JFSA Journal of Function Spaces and Applications 1758-4965 0972-6802 Hindawi Publishing Corporation 189235 10.1155/2013/189235 189235 Research Article Accurate Numerical Method for Pricing Two-Asset American Put Options Wu Xianbin Ruiz Galan Manuel Junior College, Zhejiang Wanli University, Ningbo 315100 China zwu.edu.cn 2013 26 2 2013 2013 12 10 2012 27 11 2012 2013 Copyright © 2013 Xianbin Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We develop an accurate finite difference scheme for pricing two-asset American put options. We use the central difference method for space derivatives and the implicit Euler method for the time derivative. Under certain mesh step size limitations, the matrix associated with the discrete operator is an M-matrix, which ensures that the solutions are oscillation-free. We apply the maximum principle to the discrete linear complementarity problem in two mesh sets and derive the error estimates. It is shown that the scheme is second-order convergent with respect to the spatial variables. Numerical results support the theoretical results.

1. Introduction

An option is a financial instrument that gives the holder the right, but not the obligation, to buy (call option) or to sell (put option) an agreed quantity of a specified asset at a fixed price (exercise or strike price) on (European option) or before (American option) a given date (expiry date). It was shown by Black-Scholes  that the value of a European option is governed by a second-order parabolic partial differential equation with respect to the time and the underlying asset price. The value of an American option is determined by a linear complementarity problem involving the Black-Scholes operator [2, 3]. Since this complementarity problem is, in general, not analytically solvable, numerical approximation to the solution is normally sought in practice.

Various numerical methods have been proposed for the valuation of single-factor American options. Among them, the lattice method , the Monte Carlo method , the finite difference method , the finite element method [9, 10], and the finite volume method  are the most popular ones in both practice and research.

Finite difference methods applied to the multifactor American option valuation have also been developed. S. O'Sullivan and C. O’Sullivan  presented explicit finite difference methods with an acceleration technique for option pricing. Clarke and Parrott  and Oosterlee  used finite difference schemes along with a projected full approximation scheme (PFAS) multigrid for pricing American options under stochastic volatility. Ikonen and Toivanen  proposed finite difference methods with componentwise splitting methods on nonuniform grids for pricing American options under stochastic volatility. Hout and Foulon  and Zhu and Chen  applied finite difference schemes based on the ADI method to price American options under stochastic volatility. Le et al.  presented an upwind difference scheme for the valuation of perpetual American put options under stochastic volatility. Yousuf  developed an exponential time differencing scheme with a splitting technique for pricing American options under stochastic volatility. Nielsen et al.  and Zhang et al.  analyzed finite difference schemes with penalty methods for pricing American two-asset options, but their difference methods are first-order convergent.

In part of the domain, the differential operator of the two-asset American option pricing model becomes a convection-dominated operator. The differential operator also contains a second-order mixed derivative term. The classical finite difference methods lead to some off-diagonal elements in the coefficient matrix of the discrete operator due to the dominating first-order derivatives and the mixed derivative. These elements can lead to nonphysical oscillations in the computed solution [17, 18]. In this paper, we present an accurate finite difference scheme for pricing two-asset American options. We use the central difference method for space derivatives and the implicit Euler method for the time derivative. Under certain mesh step size limitations, we obtain a coefficient matrix with an M-matrix property, which ensures that the solutions are oscillation-free. We apply the maximum principle to the discrete linear complementarity problem in two mesh sets and derive the error estimates. We will show that the scheme is second-order convergent with respect to the spatial variables.

The rest of the paper is organized as follows. In the next section, we describe some theoretical results on the continuous complementarity problem for the two-asset American put option pricing model. In Section 3, the discretization method is described. In Section 4, we present a stability and error analysis for the finite difference scheme. In Section 5, numerical experiments are provided to support these theoretical results.

