13034 Examensarbete 15 hp Juni 2013 Discontinuous Galerkin and BDF-2 for time-integration and adaptive finite differences to price options Johan Ekegren Gunnarsson Martin Pettersson Abstract Discontinuous Galerkin and BDF-2 for time-integration and adaptive finite differences to price options Johan Ekegren Gunnarsson, Martin Pettersson Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student In this paper we investigate different types of time integrators for the parabolic Black-Scholes equation used for option pricing. This problem has been solved in one and two dimensions. In one dimension it is shown that discontinuous Galerkin (dG) with uniform time steps gives better results than different versions of backward differentiation of second order (BDF2). BDF2 was implemented with two types of time adaptive grids to see if the method could be improved compared to equidistant grids. Although the local truncation error was minimized when using time adaptivity, the global errors grew and the methods failed. For the two-dimensional problem which occur when pricing an option written on two underlying assets, we compare dG with equidistant BDF2. The result shows that dG was superior both with respect to the amount of time steps and the time of computation for a certain error. Handledare: Lina Von Sydow Ämnesgranskare: Petter Tammela Examinator: Martin Sjödin ISSN: 1401-5757, UPTEC F** *** Discontinuous Galerkin and BDF-2 for time-integration and adaptive finite differences to price options Department of Scientific Computing, Uppsala University Martin Pettersson, Johan Gunnarsson June 20, 2013 Abstract In this paper we investigate different types of time integrators for the parabolic Black-Scholes equation used for option pricing. This problem has been solved in one and two dimensions. In one dimension it is shown that discontinuous Galerkin (dG) with uniform time steps gives better results than different versions of backward differentiation of second order (BDF2). BDF2 was implemented with two types of time adaptive grids to see if the method could be improved compared to equidistant grids. Although the local truncation error was minimized when using time adaptivity, the global errors grew and the methods failed. For the two-dimensional problem which occur when pricing an option written on two underlying assets, we compare dG with equidistant BDF2. The result shows that dG was superior both with respect to the amount of time steps and the time of computation for a certain error. Keywords: Black-Scholes, Time adaption, Adjoint equation, dG, BDF2. 3 Contents 1 Introduction 4 2 Discontinuous Galerkin for Time Integration 6 3 Space Discretization 8 4 Time Adaption 4.1 Time Adaption minimizing the local truncation error . . . . . 4.2 Time Adaption using the Adjoint Equation . . . . . . . . . . 8 9 9 5 Numerical Result 12 5.1 Numerical Result for One-dimensional Problem . . . . . . . . 13 5.2 Numerical Result for the Two-dimensional Problem . . . . . . 15 6 Conclusion 17 4 1 Introduction An option is in financial terminology a contract between two parties written on some underlying financial assets, e.g. stocks or minerals. The option gives the holder the right but not the obligation to sell/buy (exercise) the underlying asset S prior to or at maturity time T at a predefined strike price K. There are numerous different types of options and one of them is the European option. The European option can only be exercised exactly at the time of maturity. In this paper we consider European call options. This option gives the holder the right to buy the underlying asset from the other party (the writer) at the time of maturity for price K. Since the value of an option depends on the value of the underlying asset, options and other related financial instruments are called derivatives [1]. Options have become a very important financial instrument and how to calculate a fair price, the premium, is of great interest for the financial market. Considering a European call option with price V , you have at time 0 purchased this option, giving you the right to buy the underlying asset at time T for price K. The option only has a value when the price of the asset is greater than the strike price at the time of maturity, i.e. ST > K. In summary, the value V (ST , T ) of a call option at expiration date T is given by ( 0, if ST ≤ K, V (ST , T ) = (1) ST − K, if ST > K. Hence the payoff function Φ(S) for V is given by Φ(S) = V (ST , T ) = max(ST − K, 0). (2) Figure 1: The payoff of the European call option at the time of maturity as a function of the value of the underlying asset. 