# Accurate Numerical Method for Pricing Two-Asset American Put Options

Authors: Xianbin Wu
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

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.


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## Body

## 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 [1] 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 [4], the Monte Carlo method [5], the finite difference method [6–8], the finite element method [9, 10], and the finite volume method [11–13] 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 [14] presented explicit finite difference methods with an acceleration technique for option pricing. Clarke and Parrott [15] and Oosterlee [16] used finite difference schemes along with a projected full approximation scheme (PFAS) multigrid for pricing American options under stochastic volatility. Ikonen and Toivanen [17–19] proposed finite difference methods with componentwise splitting methods on nonuniform grids for pricing American options under stochastic volatility. Hout and Foulon [20] and Zhu and Chen [21] applied finite difference schemes based on the ADI method to price American options under stochastic volatility. Le et al. [22] presented an upwind difference scheme for the valuation of perpetual American put options under stochastic volatility. Yousuf [23] developed an exponential time differencing scheme with a splitting technique for pricing American options under stochastic volatility. Nielsen et al. [24] and Zhang et al. [25] 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 Section3, 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)ℓP(S1,S2,t)≥0,S1,S2>0,t∈[0,T),ℓP(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,S2≥0,t∈[0,T],P(S1,S2,T)=ϕ(S1,S2),S1,S2≥0,P(0,S2,t)=g2(S2,t),S2≥0,t∈[0,T],P(S1,0,t)=g1(S1,t),S1≥0,t∈[0,T],limS1→∞P(S1,S2,t)=0,S2≥0,t∈[0,T],limS2→∞P(S1,S2,t)=0,S1≥0,t∈[0,T],
where ℓ denotes the two-dimensional Black-Scholes operator defined by
(2)ℓP(S1,S2,t)≡-∂P∂t-12σ12S12∂2P∂S12-ρσ1σ2S1S2∂2P∂S1∂y-12σ22S22∂2P∂S22-rS1∂P∂S1-rS2∂P∂S2+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)ℓ-igi(Si,t)≥0,Si>0,t∈[0,T),ℓ-igi(Si,t)·[gi(Si,t)-max(E-αiSi,0)]=0,Si>0,t∈[0,T),gi(Si,t)-max(E-αiSi,0)≥0,Si≥0,t∈[0,T],gi(Si,T)=max(E-αiSi,0),Si≥0,gi(0,t)=E,limSi→∞gi(Si,t)=0,t∈[0,T],
where ℓ-i denotes the one-dimensional Black-Scholes operator defined by
(5)ℓ-igi(Si,t)≡-∂gi∂t-12σi2Si2∂2gi∂Si2-rSi∂gi∂Si+rgi,i=1,2.Introducing the logarithmic pricesx=lnS1 and y=lnS2, the linear complementarity problem (1) is transformed as
(6)Lu(x,y,t)≥0,(x,y,t)∈ℝ×ℝ×[0,T),Lu(x,y,t)·[u(x,y,t)-φ(x,y)]=0,(x,y,t)∈ℝ×ℝ×[0,T),u(x,y,t)-φ(x,y)≥0,(x,y,t)∈ℝ×ℝ×[0,T],u(x,y,T)=φ(x,y),(x,y)∈ℝ×ℝ,u(0,y,t)=g2(ey,t),(y,t)∈ℝ×[0,T],u(x,0,t)=g1(ex,t),(x,t)∈ℝ×[0,T],limx→∞u(x,y,t)=0,(y,t)∈ℝ×[0,T],limy→∞u(x,y,t)=0,(x,t)∈ℝ×[0,T],
where