2. The Continuous Problem

We consider the following two-asset American put option pricing model [24, 25]: (1)P(S1,S2,t)0,S1,S2>0,t[0,T),P(S1,S2,t)·[P(S1,S2,t)-ϕ(S1,S2)]=0,S1,S2>0,t[0,T),P(S1,S2,t)-ϕ(S1,S2)0,S1,S20,t[0,T],P(S1,S2,T)=ϕ(S1,S2),S1,S20,P(0,S2,t)=g2(S2,t),S20,t[0,T],P(S1,0,t)=g1(S1,t),S10,t[0,T],limS1P(S1,S2,t)=0,S20,t[0,T],limS2P(S1,S2,t)=0,S10,t[0,T], where denotes the two-dimensional Black-Scholes operator defined by (2)P(S1,S2,t)-Pt-12σ12S122PS12-ρσ1σ2S1S22PS1y-12σ22S222PS22-rS1PS1-rS2PS2+rP, and ϕ(S1,S2) is the final (payoff) condition defined by (3)ϕ(S1,S2)=max{E-(α1S1+α2S2),0}. Here, P is the value of the option, Si is the value of the ith underlying asset, ρ[-1,0)(0,1] is the correlation of two underlying assets, r is the risk-free interest rate, and gi(·,·) is a given function providing suitable boundary conditions. Typically, gi(·,·) is determined by solving the associated one-dimensional American put option problem (4)-igi(Si,t)0,Si>0,t[0,T),-igi(Si,t)·[gi(Si,t)-max(E-αiSi,0)]=0,Si>0,t[0,T),gi(Si,t)-max(E-αiSi,0)0,Si0,t[0,T],gi(Si,T)=max(E-αiSi,0),Si0,gi(0,t)=E,limSigi(Si,t)=0,t[0,T], where -i denotes the one-dimensional Black-Scholes operator defined by (5)-igi(Si,t)-git-12σi2Si22giSi2-rSigiSi+rgi,i=1,2.

Introducing the logarithmic prices x=lnS1 and y=lnS2, the linear complementarity problem (1) is transformed as (6)Lu(x,y,t)0,(x,y,t)××[0,T),Lu(x,y,t)·[u(x,y,t)-φ(x,y)]=0,(x,y,t)××[0,T),u(x,y,t)-φ(x,y)0,(x,y,t)××[0,T],u(x,y,T)=φ(x,y),(x,y)×,u(0,y,t)=g2(ey,t),(y,t)×[0,T],u(x,0,t)=g1(ex,t),(x,t)×[0,T],limxu(x,y,t)=0,(y,t)×[0,T],limyu(x,y,t)=0,(x,t)×[0,T], where (7)Lu(x,y,t)-ut-12σ122ux2-ρσ1σ22uxy-12σ222uy2-(r-12σ12)ux-(r-12σ22)uy+ru,φ(x,y)=max{E-(α1ex+α2ey),0}.

For applying the numerical method, we truncate the infinite domain into Ω(xmin,xmax)×(ymin,ymax), where the boundaries xmin,xmax,ymin, and ymax are chosen so as not to introduce huge errors in the value of the option . Based on Willmott et al.'s estimate  that the upper bound of the asset price is typically three or four times the strike price, it is reasonable for us to set xmax=ln(4E) and ymax=ln(4E). The artificial boundary conditions at x=xmin and x=xmax are chosen to be u(xmin,y,t)=g2(ey,t),u(xmax,y,t)=0. The artificial boundary conditions at y=ymin and y=ymax are chosen to be u(x,ymin,t)=g1(ex,t),u(x,ymax,t)=0. Therefore, in the rest of this paper, we will consider the following linear complementary problem: (8)Lu(x,y,t)0,(x,y,t)Ω×[0,T),Lu(x,y,t)·[u(x,y,t)-φ(x,y)]=0,(x,y,t)Ω×[0,T),u(x,y,t)-φ(x,y)0,(x,y,t)Ω×[0,T],u(x,y,T)=φ(x,y),(x,y)Ω,u(xmin,y,t)=g2(ey,t),(y,t)[ymin,ymax]×[0,T],u(xmax,y,t)=0,(y,t)[ymin,ymax]×[0,T],u(x,ymin,t)=g1(ex,t),(x,t)[xmin,xmax]×[0,T],u(x,ymax,t)=0,(x,t)[xmin,xmax]×[0,T].