5 A commonly used method for option pricing is the Black-Scholes partial differential equation which was formulated by Fischer Black and Myron Scholes in [2] and Merton in [9] both in 1973. For a dividend free asset in an efficient market the arbitrage free price of the option satisfies the following equation ∂V σ2S 2 ∂ 2V ∂V + + rS − rV = 0, 2 ∂t 2 ∂S ∂S V (ST , T ) = Φ(S), (3) where σ is the volatility of S, r is the short rate of interest and Φ(S) is the payoff function. If the option is written on several underlying assets we have to solve the multi-dimensional Black-Scholes partial differential equation d d X ∂V ∂V 1 X ∂2V + rsi + [σσ ∗ ]ij si sj − rV = 0, ∂t ∂si 2 ∂si ∂sj i=1 i,j=1 (4) F (T, s) = Φ(s). Instead of working with a final value problem a transformation into an initial value problem can be made. This gives the advantage that standard form of time integrators are applicable. This transformation together with transforming all variables to dimensionless quantities is shown in [3] and is given by Kx = S, KP (t̂, x) = V (t, s), KΨ(x) = Φ(s), 1 r [σ̄σ̄ ∗ ] = σ −2 [σσ ∗ ], t̂ = σ̂ 2 (T − t), r̄ = 2 , 2 σ̂ (5) where σ̂ 2 is chosen as σ̂ 2 = maxij [σσ ∗ ]ij . Other choices of σ̂ to scale σ can be used. These transformations result in the following linear partial differential equation: d d X X ∂P ∂P ∂2P = 2r̄ xi + [σσ ∗ ]ij xi xj − 2r̄P, ∂xi ∂xi ∂xj ∂ t̂ i=1 i,j=1 (6) P (0, x) = Ψ(x). And by letting L be the operator L = 2r̄ d X i=1 d X ∂ ∂2 + [σσ ∗ ]ij xi xj − 2r̄, xi ∂xi ∂xi ∂xj i,j=1 (6) can be written as 6 (7) ∂P = LP. (8) ∂ t̂ Pricing options is of great importance in the financial market and the computations must be both accurate and time efficient. One approach to solve the Black-Scholes PDE numerically is to use finite differences in both time and space. The disadvantage of this method is that for multi dimensional problems the number of discrete points grows exponentially with the number of dimensions. This problem is called ”the curse of dimensionality”. To make the computations more efficient, an adaptive technique to reduce the number of grid-points can be used. Adaptivity can be used in both time and space. In [4] the above described problem is solved with adaptivity in space using backward differentiation formula of second order (BDF2) and the discontinuous Galerkin (dG) method as time integrator. It is shown that dG as time integrator with an adapted grid in space provides a faster and more accurate result with less steps in time compared to the BDF2 with the same grid in space. The Galerkin method is a finite element method that has been used extensively for solving problems similar to the Black-Scholes equation. The dG method also allows the use of decoupling when choosing the temporal shape functions to be the normalized Legendre polynomials defined on (−1, 1), which is further described in [4]. This speeds up the computational time even more and less memory is required. The main focus in this paper is to investigate the temporal error for different time integrators when solving (6). The first experiment was to implement two different types of time adaptivities on BDF2 and see if it is better than dG when solving Black-Schoels in one dimension. The two adaptivity methods is described in section 4.1 and 4.2. The second experiment was to implement dG for two dimensions and to see if dG is superior to BDF2 in two dimensions. The outline of this paper is the following. In Section 2, the discontinuous Galerkin as time integrator is presented. In Section 3, the method for space discretization is given. In Section 4, the methods for time adaption is presented and all numerical results are given in Section 5 and discussed in Section 6. 2 Discontinuous Galerkin for Time Integration We establish the dG formulation for a system of ordinary differential equations on the form u̇(t) = Au(t) + f (t), u(0) = uo , 0 ≤ t ≤ T, (9) where A is the discretization of the spatial operator, u0 is the initial datum, 7 and f is the forcing term. Define P r (Im ) as the space of polynomials of degree r or less at the on the interval Im . We further define the finite element space containing the piecewise polynomials to be U = {Im ∈ P r (Im ). The finite element solution U is continuous within each time interval Im . In other terms Um (t) = U |Im for m = 0, 1, ...., M − 1, which is continuous in time element Im = (tm , tm+1 ) of size km = tm+1 − tm . At each time node tm , the − := U − limiting values of the approximative