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Some new computational methods for pricing financial derivatives with jumps

English Abstract

In recent years, efficient computational methods for pricing financial derivatives with jumps have become a widely discussed problem. It can be subdivided into three main parts: numerical partial differential equation methods, numerical integration methods and Monte Carlo simulation methods. This thesis will consider these three approaches to price some financial derivatives under the jump processes. The first part of this thesis is devoted to the options pricing problem under fractional diffusion models using the numerical FPDE methods. Barrier options pricing problem will be first introduced. Next, American options pricing problem will be discussed, which is the focus of this part. A fast preconditioned penalty method is developed for a system of parabolic linear complementarity problems (LCPs) involving tempered fractional order partial derivatives governing the price of American options whose underlying asset follows a geometry Levy process with multi-state regime switching. By ´ means of the penalty method, the system of LCPs is approximated with a penalty term by a system of nonlinear fractional partial differential equations (FPDEs) coupled by a finite-state Markov chain. The system of nonlinear FPDEs is discretized with the shifted Grnwald approximation by an upwind finite difference scheme which is shown to be unconditionally stable. Newton’s method is utilized to solve the difference scheme as an outer iterative method in which the Jacobi matrix is found to possess Toeplitzplus diagonal structure. Consequently, the resulting linear system can be fast solved by the Krylov subspace method as an inner iterative method via fast Fourier transform (FFT). Furthermore, a novel preconditioner is proposed to speed up the convergence rate of the inner Krylov subspace iteration with theoretical analysis. With the abovementioned preconditioning technique via FFT, the operation cost in each Newton’s step can be expected to be O(NlogN), where N is the size of the coefficient matrix. Numerical examples are given to demonstrate the accuracy and efficiency of our proposed fast preconditioned penalty method. In the second part, we consider the accumulator pricing problems under different setting of the contract. First, we review pricing an accumulator in which the barrier is applied continuously. Second, without analytical formulae, the price of an accumulator with barrier applied discretely has to be determined by approximation or numerical methods. The Fourier cosine expansions method, initiated by Fang and Oosterlee iii (2008), is applied to present a numerical method to solve it. The numerical results, compared with Barrier Correction method and Monte Carlo simulation method, are given to show the efficiency of the presented method. The last part gives the financial analysis about the risk hidden in an accumulator contract. The third part of this thesis is concerned about the pricing problems for options embedded in fixed rate mortgages by simulation. The least-squares Monte Carlo method, which was initiated by Longstaff and Schwartz (Rev. Financ. Stud. 14(1): 113-147, 2001), is applied to price the mortgage default and prepayment options in a financial environment with two stochastic factors: house price and short term interest rate. A series of numerical comparisons for presented methods with the PDE analytical approximation method in (IAENG Int. J. Appl. Math. 39(1): 9, 2009) and the binomial tree method (BTM) (Decis. Econ. Financ. 35(2): 171-202, 2012) are given. The simulation experiments show the efficiency of presented methods and some cross-validation of the obtained simulation results are given. Finally, we make a conclusion of this thesis, and list some possible research topics in my further study

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Wang, Wen Fei


Faculty of Science and Technology


Department of Mathematics




Derivative securities -- Prices -- Mathematical models

Options (Finance) -- Prices -- Mathematical models


Ding, Deng

Liu, Zhi

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