3. Discretization

The operator L contains a second-order mixed derivative term. Usual finite difference approximations lead to some positive off-diagonal elements in the matrix associated with the discrete operator due to the mixed derivative, which may lead to nonphysical oscillations in the computed solution. Hence, it is not easy to construct a discretization with good properties and accuracy for problems with mixed derivatives. There are some works dealing with stable difference approximations of mixed derivatives [27, 28]. In this paper, we present an accurate finite difference scheme to discretize the operator L. We use the technique of  to give the mesh step size limitation, which guarantees that the coefficient matrix corresponding to the discrete operator is an M-matrix.

The discretization is performed using a uniform mesh ΩN,M,K for the computational domain Ω×[0,T]. The mesh steps to the x direction, y direction, and t direction are denoted by Δx=(xmax-xmin)/N, Δy=(ymax-ymin)/M, and Δt=T/K. The mesh point values of the finite difference approximation are denoted by (9)Ui,jku(xi,yj,tk)fori=0,1,,N;j=0,1,,M;k=0,1,,K.

We discretize the differential operator L using the central difference scheme on the previous uniform mesh. We set (10)LN,M,KUi,jk-Dt+Ui,jk-12σ12δx2Ui,jk-σ1σ2(ρ~+δxy+Ui,jk+ρ~-δxy-Ui,jk)-12σ22δy2Ui,jk-(r-12σ12)DxUi,jk-(r-12σ22)DyUi,jk+rUi,jk, where (11)δx2Ui,jk=Dx+-Dx-ΔxUi,jk,  δy2Ui,jk=Dy+-Dy-ΔyUi,jk,δxy+Ui,jk=Dx+Dy++Dx-Dy-2Ui,jk,δxy-Ui,jk=Dx+Dy-+Dx-Dy+2Ui,jk,Dx+Ui,jk=Ui+1,jk-Ui,jkΔx,Dy+Ui,jk=Ui,j+1k-Ui,jkΔy,Dx-Ui,jk=Ui,jk-Ui-1,jkΔx,Dy-Ui,jk=Ui,jk-Ui,j-1kΔy,DxUi,jk=Ui+1,jk-Ui-1,jk2Δx,DyUi,jk=Ui,j+1k-Ui,j-1k2Δy,Dt+Ui,jk=Ui,jk+1-Ui,jkΔt,ρ~±=12[ρ±|ρ|].

Denote (12)Ω-h={(i,j,k)0iN,  0jM,  0kK},Ω~h={(i,j,k)1iN-1,  1jM-1,Ω~h=a1kK-1},Ωh=Ω¯hΩ~h.

Thus, we apply the central difference scheme on the uniform mesh to approximate the parabolic complementarity problem (8) as follows: (13)LN,M,KUi,jk0,(i,j,k)Ω~h,Ui,jk-φi,j0,(i,j,k)Ω~h,LN,M,KUi,jk·[Ui,jk-φi,j]=0,(i,j,k)Ω~h,Ui,jK=φi,j,0iN,0jM,U0,jk=(g2)jk,UN,jk=0,0jM,0k<K,Ui,0k=(g1)ik,Ui,Mk=0,0<i<N,0k<K. Here, (g1)ik  and  (g2)jk are discrete approximates of g1(ex,t)  and    g2(ey,t), respectively. Hence, (g1)ik and (g2)jk can be obtained by solving the corresponding one-dimensional Black-Scholes equations . In the next section, we will prove that the system matrix corresponding to the discrete operator LN,M,K is an M-matrix. Hence, from the uniqueness theorem of Goeleven , we can obtain that there exists a unique solution U for the previous linear complementarity problem (13).

4. Analysis of the Method

First, we give the stability analysis for the difference scheme (13).

Lemma 1.