solution from left Um m−1 (tm ) and + := U (t+ ) are usually different [5]. The jump which can be the right Um m m seen in figure 2 is denoted by − [U ]m = Um (t+ m ) − Um−1 (tm ). (10) 0 The dG method is as follows. Find U ∈ U, satisfying U0 (t+ 0 ) = u = u0 , such that N −1 Z X n=0 (U̇ (t) − AU )ω(t)dt + In N −1 X N −1 Z X n=0 n=0 [U ]n+1 ω(tn+1 ) = f ω(t)dt (11) In for all ω(t) ∈ U. U can be computed locally in each interval Im . Z Z (U̇ (t) − AU )ω(t)dt + [U ]n+1 ω(tn+1 ) = f ω(t)dt (12) In In Figure 2: Illustration of the discontinuous behaviour of piecewise polynomials with one sided limits With a space discretization of N discrete grid points (12) results in having to solve a linear system of (r + 1)N equations in each time step [6]. In [4] it is shown that by using Legendre polynomials as basis functions, this can be expressed as (r + 1) linear systems of equations of order N . 8 3 Space Discretization To obtain high accuracy and keep the discretization error at a certain level, an adaptive technique in space has been used, which is thoroughly explained in [3]. This adaptive method places the discrete nodes where they are most needed and helps avoiding dense grids for regions where the solution is smooth [8]. It also makes sure that the number of grid-points is at a minimum. The summary of the adaptive algorithm in [3]: 1. Solve the PDE once with a coarse grid giving low accuracy. 2. Create a new spatial grid aiming to capture the required accuracy. 3. Solve the PDE with the new grid-points. This means that we have to solve the problem two times. During the first round, the problem will be solved quickly with low accuracy but giving a good estimation how to place the grid points for the second solve. The semi-discretization in space is made by centred second order differences. When using the adaptive technique in [3] it results in a system of ordinary differential equations ∂Ph = Ah Ph , dt̂ (13) where Ah is the discrete approximation of L. Ah is a very large sparse matrix with the number of non-zeros of each row depending on the number of space dimensions, i.e., the number of underlying assets. The boundary condition we use on all boundaries is ∂ 2 P (x, t̂) = 0, ∂x2i (14) which implies that the option price is nearly linear with respect to the spot price at the boundaries [3]. 4 Time Adaption The time-discretization can be made in several different ways. The most basic is to use equidistant time-steps. However, since the benefits of minimizing the number of grid-point is of great value, especially in several dimensions, other alternatives of discretization will be investigated. 9 4.1 Time Adaption minimizing the local truncation error Using the backward differentiation formula of second order, adaption can be made by controlling that the local discretization error is at a predescribed level. The method fully described in [3] is similar to the space-adaption, the problem is first solved with a coarse grid in time and space. And from this solution the local truncation error is approximated for every time-step. Then the new step-sizes will be calculated and they will be as large as possible still within the prescribed tolerance level. 4.2 Time Adaption using the Adjoint Equation The second approach to time adaption is to formulate the adjoint equation to the problem. From [7] we have that if we let P̄ denote a smooth reconstruction of the discrete data in P so that they agree at t = tn and at the grid points. The solution error E = P − P̄ approximately satisfies the following boundary value problem Et − 2rxEx − σ 2 x2 Exx + 2rE = Et − L = τ, E(0, x) = 0, x ∈ D E(t, x) = 0, x ∈ ∂D. (15) Where D is the domain in which we solve for. The local discretization error τ consists of two parts, the temporal discretization error τk and the spatial discretization error τh τ = τk + τh . (16) To control the temporal discretization error we use the method described in [7]. By using a fine grid in space we can make the assumption that |τh | |τk |. To do this we introduce the adjoint equation to (15): wt + L∗ w = 0, ∂ ∂2 (wx) + σ 2 2 (x2 w) − 2rw, ∂x ∂x w(T, x) = g(x), L∗ w = −2r (17) with boundary conditions on the adjoint equation where w = 0 on the boundary of x and letting g(x) be a test function by choice. Since we know the solution to the adjoint equation at time T it is a final value problem. Deriving the derivatives in (17) we get L∗ w = −2rw − 2rxwx + σ 2 (2w + 2xwx + 2xwx + x2 wxx ) − 2rw, wt = 4rw + 2rxwx − σ 2 (2w + 4xwx + x2 wxx ). (18) With centred difference approximation of