(7)Lu(x,y,t)≡-∂u∂t-12σ12∂2u∂x2-ρσ1σ2∂2u∂x∂y-12σ22∂2u∂y2-(r-12σ12)∂u∂x-(r-12σ22)∂u∂y+ru,φ(x,y)=max{E-(α1ex+α2ey),0}.For applying the numerical method, we truncate the infinite domain intoΩ≡(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 [26]. Based on Willmott et al.'s estimate [3] 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 operatorL 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 [22] 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 meshΩN,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,jk≈u(xi,yj,tk)fori=0,1,…,N;j=0,1,…,M;k=0,1,…,K.We discretize the differential operatorL 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)∣0≤i≤N,0≤j≤M,0≤k≤K},Ω~h={(i,j,k)∣1≤i≤N-1,1≤j≤M-1,Ω~h=a1≤k≤K-1},∂Ωh=Ω¯h∖Ω~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,jk≥0,(i,j,k)∈Ω~h,Ui,jk-φi,j≥0,(i,j,k)∈Ω~h,LN,M,KUi,jk·[Ui,jk-φi,j]=0,(i,j,k)∈Ω~h,Ui,jK=φi,j,0≤i≤N,0≤j≤M,U0,jk=(g2)jk,UN,jk=0,0≤j≤M,0≤k<K,Ui,0k=(g1)ik,Ui,Mk=0,0<i<N,0≤k<K.
Here, (g1)ikand(g2)jk are discrete approximates of g1(ex,t)andg2(ey,t), respectively. Hence, (g1)ik and (g2)jk can be obtained by solving the corresponding one-dimensional Black-Scholes equations [29]. 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 [30], 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)Δx≤σ12|2r-σ12|,Δy≤σ22|2r-σ22|,(15)2|ρ|σ1σ2≤ΔxΔy≤2σ1|ρ|σ2,
then the system matrix corresponding to the discrete operator LN,M,K is an M-matrix.Proof.
The difference operatorLN,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)2≥0.
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|2Δx≥0,σ224(Δy)2-|r-(1/2)σ22|2Δy≥0,σ1σ2ρ~+-ρ~-2ΔxΔy-σ224(Δy)2≤0,σ1σ2ρ~+-ρ~-2ΔxΔy-σ124(Δx)2≤0.
Together, they require that the following inequalities hold:
(19)Δx≤σ12|2r-σ12|,Δy≤σ22|2r-σ22|,2σ1(ρ~+-ρ~-)σ2≤ΔxΔy≤2σ1σ2(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.
Letu(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.
Letu(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~h∣u(xi,yj,tk)=φ(xi,yj)},Ω(2)=Ω~h∖Ω(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~h∣Ui,jk=φ(xi,yj)},Ωh(2)=Ω~h∖Ωh(1).
Obviously,
(25)LN,M,KUi,jk=0,(i,j,k)∈Ωh(2).
Define the function onΩ~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,jk≥0,
and the nodes (i,j,k)∈∂Ωh,
(29)u(xi,yj,tk)-Ui,jk+Wi,jk=Wi,jk≥0.
Applying the maximum principle to Ωh(2), we get
(30)u(xi,yj,tk)-Ui,jk+Wi,jk≥0,(i,j,k)∈Ωh(2).
Thus,
(31)u(xi,yj,tk)-Ui,jk+Wi,jk≥0,(i,j,k)∈Ω-h.
For(i,j,k)∈Ω(2),Lu(xi,yj,tk)=0, but LN,M,KUi,jk≥0, 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,jk≤0.
On the “boundary” of Ω(2), the nodes (i,j,k)∈Ω(1), so u(xi,yj,tk)=φ(xi,yj), but Ui,jk≥φ(xi,yj), therefore
(33)u(xi,yj,tk)-Ui,jk-Wi,jk=φ(xi,yj)-Ui,jk-Wi,jk≤0,
and the nodes (i,j,k)∈∂Ωh,
(34)u(xi,yj,tk)-Ui,jk-Wi,jk=-Wi,jk≤0.
Applying the maximum principle to Ω(2), we get
(35)u(xi,yj,tk)-Ui,jk-Wi,jk≤0,(i,j,k)∈Ω(2).
Thus,
(36)u(xi,yj,tk)-Ui,jk-Wi,jk≤0,(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,jk≤C[(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 Tables1 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. [24] and Zhang et al. [25] verify that their schemes are only first-order convergent. Hence, our scheme is more accurate.Table 1
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
1.837Table 2
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



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