If mesh steps satisfy the inequalities (14)Δxσ12|2r-σ12|,Δyσ22|2r-σ22|,(15)2|ρ|σ1σ2ΔxΔy2σ1|ρ|σ2, then the system matrix corresponding to the discrete operator LN,M,K is an M-matrix.

Proof.

The difference operator LN,M,K can be written as follows: (16)LN,M,KUi,jk=-ρ~+σ1σ22ΔxΔyUi-1,j-1k+[σ1σ2ρ~+-ρ~-2ΔxΔy-σ222(Δy)2+r-(1/2)σ222Δy]Ui,j-1k+ρ~-σ1σ22ΔxΔyUi+1,j-1k+[σ1σ2ρ~+-ρ~-2ΔxΔy-σ122(Δx)2+r-(1/2)σ122Δx]Ui-1,jk+[1Δt+σ12(Δx)2-σ1σ2ρ~+-ρ~-ΔxΔy+σ22(Δy)2+r]Ui,jk+[-σ122(Δx)2+σ1σ2ρ~+-ρ~-2ΔxΔy-r-(1/2)σ122Δx]Ui+1,jk+ρ~-σ1σ22ΔxΔyUi-1,j+1k+[σ1σ2ρ~+-ρ~-2ΔxΔy-σ222(Δy)2-r-(1/2)σ222Δy]Ui,j+1k-ρ~+σ1σ22ΔxΔyUi+1,j+1k-1ΔtUi,jk+1. The coefficient of Ui,j in the previous expression (which corresponds to the diagonal of the system matrix) is positive since (17)σ12(Δx)2-σ1σ2ρ~+-ρ~-ΔxΔy+σ22(Δy)20. All the coefficients of the other U in the previous expression (which correspond to off-diagonal elements in the system matrix) will be nonpositive once the following inequalities are satisfied: (18)σ124(Δx)2-|r-(1/2)σ12|2Δx0,σ224(Δy)2-|r-(1/2)σ22|2Δy0,σ1σ2ρ~+-ρ~-2ΔxΔy-σ224(Δy)20,σ1σ2ρ~+-ρ~-2ΔxΔy-σ124(Δx)20. Together, they require that the following inequalities hold: (19)Δxσ12|2r-σ12|,Δyσ22|2r-σ22|,2σ1(ρ~+-ρ~-)σ2ΔxΔy2σ1σ2(ρ~+-ρ~-), which are (14) and (15), respectively. Thus, we have shown that the system matrix, corresponding to the discrete operator LN,M,K is an M-matrix and the result follows.

There are only few error estimates for the direct application of finite difference method to linear complementarity problems. Here, we apply the maximum principle to the linear complementarity problem (13) in two mesh sets and derive the error estimates [29, 31].

By using Taylor's formula, we can easily obtain the following truncation error estimate.

Lemma 2.

Let u(x,y,t) be a smooth function defined on ΩN,M,K. Then the truncation error of the difference scheme (10) satisfies (20)|LN,M,Kui,jk-Lui,jk|=O((Δx)2+(Δy)2+ΔxΔy+Δt), for all (i,j,k)Ω~h.

Now we can derive our main result for the difference scheme.

Theorem 3.

Let u(x,y,t) be the solution of the problem (8) and let Ui,jk be the solution of the problem (13). If mesh steps satisfy conditions (14) and (15), the difference scheme (13) satisfies the following error estimate: (21)max(i,j,k)Ω-h|u(xi,yj,tk)-Ui,jk|C[(Δx)2+(Δy)2+ΔxΔy+Δt], where C is a constant independent of Δx,Δy, and Δt.

Proof.

Denote (22)Ω(1)={(i,j,k)Ω~hu(xi,yj,tk)=φ(xi,yj)},Ω(2)=Ω~hΩ(1). From (8), we have the result (23)Lu(xi,yj,tk)0,(i,j,k)Ω(1),Lu(xi,yj,tk)=0,(i,j,k)Ω(2). Denote (24)Ωh(1)={(i,j)Ω~hUi,jk=φ(xi,yj)},Ωh(2)=Ω~hΩh(1). Obviously, (25)LN,M,KUi,jk=0,(i,j,k)Ωh(2).

