Abstract
In this paper a robust numerical method is proposed for pricing American put options. The Black-Scholes differential operator in the original form is discretized by using a quadratic spline collocation method on a piecewise uniform mesh for the spatial discretization and the implicit Euler scheme for the time discretization. The position of collocation points is chosen so that the spline difference operator satisfies the discrete maximum principle, which guarantees that the scheme is maximum-norm stable. The error estimation is derived by applying the maximum principle to the discrete linear complementarity problem in two mesh sets. It is proved that the scheme is second-order convergent with respect to the spatial variable and first-order convergent with respect to the time variable. Numerical results demonstrate that the scheme is stable and accurate.
1. Introduction
The American option is an important financial instrument that gives the holder the right, but not obligation, to buy (call option) or sell (put option) an asset at any time prior to its maturity date. One way to price American options is to solve a linear complementarity problem involving the Black-Scholes differential operator [1]. Since this complementarity problem is, in general, not analytically solvable, numerical methods are required to obtain the approximate solution.
The spline approximation methods have become interesting and very promising in solving differential equations due to their flexibility in practical applications. The spline solution has its own advantages, for example, once the solution has been computed, the information required for the spline interpolation between mesh points is easy to obtain. A few papers have used the spline approximation methods to solve option pricing problems. Khabir and Patidar [2] applied a B-spline collocation method to solve the heat equation which is obtained from the Black-Scholes equation by an Euler transformation. Kadalbajoo et al. [3, 4] used cubic B-spline collocation methods for the Euler transformed generalized Black-Scholes equation. Mohammadi [5] developed a quintic B-spline collocation method for solving the generalized Black-Scholes equation governing option pricing. Christara and Leung [6] derived a quadratic spline collocation method on a uniform mesh in the pricing problem under a finite activity jump-diffusion model. Rashidinia and Jamalzadeh [7] proposed a modified B-spline collocation approach for pricing American style Asian options.
In this paper we adopt a quadratic spline collocation method on a piecewise uniform mesh to solve the continuous linear complementarity problem arising from American put option pricing. The Black-Scholes differential operator in the original form is discretized. By applying the technique of Surla et al. [8, 9], the position of collocation points is chosen so that the spline difference operator on a piecewise uniform mesh satisfies the discrete maximum principle, which guarantees that the discretization scheme is maximum-norm stable. The error estimation is derived by applying the maximum principle to the discrete linear complementarity problem in two mesh sets. It is proved that the convergence order of the scheme is second-order with respect to the spatial variable and first-order with respect to the time variable. The scheme examined in the present paper overcomes the difficulty in the derivation of the error estimation for the linear complementarity problem under the difference schemes. Numerical results are presented to support these theoretical results.
The outline of the paper is as follows. In Section 2 the continuous linear complementarity problem for pricing American put options is given. In Section 3 the spline difference scheme is derived. In Section 4 the stability and error analysis for the spline difference scheme are proved. In Section 5 numerical experiments are carried out. Finally conclusion and discussion are indicated in Section 6.
2. The Continuous Problem
It is well known that the value of an American put option satisfies the following continuous linear complementarity problem [1, 10]: where represents the Black-Scholes differential operator defined by is the underlying asset price, is the time, is the volatility of the underlying asset price, is the risk-free interest rate, is the continuous dividend rate, is the strike price, is the maturity date, and is the payoff function given by Here we assume that .
The infinite domain needs to be truncated into a bounded domain for solving the problem by the numerical method, where is taken as based on Willmott et al.’s estimate [1]. The boundary condition at is set to be . A detailed discussion of the choice of the linear boundary conditions can be found in [11]. Normally, the error in the computed option price due to the domain truncation is negligible [12]. Therefore, in the remaining of this paper we will focus on solving the following linear complementarity problem
3. Discretization
Since the Black-Scholes differential operator at is degenerative, the Euler transformation is usually used to remove the singularity of the differential operator at (corresponding to in the transformed domain); see, e.g., Schwartz [13], Tangman et al. [14], and Zhao et al. [15]. However, this will result in additional computational errors due to the truncation on the left-hand side of the domain. Furthermore, the originally mesh points will concentrate around inappropriately by using the uniform mesh on the transformed interval. In this paper we will discretize the Black-Scholes equation in the original form, but with a quadratic spline collocation method for the spatial discretization and the implicit Euler scheme for the temporal discretization. It should be remarked that if the commonly quadratic spline collocation method is adopted on a uniform mesh for the original Black-Scholes equation, the stability of the scheme could not be guaranteed. The analogous problems have been discussed in [16, 17].
For the spatial discretization, a piecewise uniform mesh is constructed to guarantee the stability of the discretization scheme:whereFor the temporal discretization, a uniform mesh on with mesh elements is used. Then the domain is divided into the piecewise uniform mesh . The mesh sizes and , respectively, satisfyand It will be shown that the matrix associated with the spline difference operator on the above piecewise uniform mesh is an M-matrix.
The implicit Euler scheme on is used for the time discretization to get the following linear complementarity problem: where and denotes the approximation of the exact solution at -th time level.
The solution of the problem (18)-(22) is approximated by a quadratic spline function which on each subinterval has the following form:The collocation points are chosen in a nonstandard way as that in [8, 9]: for , where . Then the collocation equations are defined by
We can obtain , from the quadratic spline function (24). Since , , we have Substituting the above expressions for , and into (27) on the interval with we havewhereUsing the same technique on the interval with the collocation equation (26) and replacing by in (31), we also can obtainwhere Combining (30) with (33), we have the following spline difference schemewhere Two degrees of freedom are provided by parameters and , which can be used to guarantee that the matrix associated with the discrete operator is an M-matrix. Here we choose and for . In the next section we will prove that and .
