Quarter-Sweep Projected Modified Gauss-Seidel Algorithm Applied to Linear Complementarity Problem

Problem statement: Modified Gauss-Seidel (MGS) was developed in order to improve the convergence rate of classical iterative method in s olving linear system. In solving linear system iteratively, it takes longer time when many computa tional points involved. It is known that by applyin g quarter-sweep iteration scheme, it can decrease the computational operations without altering the accuracy. In this study, we investigated the effect iv ness of the new Quarter-Sweep Projected Modified Gauss-Seidel (QSPMGS) iterative method in solving a Linear Complementarity Problem (LCP). Approach: The LCP we looked into is the LCP arise in American option pricing problem. Actually, American option is a Partial Differential Complementarity Problem (PDCP). By using full-, halfand quarter-sweep Crank-Nicolson finite diffe rence schemes, the problem was reduced to Linear Complementarity Problem (LCP). Results: Several numerical experiments were carried out to t est the effectiveness of QSPMGS method in terms of number o f iterations, computational time and Root Mean Square Error (RMSE). Comparisons were made wit h full-, halfand quarter-sweep algorithm based on Projected Gauss-Seidel (PGS) and Projected Modified Gauss-Seidel (PMGS) methods. Thus, the experimental results showed that the QSPM GS iterative method has the least number of iterations and shortest computational time. The RMS E of all tested methods are in good agreement. Conclusion: QSPMGS is the most effective among the tested itera tiv methods in solving LCP whereby it is fastest and the accuracy remains the ame.


INTRODUCTION
The Linear Complementarity Problem (LCP) is normally applied in the area of computational mechanics, financial engineering and other disciplines in engineering, science and economics. The widely applications of LCP are because it corresponds to the notion of equilibrium and constraint optimization problems (Ferris and Pang, 1997).
In order to define the LCP, consider a matrix M, vector q and unknown vector z. Then, the unknown vector z will be solved in the following conditions: The above formulations are the standard LCP. In this study, we will look into an implicit type of LCP whereby there exists another function y which plays an important role (Koulisianis and Papatheodorou, 2003): Actually, we can solve the LCP by using either direct or iterative methods. However, when we deal with a large sparse linear system, iterative method is preferable. Moreover, it does not consume much memory compared to direct method.
The aim of this study is to introduce a new iterative method known as Quarter-Sweep Projected Modified Gauss-Seidel (QSPMGS) algorithm which will accelerate the convergence rate. It is the combination of Quarter-Sweep approximation scheme with Projected Modified Gauss-Seidel (PMGS) algorithm. Quartersweep iteration scheme is known to be effective to reduce the computational operations and thus speeds up the computational time without altering the accuracy; Sulaiman et al. (2004;. The PMGS algorithm is a preconditioned iterative method based on the Modified Gauss-Seidel (MGS) method, established by Gunawardena et al. (1991) for the purpose of improving the convergence rate of classical iterative methods. Since then, many studies about the MGS method have been carried out like Li and Sun (2000) and Li (2005). Actually, Quarter-Sweep Modified Gauss-Seidel (QSMGS) had been applied to solve PDE in European option pricing problem, Koh and Sulaiman (2009). For verification of the new QSPMGS algorithm in solving LCP, we will examine it in the case of American option pricing.
As American option pricing model involves Partial Differential Complementarity Problem, (PDCP), Crank-Nicolson (CN) scheme will be applied to discretize the PDCP into a LCP. Full-, half-and quarter-sweep CN schemes for approximation of the PDCP will be presented. Then, we will show how the generated LCP solved by PMGS method. Several numerical experiments will be carried out in a family of PGS methods, namely Full-Sweep Projected Gauss-Seidel Case study: American option pricing model: Option is a financial instrument which allows the holder to trade a certain asset in future time with a certain price. The two major styles of options are European and American options. Generally, the difference between them is in the trading aspect as European option can only be traded at the expiration time while American option can be traded at any time before or on the maturity time. Due to this reason, the pricing of American option involved PDCP. The PDCP is shown as follows (Tavella and Randall, 2000): Where: v = Value of the options t = Time s = Underlying asset price σ = Volatility of the asset price r = Risk free interest rate g = Payoff function of the option The final time condition can be defined as follows (Ikonen and Toivanen, 2007): Where: K = The exercise price T = Maturity time The boundary conditions for the American option will be as (Ikonen and Toivanen, 2007): where, S is the maximum asset price whereby it is sufficiently large. The boundary conditions given in (11) and (12) correspond to American call and put options respectively.

