Abstract
Problem statement: This study presents the numerical solution of two-dimensional European option pricing problem based on Quarter-Sweep Modified Gauss-Seidel (QSMGS) iterative method. In fact, the pricing of European option with two-underlying assets can be governed by two-dimensional Black-Scholes Partial Differential Equation (PDE). Approach: The PDE needs to be discretized by using full-, half- and quarter-sweep second-order Crank-Nicolson schemes to generate a system of linear equations. Then, the Modified Gauss-Seidel, a preconditioned iterative method is applied to solve the generated linear system. Results: In order to examine the effectiveness of QSMGS method, several numerical experiments of Full-Sweep Gauss-Seidel (FSGS), Half-Sweep Gauss-Seidel (HSGS) and Quarter-Sweep Gauss-Seidel (QSGS) methods are also included for comparison purpose. Thus, the numerical experiments show that the QSMGS iterative method is the fastest in computing as well as having the least number of iterations. In the error analysis, QSMGS method shows good and consistent results. Conclusion: Finally, it can be concluded that QSMGS method is superior in increasing the convergence rate.
Highlights
Option is a financial derivative which gives the holder the right to trade the underlying asset by a certain date for a certain price
Several numerical experiments are performed to test the effectiveness of Full-Sweep Gauss-Seidel (FSGS), Half-Sweep Gauss-Seidel (HSGS), Quarter-Sweep Gauss-Seidel (QSGS) and Quarter-Sweep Modified Gauss-Seidel (QSMGS) iterative methods
3 they clearly show that QSMGS method has the least number of iterations and execution time among the tested iterative methods
Summary
Option is a financial derivative which gives the holder the right to trade the underlying asset by a certain date for a certain price. The right to buy is known as call option while the vice versa is put option. Black and Scholes (1973) and Merton (1973) derived the Black-Scholes Partial Differential Equation (PDE) for option pricing which earned them the 1997 Nobel Prize in Economics. We focus on a two-dimensional Black-Scholes PDE as follows Eq 1 (Stulz, 1982; Jeong et al, 2009): ∂v ∂t = − σ12s12 ∂2v ∂s12 σ22s ∂2v ∂s22 (1) − ρσ1σ2s1s2 ∂2v ∂s1∂s2
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