Combining (18)-(22) with (35) we can obtain the following fully discretization schemewhereHere we have used the linear interpolation to get the approximated solutions at mesh points and since we only know the numerical solution on mesh points at -th time level.
There exists a unique solution for the above linear complementarity problem (37)-(41). A detailed discussion of existence and uniqueness of the solution for problem (37)-(41) can be found in [18]. Then from the above solution we can obtain the optimal stopping price which is the maximum asset price such that for each .
Remark 1. In general the points are used as the collocation points [19, 20], i.e., . But for the problem (18)-(22) the discrete operator with does not satisfy the discrete maximum principle. Hence, in order to apply the discrete maximum principle and barrier function technique to analyze the stability and error estimate we use collocation points on and on for .
4. Analysis of the Method
The discrete maximum principle, truncation error analysis techniques, and barrier function techniques are used for analysis of the scheme.
Lemma 2 (discrete maximum principle). When and for , the operator on satisfies a discrete maximum principle; i.e., if is a mesh function satisfying , , and , , then for all .
Proof. When and , it is easy to check that for sufficiently small and . We also have Hence we can conclude that and for and . Hence the matrix associated with is an M-matrix, which reveals that the operator satisfies the discrete maximum principle [21].
By using Taylor expansions we can get the following bound of the truncation error where and Thus, when and for , from (47) we havewhere is a positive constant independent of the mesh.
The following theorem is our main result for the quadratic spline collocation scheme.
Theorem 3. Let be the solution of problem (9)-(13) and the solution of problem (37)-(41). Then if we set and for , there exists a positive constant independent of the mesh such that
Proof. The error estimate will be derived by applying the maximum principle to the discrete linear complementarity problem in two mesh sets as that in [22]. Set From (9)-(13) we have Let It is easy to check that Define the barrier functionwhere is a sufficiently large constant independent of and .
For we have by using and (49), (54), and (55). On the “boundary” of the we have and Then we have by using the maximum principle (Lemma 2) to . Hence, we haveFor , but ; thus for the sufficiently large constant we have where we have used and (49). On the “boundary” of the we have and Then we have by using the maximum principle (Lemma 2) to . HenceCombining (60) with (65) we have which completes the proof.
5. Numerical Experiments
In this section we perform the numerical experiments to illustrate the applicability and accuracy of the method obtained in the preceding section. Errors and convergence rates for the spline difference scheme are presented for two examples.
Example 1. Compute the value of an American put option with , and .
Example 2. Compute the value of an American put option with , and .
The projection scheme [23] is used to solve the linear inequality system (37)-(41). Assume that are known; then we first solve the following discretization problem for : Next we set which ensure that (38) and (39) are satisfied.
The computed option values and the constraints with and are plotted in Figures 1–4 for Examples 1 and 2, respectively.
Since the exact solutions of our examples are not available, the quadratic spline approximated solution of is used as the exact solution denoted by . The error estimates for different at are presented. The accuracy in the discrete maximum-norm, and the convergence rate are computed. The error estimates and convergence rates for Examples 1 and 2 are listed in Tables 1 and 2, respectively.
From Figures 1 and 3 we can conclude that the numerical solutions are nonoscillatory, and from Figures 2 and 4 we also can conclude that the numerical solutions satisfy the constraint conditions (38) and (39). Tables 1 and 2 show that is close to 2 for sufficiently large . These results support the convergence estimate of Theorem 3.
Finally, our scheme is compared with the binomial method, analytic approximation method, and compact finite difference methods. The numerical example and numerical results in Zhao [15] are used. The parameters for the American put option are , and . The results for the binomial method are obtained with the time mesh size . The results for the analytical approximation method are obtained with the time mesh size . The results for the compact finite difference methods 1, 2, and 3 are obtained with the spatial mesh size and the time mesh size . For our spline collocation scheme we use the time mesh size and mesh points for the spatial discretization which has almost same number of mesh points as that of the compact methods. The true option values are obtained with the trinomial method using the time mesh size . Table 3 shows that our scheme is more accurate than other methods.
As discussed above, the optimal stopping boundary can be obtained as the maximum asset price such that for each . We plot the optimal stopping boundary for the two examples with in Figures 5 and 6, respectively.
6. Conclusion and Discussion
In this paper the linear complementarity problem arising from American put option pricing is treated by a quadratic spline collocation method on a piecewise uniform mesh. The main advantage of the spline collocation method examined in the present paper lies in the possibility of simple and direct constructing of a continuous approximate solution and normalized flux between the mesh points. The suitable choice of collocation points guarantees that the spline difference operator on the piecewise uniform mesh satisfies the discrete maximum principle. Hence the scheme is maximum-norm stable. The error estimation is derived by applying the maximum principle to the discrete linear complementarity problem in two mesh sets. It is proved that the scheme is second-order convergent with respect to the spatial variable and first-order convergent with respect to the time variable. Numerical results demonstrate that the scheme is stable and accurate.
For the time discretization, although the Crank-Nicolson scheme can be used to improve the truncation error scheme, the discretization scheme does not satisfy the maximum principle without additional constraints about the time sizes, which leads to the difficulty in the derivation of the error estimation for the linear complementarity problem under the difference schemes. In future we plan to overcome this difficulty in the error analysis for the Crank-Nicolson scheme in discretizing the linear complementarity problem.
Since the probability distribution of realized asset returns may exhibit features such as heavy tails, volatility clustering, and volatility smile, two different classes of models have been studied in the finance literature: the jump-diffusion models [24, 25] and the stochastic volatility models [26, 27]. In future we extend this technique to construct spline collocation schemes and study the stability and error analysis of these schemes for the jump-diffusion models and the stochastic volatility models.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work was supported by Major Humanities and Social Sciences Projects in Colleges and Universities of Zhejiang Province of China (Grant no. 2018GH020).