Quarter-sweep Crank-Nicolson scheme:
The finite grid network for the full-, half-and quarter-sweep approximation schemes are illustrated in Fig. 1. The solid node points shown in Fig. 1 are the points that will be considered by using full-, half-and quarter-sweep iterative methods. However, the values for the remaining points will be approximated by using direct method, Sulaiman et al. (2004;) and Koh and Sulaiman (2009). The PDE in (9) that is Black-Scholes PDE (Black and Scholes, 1973) can be discretized as follows (Tavella and Randall, 2000;Koh and Sulaiman, 2009): Fig. 1: (a-c) the node points for the full-, half-and quarter-sweep cases respectively Then the approximation Eq. 13 can be simplified in the following equation: Where: ( ) If p is equal to 1, 2 or 4, it represents the full-, half-, or quarter-sweep schemes respectively. Then, we can rewrite (14) in a matrix form as: Where: A family of projected Gauss-Seidel iterative methods: As mentioned before, the Projected Modified Gauss-Seidel (PMGS) method is based on a preconditioned iterative method, namely Modified Gauss-Seidel (MGS) method (Gunawardena et al., 1991). In order to develop and implement a family of PGS algorithm, consider (15) and multiply both sides of the equation with preconditioned such as: Where:

I = Identity matrix
When α = 0, it is the classical Gauss-Seidel (GS) iteration, while if α = 1, it will become MGS method (Gunawardena et al., 1991;Koh and Sulaiman, 2009). Based on (16), the linear system can be rewritten as: Where: A* PA = f * Pf = By using the linear system generated in (17), the PDCP in (9) can be shown as: Now, a LCP has been generated from the PDCP and has the similar form as LCP in (4-6). By considering (18) and the PDCP defined in (7-9), the algorithms of the family of PGS methods will be generally described in Algorithm 1:

Algorithm 1:
i. Initializing all the parameters. Set k = 0. ii. General iteration If the error tolerance is satisfied, the value option at that time is k 1 i v + and the algorithm end. Else, set k = k+1 and go to Step ii.

RESULTS
Several numerical experiments will be performed to examine the effectiveness of FSPGS, HSPGS, QSPGS, FSPMGS, HSPMGS and QSPMGS. The criteria concerned in these experiments include the number of iterations, computational time and Root Mean Square Error (RMSE). The parameters used in these experiments are taken from Hon (2002) whereby K = 100, r = 0.1, σ = 0.30, T = 1(year), s∈[e −5 , e 7 ]. The matrix sizes tested are 512, 1024, 2048, 4096, 8192 and 16384. As for the time steps, we will have 100 time steps which means ∆t will be 0.01. The error tolerance ε = 10 −10 is selected for the convergence test. For comparison, the numerical results obtained will be compared with the results of Binomial method (Hon, 2002). Table 1 presents the experimental results. The results are also illustrated in Fig. 2 and 3.

DISCUSSION
In the computational experiments, we have tested the iterative methods with different mesh sizes in terms of number of iterations, computational time and RMSE.
As the mesh sizes go larger, the more points could be considered, which means option with more precise underlying assets price could be priced. Based on the results for different mesh sizes, the accuracies of all iterative methods are in good agreement. This means that half-and quarter-sweep algorithms computed only parts of the entire node points and their accuracies don't alter.
According to Fig. 2 and 3, QSPMGS has the lowest computational time as while as the least number of iterations. Through numerical results in Table 1

CONCLUSION
In this study, the effectiveness of the Quarter-Sweep Projected Modified Gauss-Seidel (QSPMGS) algorithm has been examined in solving Linear Complementarity Problem (LCP). In the experiments involved full, halfand quarter-sweep algorithm based on Projected Gauss-Seidel (PGS) and Projected Modified Gauss-Seidel (PMGS) methods, QSPMGS proved to be the most effective iterative method. QSPMGS converges faster by having the least number of iterations and thus speed up the execution time.
For future work, further investigation for the capability of the combination of quarter-sweep iteration with MGS method needs to be performed for solving various multidimensional problems (Ibrahim and Abdullah, 1995;Tavella and Randall, 2000;Sulaiman et al., 2009). In fact, we can consider improving the proposed method by implementing block iterative approach.