second order, from Taylor expansion we get 10 ∂w(xj ) w(xj+1 ) − w(xj−1 ) = , ∂x 2h w(xj+1 ) − 2w(x) + w(xj−1 ) ∂w(xj ) = , 2 ∂x h2 (19) where h is the step size in space. Since the final value is a function by choice we can integrate backwards in time. Here we use Euler forward wn−1 = wn − kwt , (20) where k is the time-step. By combining the centred difference approximation in (19) with (20) we have 2x2j σ 2 k xj kr 2xj σ 2 k x2j σ 2 k n 2 + + ) + w (1 − 4kr + 2σ k − ) j h h h2 h2 xj kr 2xj σ 2 k x2j σ 2 k n +wj−1 ( − + ). h h h2 (21) If we let n wjn−1 =wj+1 (− 2x2j σ 2 k , h2 xj kr 2xj σ 2 k x2 σ 2 k + + , ϕ=− h h h2 xj kr 2xj σ 2 k x2j σ 2 k θ= − + , h h h2 ρ = 1 − 4kr + 2σ 2 k − (22) we can write this as a system of equations. With matrix A, vector ŵn with inner values and vector w̄n with boundary values we get wn−1 = Aŵn + w̄n , (23) where ρ ϕ θ ρ ϕ ... A= θ ρ ϕ θ ρ n , ŵ = 11 w2n w3n . . . n wN n , w̄ = θw1n 0 0 . . n ϕwN +1 . (24) Integrating backwards in time beginning at T we can solve w for the entire time interval [0, T ]. From [7] we then have Z T T Z Z Z wE(T, x)dxdr. wτ dxdt = 0 0 D (25) D . The function g(x) in (17) is chosen to be non-negative and with a compact support around the strike price where we are most interested in having an accurate solution. Taking the absolute value of (25) we get the following Z T Z Z T Z wτ (t, x)dxdt ≤ sup|τ (t, x)|x∈D |w|dxdt 0 D 0 Z D T sup|τ (t, x)|x∈D ||w||n dt = ≈ 0 N X sup|τ (t, x)| · ||w||n ∆tn , x∈D n=1 with the definition (26) Z ||w||n = |w|n dx. (27) D And to control the discretization error we prescribe that Z T Z wE(T, x)dxdr ≤ , 0 (28) D where is the error tolerance. From (28),(25) and (26) we then get that if N X n=1 sup|τ (t, x)| · ||w||n ∆tn ≤ , (29) x∈D then (28) holds. Defining η = sup|τ (t, x)| · ||w||, (30) x∈D and demanding η to be of the same size in every time-step we get η = PN n=1 ∆tn 12 = . T (31) Computing the right hand side sup|τ (t, x)| ≤ η ||w||n (32) we thereby get the tolerance for the local truncation error for every step. Using this tolerance we then use the same method as in section 4.1, taking the largest time step possible still within the computed tolerance. In summary we have the following: 1. The adjoint equation is derived and the solution at time T is a function by choice. 2. With the known solution at time T we solve the adjoint equation backwards in time with Euler forward on the whole interval [0, T ]. 3. With solution of the adjoint equation we compute the tolerance for the local discretization error in every step as in equation (32). We then take the largest time step possible with this new tolerance. 5 Numerical Result In this section, we investigate the performance of the adaptive part in time in one dimension and the performance of discontinuous Galerkin in two dimensions. The algorithms presented in this paper have been implemented in Matlab and performed on a system with Scientific Linux as operation system with AMD Opteron (Bulldozer) 6282SE, 2.6 GHz, 16-core, dual socket processors and 128GB memory. In all of the following experiments we consider the European call option with short rate of interest 5%, i.e. r = 0.05, a volatility σ of 0.3, strike price K of 30 and the time of maturity T is set to 2.22 years. In two dimensions we consider two underlying assets with the same specifications as above, this results in a total strike price K of 60. In Section 1 it is mentioned that the use of decoupling speeds up the computational time for the Galerkin method, therefore the decoupled Galerkin in [4] has been used for these experiments. When calculating these problems we have to restrict the spatial domain. A common rule for this is to set Smax = 4dK, i.e. we truncate the domain at four times the strike price multiplied by the number of dimensions d. The errors have been calculated using the L1 -norm of the difference between the reference solution and the computed solution. The Error computed in two dimensions: Error = ||x||1 · mean(h) = N X i,j=1 13 |xi,j | · mean(h), h ∈ D̃ (33) where xi,j is the difference between the reference solution and the computed, D̃ is the spatial domain we are interested of and h is the vector containing the step lengths in space. When the error is computed in one dimension, we summarize for i only. Furthermore, the area of interest is when the price is close to the strike price, therefore we let D̃ be a domain around strike price in which the error is calculated. The two different types of time-integrators that has been used is the backward differentiation