Define the function on Ω~h by (26)Wi,jk=C[(Δx)2+(Δy)2+ΔxΔy+Δt]>0, where C is a sufficiently large constant.

For (i,j,k)Ωh(2), by the fact that Lu(xi,yj,tk)0, (25), (26), and Lemma 2, we obtain (27)LN,M,K(u(xi,yj,tk)-Ui,jk+Wi,jk)=LN,M,Ku(xi,yj,tk)+LN,M,KWi,jk=[LN,M,Ku(xi,yj,tk)-Lu(xi,yj,tk)+LN,M,KWi,jk]+Lu(xi,yj,tk)0. On the “boundary” of Ωh(2), the nodes (i,j,k)Ωh(1), so Ui,jk=φ(xi,yj), but u(xi,yj,tk)φ(xi,yj), therefore (28)u(xi,yj,tk)-Ui,jk+Wi,jk=u(xi,yj,tk)-φ(xi,yj)+Wi,jk0, and the nodes (i,j,k)Ωh, (29)u(xi,yj,tk)-Ui,jk+Wi,jk=Wi,jk0. Applying the maximum principle to Ωh(2), we get (30)u(xi,yj,tk)-Ui,jk+Wi,jk0,(i,j,k)Ωh(2). Thus, (31)u(xi,yj,tk)-Ui,jk+Wi,jk0,(i,j,k)Ω-h.

For (i,j,k)Ω(2),Lu(xi,yj,tk)=0, but LN,M,KUi,jk0, thus, (32)LN,M,K(u(xi,yj,tk)-Ui,jk-Wi,jk)=[LN,M,Ku(xi,yj,tk)-Lu(xi,yj,tk)-LN,M,KWi,jk]-LN,M,KUi,jk0. On the “boundary” of Ω(2), the nodes (i,j,k)Ω(1), so u(xi,yj,tk)=φ(xi,yj), but Ui,jkφ(xi,yj), therefore (33)u(xi,yj,tk)-Ui,jk-Wi,jk=φ(xi,yj)-Ui,jk-Wi,jk0, and the nodes (i,j,k)Ωh, (34)u(xi,yj,tk)-Ui,jk-Wi,jk=-Wi,jk0. Applying the maximum principle to Ω(2), we get (35)u(xi,yj,tk)-Ui,jk-Wi,jk0,(i,j,k)Ω(2). Thus, (36)u(xi,yj,tk)-Ui,jk-Wi,jk0,(i,j,k)Ω-h.

From (31) and (36), we obtain (37)max(i,j,k)Ω-h|u(xi,yj,tk)-Ui,jk|max(i,j,k)Ω-hWi,jkC[(Δx)2+(Δy)2+ΔxΔy+Δt], where C is a sufficiently large constant. From this we complete the proof.

5. Numerical Experiments

In this section, we verify experimentally the theoretical results obtained in the preceding section. Errors and convergence rates for the second-order finite difference scheme are presented for two test problems.

Test  1. American put option with parameters: T=1, r=0.1, σ1=0.4, σ2=0.5, ρ=0.5, xmin=-ln(30), xmax=ln(30), ymin=-ln(40), ymax=ln(40), α1=0.3, α2=0.7, and E=10.

Test  2. American put option with parameters: T=1, r=0.08, σ1=0.3, σ2=0.4, ρ=-0.6, xmin=-ln(30), xmax=ln(30), ymin=-ln(40), ymax=ln(40), α1=0.3, α2=0.7, and E=10.

To solve the linear inequality system (13), we use the projection scheme used in [32, page 433]. Since mesh steps need to satisfy conditions (14) and (15), we choose the number of mesh steps in the y direction (38)M=[(2|ρ|σ1/σ2+2σ1/|ρ|σ2)N2(xmax-xmin)], where N is the number of mesh steps in the x direction. The exact solutions of the test problems are not available. Therefore, we use the double mesh principle to estimate the errors and compute the experiment convergence rates in our computed solution. We measure the accuracy in the discrete maximum norm (39)eN,M,K=maxi,j,k|Ui,j,kN,M,K-Ui,j,k2N,2M,K|, and the convergence rate (40)RN,M,K=log2(eN,M,Ke2N,2M,K).

