarantee convergence would be very beneficial. References Andersen, L. and J. Andreasen (2). Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing. Review of Derivatives Research 4, Barndorff-Nielsen, O. E., J. L. Jensen, and M. Sorensen (998, Dec.). Some stationary processes in discrete and continuous time. Advances in Applied Probability 3 (4), Black, F. and M. Scholes (973). The valuation of options and corporate liability. Journal of Political Economy 8,

13 September 2, 2 7:6 Quantitative Finance methodsqfstyle REFERENCES F 4 F σ 5 S 5.5 σ 5 S 5 (a) λ = (b) λ = 2 5 F F 5.5 σ 5 S 5.5 σ 5 S 5 (c) λ = (d) λ = Figure 6. Changes in the solution at t = when the importance of the integral term increases Chesney, M. and L. Scott (989). Pricing european currency options: a comparison of the modified black-scholes model and a random variance model. J. Finan. Quant. Anal. 24, Cont, R. and P. Tankov (23). Financial modeling with jumps processes. CRC Financial mathematics series. Chapman & Hall. Cont, R. and E. Voltchkova (25). A finite difference scheme for option pricing in jump diffusion and. SIAM Journal of Numerical Analysis 43 (4), Delbaen, F. and W. Schachermayer (994). A general version of the fundamental theorem of asset pricing. Mathematische Annalen (3), Duffie, D. (2). Dynamic Asset Pricing Theory (3rd ed.). Princeton University Press. Fan, J. and Y. Wang (27). Multi-scale jump and volatility analysis for high-frequency financial data. Journal of the American Statistical Association 2, Florescu, I. and M. Mariani (2, May). Solutions to integro-differential parabolic problems arising in the pricing of financial options in a levy market. Electronic Journal of Differential Equations 2 (2)(62),. Florescu, I. and F. Viens (28, April). Stochastic volatility: option pricing using a multinomial recombining tree. Applied Mathematical Finance 5 (2), 5 8.

14 September 2, 2 7:6 Quantitative Finance methodsqfstyle 4 REFERENCES Fouque, J.-P., G. Papanicolaou, and K. R. Sircar (2). Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press. Geman, H. (22). Pure jump Levy processes for asset price modeling. Journal of Banking and Finance 26, Hagan, P., D. Kumar, A. Lesniewski, and D. Woodward (22). Managing smile risk. Wilmott Magazine. Harrison, M. and S. Pliska (98). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications, Heston, S. L. (993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6 (2), Hull, J. C. (28). Options, Futures, and other Derivatives (7th ed.). Prentice Hall. Hull, J. C. and A. D. White (987, June). The pricing of options on assets with stochastic volatilities. Journal of Finance 42 (2), Ikeda, N. (989). Stochastic Differential Equations and Diffusion Processes (2nd Revised ed.). North-Holland. Jarrow, R. A. (22). Modelling Fixed Income Securities and Interest Rate Options (2nd ed.). Stanford Economics and Finance. Krylov, N. (996). Lectures on elliptic and parabolic equations in Hölder Spaces, Volume 2 of Graduate Studies in Mathematics. American Mathematical Society. Lieberman, G. M. (996). Second order parabolic differential equations. World Scientific Publ. Madan, D. B., P. P. Carr, and E. C. Chang (998). The variance gamma process and option pricing. European Finance Review 2 (), Mariani, M., I. Florescu, M. B. Varela, and E. Ncheuguim (29, April). Long correlations and levy models applied to the study of memory effects in high frequency (tick) data. Physica A 388 (8), Merton, R. (992). Continuous-Time Finance. Wiley-Blackwell. Scott, L. O. (987). Option pricing when the variance changes randomly: Theory, estimation, and an application. The Journal of Financial and Quantitative Analysis 22 (4), Sewell, G. (25). The Numerical Solution of Ordinary and Partial Differential Equations (2nd ed.). John Wiley & Sons. Sewell, G. (2, May). Solving pdes in non-rectangular 3d regions using a collocation finite element method. Advances in Engineering Software 4, Stein, E. M. and J. C. Stein (99). Stock price distributions with stochastic volatility: An analytic approach. Review of Financial Studies 4 (4), Topper, J. (25). Financial Engineering with Finite Elements. John Wiley & Sons. Wiggins, J. B. (987, December). Option values under stochastic volatility: Theory and empirical estimates. Journal of Financial Economics 9 (2), Dr. Ionuţ Florescu ( Ionut.Florescu@stevens.edu) Department of Mathematical Sciences, Stevens Institute of Technology Castle Point on Hudson, Hoboken, NJ 73 USA Dr. Maria Christina Mariani ( mcmariani@utep.edu) Department of Mathematical Sciences, The University of Texas at El Paso Bell Hall 24, El Paso, Texas , USA

15 September 2, 2 7:6 Quantitative Finance methodsqfstyle REFERENCES 5 Dr. Granville Sewell ( sewell@utep.edu) Department of Mathematical Sciences, The University of Texas at El Paso Bell Hall 24, El Paso, Texas , USA