formula of second order (BDF2) and the discontinuous Galerkin method (dG). The dG has been implemented with an implicit single step scheme which allows for arbitrary variations of the step size and polynomial order. As basis functions the Legendre polynomials of order one and two has been used. 5.1 Numerical Result for One-dimensional Problem In one dimension we examine the performance of equidistant BDF2, BDF2 with time adaption (section 4.1), BDF2 with time adaption using the adjoint equation (section 4.2) and discontinuous Galerkin. To measure the error we compare the computed solutions with the exact solution of the problem which is known for the one dimensional problem. The L1 norm, as described above, of the difference is then used to estimate the error. This will be done on the domain D̃ = [K/3, 5K/3]. The aim is to measure the time discretization error, therefore a fine spatial mesh has been used. The tolerance for the spatial discretization error has been set to 3 · 10− 7 which resulted in a spatial grid with 3353 grid points. As can be seen in Figure 3 the dG method only need a few time steps before it converges to the spatial error. The different versions of BDF2 needs much more time steps to obtain the same result. It can be seen that the time adaptive versions also have a larger error then the equidistant when using the same amount of time steps. 14 1 10 Adjoint Eq Time−adaptive BDF2 Time−adaptive BDF2 dG r=1 dG r=2 BDF2 0 10 −1 10 −2 Error 10 −3 10 −4 10 −5 10 −6 10 −7 10 0 10 1 10 2 10 steps 3 10 4 10 Figure 3: The error of the time integrators BDF2, BDF2 with time adaption and dG with polynomial order dG(1), r = 1 and dG(2), r = 2 for different amounts of time steps. The time of computation can be seen in Figure 4 and we see that the dG converges to the spatial error a lot faster than the BDF2 methods. Also the time adaptive versions are slower and has a larger error than the equidistant BDF2. 15 2 10 Adjoint Eq Time−adaptive BDF2 Time−adaptive BDF2 dG r=1 dG r=2 BDF2 1 Time 10 0 10 −1 10 −2 10 −8 10 −6 10 −4 −2 10 10 0 10 2 10 Error Figure 4: The computational time of the time integrators BDF2, BDF2 with time adaption and dG with polynomial order dG(1), r = 1 and dG(2), r = 2 for different size of errors. To make sure that the adjoint equation formula works we control that equation (28) holds. This can be seen in Table 1, were the absolute value of the integral of the error times the test function should be less than the prescribed tolerance. This is valid for all except the last column. The reason for this is that when using a time tolerance of 1 · 10−9 we hit the spatial error and the major influence on the total error will be the error in space. Table 1: The integral of the error E at time T , times the test function on the domain D and the time tolerance used when calculating the different errors. | R D 5.2 g(x)E(T, x)dx| Time tol 6 · 10−4 0,01 5.2 · 10−6 1 · 10−3 5.6 · 10−7 1 · 10−5 8.5 · 10−9 1 · 10−7 1.4 · 10−8 1 · 10−9 Numerical Result for the Two-dimensional Problem The Black-Scholes equation in two dimensions (4) has no analytical solution. Therefore, to be able to measure the error for the different time integrators presented in this paper an accurate reference solution had to be 16 made. The reference solution for the comparisons is computed with BDF2 with an adapted grid in space and using 1000 steps in time. The adapted grid in space is made with the method presented in Section 3 starting with 41 grid-points and the space tolerance set to 3 · 10−4 . This results in an adapted grid with 163 points in both dimensions. Since we are focusing on the temporal error, the grid used for the reference solution has also been used for the tests. Since we are solving Black-Scholes for an option based on two underlying assets, the spatial domain we solve for is [0, 8K] in each dimensions, as mentioned in Section 5. In this tests we compute the error on the interval D̃ = [K/3, 10K/3] in each dimension. BDF2 and dG with Legendre polynimials of order 1 and 2 is the three methods compared in two dimensions. 