The error estimates and convergence rates in our computed solutions of Tests 1 and 2 are listed in Tables 1 and 2, respectively. From Tables 1 and 2, we see that eN,M,K/e2N,2M,K is close to 4 for sufficiently large K, which supports the convergence estimate of Theorem 3. However, the numerical results of Nielsen et al.  and Zhang et al.  verify that their schemes are only first-order convergent. Hence, our scheme is more accurate.

Numerical results for Test 1.

K N Error Rate
128 6 1.8124 e - 1
12 5.1852 e - 2 1.805
24 1.4761 e - 2 1.813
48 4.1312 e - 3 1.837

Numerical results for Test 2.

K N Error Rate
128 6 1.2256 e - 1
12 3.3776 e - 2 1.859
24 9.2571 e - 3 1.867
48 2.5124 e - 3 1.882

Acknowledgments

The authors would like to thank the anonymous referees for several suggestions for the improvement of this paper. The work was supported by Zhejiang Province Natural Science Foundation of China (Grant no. Y2111160).

Black F. Scholes M. The pricing of options and corporate liabilities Journal of Political Economy 1973 81 3 637 654 10.1086/260062 ZBL1092.91524 Huang J. Pang J. S. Option pricing and linear complementarity Journal of Computational Finance 1998 2 3 31 60 Wilmott P. Dewynne J. Howison S. Option Pricing: Mathematical Models and Computation 1993 Oxford, UK Oxford Financial Press Cox J. C. Ross S. A. Rubinstein M. Option pricing: a simplified approach Journal of Financial Economics 1979 7 3 229 263 2-s2.0-49249142814 10.1016/0304-405X(79)90015-1 ZBL1131.91333 Ibáñez A. Zapatero F. Monte Carlo valuation of American options through computation of the optimal exercise frontier Journal of Financial and Quantitative Analysis 2004 39 2 253 275 2-s2.0-2442614937 10.1017/S0022109000003069 Vázquez C. An upwind numerical approach for an American and European option pricing model Applied Mathematics and Computation 1998 97 2-3 273 286 10.1016/S0096-3003(97)10122-9 1643127 Wu L. Kwok Y. K. A front-xing nite dierence method for the valuation of American options Journal of Financial Engineering 1997 6 2 83 97 Zhao J. Davison M. Corless R. M. Compact finite difference method for American option pricing Journal of Computational and Applied Mathematics 2007 206 1 306 321 10.1016/j.cam.2006.07.006 2337446 ZBL1151.91552 Forsyth P. A. Vetzal K. R. Zvan R. A nite element approach to the pricing of discrete lookbacks with stochastic volatility Applied Mathematical Finance 1999 6 2 87 106 10.1080/135048699334564 Zvan R. Forsyth P. A. Vetzal K. R. A general nite element approach for PDE option pricing models University of Waterloo, Canada, 1998 Angermann L. Wang S. Convergence of a fitted finite volume method for the penalized Black-Scholes equation governing European and American option pricing Numerische Mathematik 2007 106 1 1 40 10.1007/s00211-006-0057-7 2286005 ZBL1131.65301 Wang S. Yang X. Q. Teo K. L. Power penalty method for a linear complementarity problem arising from American option valuation Journal of Optimization Theory and Applications 2006 129 2 227 254 10.1007/s10957-006-9062-3 2281387 ZBL1139.91020 Zvan R. Forsyth P. A. Vetzal K. R. A finite volume approach for contingent claims valuation IMA Journal of Numerical Analysis 2001 21 3 703 731 10.1093/imanum/21.3.703 1844364 ZBL1004.91032 O'Sullivan S. O'Sullivan C. On the acceleration of explicit finite difference methods for option pricing Quantitative Finance 2011 11 8 1177 1191 10.1080/14697680903055570 2823303 Clarke N. Parrott K. Multigrid for American option pricing with stochastic volatility Applied