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Accurate Numerical Method for Pricing Two-Asset American Put Options
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.
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 [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  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 [17–19] 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]: where denotes the two-dimensional Black-Scholes operator defined by and is the final (payoff) condition defined by Here, is the value of the option, is the value of the th underlying asset, is the correlation of two underlying assets, is the risk-free interest rate, and is a given function providing suitable boundary conditions. Typically, is determined by solving the associated one-dimensional American put option problem where denotes the one-dimensional Black-Scholes operator defined by
Introducing the logarithmic prices and , the linear complementarity problem (1) is transformed as where
For applying the numerical method, we truncate the infinite domain into , where the boundaries , and 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 and . The artificial boundary conditions at and are chosen to be . The artificial boundary conditions at and are chosen to be . Therefore, in the rest of this paper, we will consider the following linear complementary problem:
The operator 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 . 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 -matrix.
The discretization is performed using a uniform mesh for the computational domain . The mesh steps to the direction, direction, and direction are denoted by , , and . The mesh point values of the finite difference approximation are denoted by
We discretize the differential operator using the central difference scheme on the previous uniform mesh. We set where
Thus, we apply the central difference scheme on the uniform mesh to approximate the parabolic complementarity problem (8) as follows: Here, are discrete approximates of , respectively. Hence, and 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 is an M-matrix. Hence, from the uniqueness theorem of Goeleven , we can obtain that there exists a unique solution 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 then the system matrix corresponding to the discrete operator is an -matrix.
Proof. The difference operator can be written as follows: The coefficient of in the previous expression (which corresponds to the diagonal of the system matrix) is positive since All the coefficients of the other in the previous expression (which correspond to off-diagonal elements in the system matrix) will be nonpositive once the following inequalities are satisfied: Together, they require that the following inequalities hold: which are (14) and (15), respectively. Thus, we have shown that the system matrix, corresponding to the discrete operator is an -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 be a smooth function defined on . Then the truncation error of the difference scheme (10) satisfies for all .
Now we can derive our main result for the difference scheme.
Theorem 3. Let be the solution of the problem (8) and let be the solution of the problem (13). If mesh steps satisfy conditions (14) and (15), the difference scheme (13) satisfies the following error estimate: where is a constant independent of , and .
From (8), we have the result
Define the function on by where is a sufficiently large constant.
For , by the fact that , (25), (26), and Lemma 2, we obtain On the “boundary” of , the nodes , so , but , therefore and the nodes , Applying the maximum principle to , we get Thus,
For , but , thus, On the “boundary” of , the nodes , so , but , therefore and the nodes , Applying the maximum principle to , we get Thus,
From (31) and (36), we obtain where 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: , , , , , , , , , , , and .
Test 2. American put option with parameters: , , , , , , , , , , , and .
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 direction where is the number of mesh steps in the 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 and the convergence rate
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 is close to 4 for sufficiently large , 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.
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).
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