0 10 BDF2 dG(1) dG(2) −1 10 −2 Size of error 10 −3 10 −4 10 −5 10 −6 10 0 10 1 10 2 10 Number of time steps 3 10 Figure 5: Comparisons for the two-dimensional problem between BDF2 and dG with polynomial order dG(1), r = 1 and dG(2), r = 2 respectively. The number of time steps used are [3:1:100] for the dG methods, [3:3:99] and then [100:20:220] for BDF2, starting from the left. The reason for the plateau in Figure 5 and 6, is that the total error cannot be reduced further by increasing the amount of time steps due to the spatial error. In figure 5 we can clearly see that dG is a better time integrator than BDF2 when solved for two dimensions. We can see that BDF2 demands much more time steps than the dG methods to reach a certain error. And when analysing it further, one can see that the slope of 17 the dG methods are steeper than BDF2. This shows that for any number of time steps, the dG will generate a smaller size of error. With that in mind, an important notation can be made when analysing figure (6). As we can see, the slope of BDF2 is steeper then dG of both orders which implies slower convergence and shows further that dG is superior to BDF2. If it had been the reversed scenario and the slope of dG was steeper than BDF2, it would have shown that even when using more time steps then dG the BDF2:s computation time would have been less than the dG methods. In other words, for a certain error BDF2 would have been the faster time integrator even though it needs more time steps. But as we can see, this is not the case and that is an important remark. 2 10 Time of computation BDF2 dG(1) dG(2) 1 10 0 10 −6 10 −5 10 −4 10 −3 10 Size of error −2 10 −1 10 Figure 6: Comparisons for the two-dimensional problem between BDF2 and dG with polynomial order dG(1), r = 1 and dG(2), r = 2 respectively. The number of time steps used are [3:1:100] for the dG methods, [9:3:99] and then [100:20:220] for BDF2 starting from the right. 6 Conclusion This paper has examined different methods of time integrators when solving the parabolic PDE obtained when pricing options using the Black-Scholes model. This has been done in both one and two dimensions. The two methods of time-integration that has been used is BDF2 and the discontinuous Galerkin method. In one dimension the BDF2 time discretization has been 18 implemented with an equidistant grid and two types of adaptive grids. The dG has been used with the Legendre polynomials of order r = 1, 2 as basis functions and solved with equidistant time steps. For the two-dimensional problem BDF2 and dG with equidistant time steps has been compared. The numerical result in one dimension shows that the dG methods gives by far the best results both with respect to error to time-step and error to computational time. The two versions of time-adaptive BDF2 fails to provide a better solution than the equidistant BDF2. The two adaptive versions need more time steps and has longer computation time compared to the equidistant version given a certain accuracy. This is a surprise since the size of the local truncation error is controlled in every step. The reason for failure is that since the truncation error is kept within the tolerance level, very small steps are used where needed. This results in increasing step lengths on other intervals and despite that the local truncation error is controlled the global error grows. As for the results obtained in the two-dimensional tests we can conclude that the dG is as a better time integrator than the equidistant BDF2. Furthermore, the dG with Legendre polynomials of second order provides significant better results than the first order. Only a few time steps are needed to obtain a high temporal accuracy. For dG of second order with Legendre polynomials, with less than ten time steps the solution converges to the spatial error tolerance. The BDF2 never reaches the spatial error although 100 steps were used. Also the time of computation for the dG method relative the error is comparably short. 19 References [1] Rüdiger U.Seydel, Tools for Computational Finance, 2002. [2] Fischer Black and Myron Scholes, The pricing of Options and Corporate liabilities, Journal of Political Economy, 81:637-659, 1973. [3] Jonas Persson and Lina von Sydow,Pricing European multi-asset options using a space-time adaptive FD-method, Comput Visual Sci (2007)10:17-183. [4] Emil Larsson, Option pricing using discontinous Galerkin method for time integration, Uppsala University, 2013. [5] Sigal Gottlieb, G. W. Wei and Shan Zhao, A unified discontinous Galerkin framework for time integration, 2010. [6] K Eriksson, D Estep, P Hansbro, C Johnson, Computational Differential Equations, 1996. [7] Per Lötstedt, Jonas Persson, Lina von Sydow, Johan Tysk, Spacectime adaptive finite difference method for Europeean multi-asset options, 2006 [8] Paria Ghafari, Dimension Reductioni and Adaptivity to Price Basket Options [9] Merton, R.C, Theory of rational option pricing, 1973 20

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