Section 3 provides the numerical result of the Frey model. We compare the option pricing under different liquidity circumstances and compute the speed of programming with direct method and iterative method. Finally, we check the condition number of the linear system in each time step from “coarse grid” to “find grid.”
3.1 Option pricing under perfect liquidity market
In section 3.1, we set the parameter
ρ
= , which means that the Frey model is 0 equivalent to the classical BS model. Our benchmark set of underlying parameters is reported in Table 1.[Insert Table 1 here]
According to Table 1, we obtain that the exact solution of BS call option price is 7.9260. Moreover, we display the profile of the numerical solution in Figure 1 and verify the numerical solution of the Frey model with the BS model in perfect liquidity market. The terminal payoff and intrinsic value of call option are showed in Figure 1.
[Insert Figure 2 here]
3.2 Option pricing under imperfect liquidity market
In section 3.2, we implement the option pricing under illiquid market. The assumption and the parameter setting dose not change except the parameter
ρ
. We want to realize the call option value with different liquidity. In figure 2, we present the outcomes for a 1-year call option with other parameter setting for different values ofthe market liquidity
ρ
ranging from 0 to 0.5. Figure 2 shows that the relationship betweenρ
and call option value. However, we observe that the large trader will spend more money to hedge a call option under worse market liquidity condition. Thus, the hedge cost of the large trader is increasing in the parameterρ
.[Insert Figure 3 here]
3.3 The computation speed of direct method and iterative method One of the important issues is to explore the efficient method for solving the tridiagonal system. While there are several methods for solving the linear system, the iterative and direct methods are explored in our study. Jacobi, Gauss-Seidel (GS) and successive over-relaxation (henceforth, SOR) are most prevailing iterative method (indirect method). For direct method, the Gauss-eliminate type is the basic routines for solving linear system and usually based on some forms of Gaussian elimination with pivoting. The LU decomposition and the Thomas algorithm are the most popular approach in direct method for solving the linear system. It notes worthy that the Thomas algorithm is the most efficient way for solving “the tridiagonal linear system.” If the linear system is not tridiagonal, the Thomas algorithm is not suitable and can not be used in non-tridiagonal type system. The setting of parameters is the same as Table 1 and the parameter
ρ
is assumed to be zero.Table 2 demonstrates some of results from numerical method and compares the computational speed in three different methods. First, the Thomas algorithm is apparently the fastest way to address the tridiagonal system and it can save a lot of time for our procedure. According to the result of the Table 2 is not significant in the column of “elapsed time” because the mesh grid is “coarse.” If we increase the
partition of time and (stock) space, the Thomas algorithm will obviously exhibit its computational power. Second, the disadvantage of GS method is time-consuming and the accuracy of the solution depends on the number of iterations. The detail of the Thomas algorithm will show in Appendix. Hence, we abandon the iterative method because it is time-wasted. In our point of view, the Thomas algorithm is most efficient and fastest method for solving the tridiagonal system but a majority of textbook still uses LU-decomposition to solve the tridiagonal system in each time step. As a result, the Thomas algorithm provides a great improvement in numerical scheme and it decreases the computational costs.
[Insert Table 2 here]
3.4 The condition number of tridiagonal system
The condition number measures the sensitivity of the linear system. As the size of the coefficient matrix increases, the condition number will increase and the solution of the linear system becomes sensitive to the numerical methods. Using a different way to take an inverse in the coefficient matrix will result in a different solution. Table 3 shows that the relationship between the condition number and market liquidity.
[Insert Table 3 here]
Obviously, the condition number ( )
κ A
is increasing in the liquidity parameterρ
when ΔS holds constant. On the other side, we increase the partition of the spaceM and the parameter ρ
holds constant at the same time. In Table 4, the condition number grows about quadruple as ΔS decreases a half and thus we claim that the solution is stable. Consequently, we conclude the numerical solution of the Freymodel is not sensitive to the size of the coefficient matrix because the relative ratio of condition number is quite stable. The following table displays the relative ratio of condition number.
[Insert Table 4 here]
In this section, we not only check the condition number of the linear system at each time step but also list the maximum value of the condition number in each time step. We consider that the sensitivity of the numerical solution is highly correlated with the condition number23. At beginning, we conjecture that the coefficient of matrix could generate a great influence on numerical solution of the option pricing model as the market liquidity becomes worse. However, we demonstrate that the condition number increases quadruply as the partition of the space increases double and thus the sensitivity of the solution is conducted.24
23 See Trefethen and Bau (1997).
24 We should be careful as doing the numerical analysis especially for the application of the option pricing model in finance issue.