Discrete Dynamics in Nature and Society
Volume 2019 (2019), Article ID 1753782, 11 pages
https://doi.org/10.1155/2019/1753782

Research Article

A Robust Spline Collocation Method for Pricing American Put Options

Zhongdi Cen, Anbo Le, and Aimin Xu

Institute of Mathematics, Zhejiang Wanli University, Ningbo 315100, Zhejiang, China

Correspondence should be addressed to Anbo Le; anbole7@hotmail.com

Received 29 January 2019; Revised 27 March 2019; Accepted 18 April 2019; Published 2 May 2019

Academic Editor: Xiaohua Ding

Copyright © 2019 Zhongdi Cen et al. 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.

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:

where

For 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, satisfy

and

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 have

where

Using the same technique on the interval with the collocation equation (26) and replacing by in (31), we also can obtain

where

Combining (30) with (33), we have the following spline difference scheme

where

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 scheme

where

Here 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 have

where 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 function

where 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 have

For , 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 . Hence

Combining (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.

Figure 1: Computed option value for Example 1.

Figure 2: The constraint for Example 1.

Figure 3: Computed option value for Example 2.

Figure 4: The constraint for Example 2.

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.

Table 1: Error estimates and convergence rates for Example 1.

K
N
error
rate

1024
128
2.3532e-2
2.009

256
5.8452e-3
1.914

512
1.5506e-3
2.342

1024
3.0592e-4
-

Table 2: Error estimates and convergence rates for Example 2.

K
N
error
rate

1024
128
8.6028e-3
2.031

256
2.1045e-3
2.097

512
4.9176e-4
2.451

1024
8.9918e-5
-

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.

Table 3: Comparison of the American put option values at for some different values of with parameters: .

Stock price

Binomial method
Analytic approx. method
Compact method 1
Compact method 2
Compact method 3
Our stable scheme
True values

75.9572

25.33949
25.4509
25.10042
25.32570
25.32739
25.32892
25.32986

error
0.00963
0.12104
0.22944
0.00416
0.00247
0.00094

83.9457

19.49101
19.6617
19.34597
19.49193
19.49383
19.49565
19.49691

error
0.00590
0.16479
0.15094
0.00498
0.00308
0.00126

92.7743

14.27957
14.4477
14.16375
14.25707
14.25914
14.26121
14.26265

error
0.01692
0.18505
0.09890
0.00558
0.00351
0.00144

102.5315

9.87092
10.0278
9.78167
9.83789
9.84000
9.84216
9.84354

error
0.02738
0.18426
0.06187
0.00565
0.00354
0.00138

113.3148

6.35580
6.53401
6.32881
6.36044
6.36241
6.36458
6.36558

error
0.00978
0.16843
0.03677
0.00514
0.00317
0.00100

125.2323

3.84473
3.97728
3.81244
3.82898
3.83064
3.83268
3.83337

error
0.01136
0.14391
0.02093
0.00439
0.00273
0.00069

138.4031

2.14801
2.25467
2.12653
2.13452
2.13578
2.13757
2.13784

error
0.01017
0.11683
0.01131
0.00332
0.00206
0.00027

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.

Figure 5: The optimal stopping boundary for Example 1.

Figure 6: The optimal stopping boundary for Example 2.

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).

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