Mathematics Finance 1999 6 3 177 195 10.1080/135048699334528 ZBL1009.91034 Oosterlee C. W. On multigrid for linear complementarity problems with application to American-style options Electronic Transactions on Numerical Analysis 2003 15 165 185 1991272 ZBL1031.65072 Ikonen S. Toivanen J. Componentwise splitting methods for pricing American options under stochastic volatility International Journal of Theoretical and Applied Finance 2007 10 2 331 361 10.1142/S0219024907004202 2300525 ZBL1137.91451 Ikonen S. Toivanen J. Efficient numerical methods for pricing American options under stochastic volatility Numerical Methods for Partial Differential Equations 2008 24 1 104 126 10.1002/num.20239 2371350 ZBL1152.91516 Ikonen S. Toivanen J. Operator splitting methods for pricing American options under stochastic volatility Numerische Mathematik 2009 113 2 299 324 10.1007/s00211-009-0227-5 2529511 ZBL1204.91126 Hout K. J. I. Foulon S. ADI finite difference schemes for option pricing in the Heston model with correlation International Journal of Numerical Analysis and Modeling 2010 7 2 303 320 2587424 Zhu S.-P. Chen W.-T. A predictor-corrector scheme based on the ADI method for pricing American puts with stochastic volatility Computers & Mathematics with Applications 2011 62 1 1 26 10.1016/j.camwa.2011.03.101 2821803 ZBL1228.91077 Le A. Cen Z. Xu A. A robust upwind difference scheme for pricing perpetual American put options under stochastic volatility International Journal of Computer Mathematics 2012 89 9 1135 1144 10.1080/00207160.2012.658379 2934800 Yousuf M. Efficient L-stable method for parabolic problems with application to pricing American options under stochastic volatility Applied Mathematics and Computation 2009 213 1 121 136 10.1016/j.amc.2009.02.060 2533368 ZBL1173.91022 Nielsen B. F. Skavhaug O. Tveito A. Penalty methods for the numerical solution of American multi-asset option problems Journal of Computational and Applied Mathematics 2008 222 1 3 16 10.1016/j.cam.2007.10.041 2462648 ZBL1152.91542 Zhang K. Wang S. Yang X. Q. Teo K. L. A power penalty approach to numerical solutions of two-asset American options Numerical Mathematics. Theory, Methods and Applications 2009 2 2 202 223 2535448 ZBL1212.65390 Kangro R. Nicolaides R. Far field boundary conditions for Black-Scholes equations SIAM Journal on Numerical Analysis 2000 38 4 1357 1368 10.1137/S0036142999355921 1790037 ZBL0990.35013 Matus P. Rybak I. Difference schemes for elliptic equations with mixed derivatives Computational Methods in Applied Mathematics 2004 4 4 494 505 2119619 ZBL1070.65110 Rybak I. V. Monotone and conservative difference schemes for elliptic equations with mixed derivatives Mathematical Modelling and Analysis 2004 9 2 169 178 2063258 ZBL1081.65098 Cen Z. Le A. A robust finite difference scheme for pricing American put options with singularity-separating method Numerical Algorithms 2010 53 4 497 510 10.1007/s11075-009-9316-x 2600921 ZBL1192.91190 Goeleven D. A uniqueness theorem for the generalized-order linear complementary problem associated with M-matrices Linear Algebra and Its Applications 1996 235 221 227 10.1016/0024-3795(94)00141-3 1374261 ZBL0845.90119 Cheng X. Xue L. On the error estimate of finite difference method for the obstacle problem Applied Mathematics and Computation 2006 183 1 416 422 10.1016/j.amc.2006.05.082 2286202 ZBL1133.65039 Glowinski R. Lions J. L. Trémolières T. Numerical Analysis of Variational Inequality 1984 Amsterdam, The Netherlands North-Holland