Numerical performance of a family of preconditioned gauss-seidel methods for one and two asset standard option pricings

Koh , Wei Sin (2012) Numerical performance of a family of preconditioned gauss-seidel methods for one and two asset standard option pricings. Masters thesis, Universiti Malaysia Sabah.

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Abstract

Development in numerical techniques has greatly influenced the advancement of quantitative finance in solving any mathematical models concerned efficiently. Recently, solving the Black-Scholes partial differential equations (PDEs), the option pricing models have attracted many mathematicians to contribute and enhance the existing analytical and numerical solutions. Option is a financial instrument which gives its holder the right without obligation, to trade a certain asset in future at a stated price. The most traded option styles in the market are European and American options. In this thesis, the scope of study covers the pricing of European and American options underlying one- and two-asset which are modelled by one and two-dimensional Black-Scholes PDEs respectively. Full-, half-, and quarter sweep Crank-Nicolson finite difference approximations are used to discretize the Black-Scholes PDEs. Hence, linear systems made up of three- and nine-point stencils for one- and two-dimensional problems respectively are generated from the corresponding approximation equations. To solve the European options pricing, a family of preconditioned Gauss-Seidel (GS) methods which consists of Full-Sweep Gauss-Seidel (FSGS), Half-Sweep Gauss-Seidel (HSGS), Quarter-Sweep Gauss-Seidel (QSGS), Full-Sweep Modified Gauss-Seidel (FSMGS), Half-Sweep Modified Gauss-Seidel (HSMGS), Quarter-Sweep Modified Gauss-Seidel (QSMGS), Full-Sweep Improving Modified Gauss-Seidel (FSIMGS), Half-Sweep Improving Modified Gauss-Seidel (HSIMGS) and Quarter-Sweep Improving Modified Gauss-Seidel (QSIMGS) iterative methods are proposed. Due to early exercise of American options, linear complementarity problems (LCPs) are formed and a family of projected preconditioned GS methods are developed. Several numerical experiments for the families of preconditioned GS and projected preconditioned GS methods are also implemented. The performances of these iterative methods are analyzed by observing the number of iterations, computational time and root mean squared error (RMSE). Based on the results for all the problems, QSIMGS and Quarter Sweep Projected Improving Modified Gauss-Seidel (QSPIMGS) iterative methods yield the fastest number of iterations and computational time among the tested methods. Moreover, the accuracies of QSIMGS and QSPIMGS methods are in good agreement with GS and projected GS (PGS) methods respectively. Overall, it can be concluded that QSIMGS and QSPIMGS iterative methods are very efficient in terms of number of iterations and computational time in solving option pricing models.

Item Type:Thesis (Masters)
Uncontrolled Keywords:numerical techniques, Black-Scholes, partial differential equations (PDEs), traded option styles, quartersweep Crank-Nicolson, pricing models
Subjects:Q Science > QA Mathematics
Divisions:SCHOOL > School of Science and Technology
ID Code:11543
Deposited By:IR Admin
Deposited On:20 Aug 2015 12:17
Last Modified:20 Aug 2015 12:17

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