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# A numerical scheme for pricing American options with transaction costs under a jump diffusion process

• In this paper we develop a numerical method for a nonlinear partial integro-differential complementarity problem arising from pricing American options with transaction costs when the underlying assets follow a jump diffusion process. We first approximate the complementarity problem by a nonlinear partial integro-differential equation (PIDE) using a penalty approach. The PIDE is then discretized by a combination of a spatial upwind finite differencing and a fully implicit time stepping scheme. We prove that the coefficient matrix of the system from this scheme is an M-matrix and that the approximate solution converges to the viscosity solution to the PIDE by showing that the scheme is consistent, monotone, and unconditionally stable. We also propose a Newton's iterative method coupled with a Fast Fourier Transform for the computation of the discretized integral term for solving the fully discretized system. Numerical results will be presented to demonstrate the convergence rates and usefulness of this method.

Mathematics Subject Classification: Primary: 65K15, 65M06; Secondary: 91G60.

 Citation: • • Figure 6.1.  Prices of the European call and put options with $a=0.01$ and $b=0.07$

Figure 6.2.  Prices of the European call and put option for different values of the transaction cost parameter

Figure 6.3.  Computed American and European put option prices

Figure 6.4.  Computed American put option prices for different values of the transaction cost parameter

Table 6.1.  Computed rates of convergence for the call option with $a = 0.01$ and $b=0.07$

 $M$ $N$ $\|\cdot\|_{h,2}$ Ratio$(\|\cdot\|_{h,2})$ 21 11 0.215680 41 21 0.116543 1.85 81 41 0.061550 1.89 161 81 0.031986 1.92 321 161 0.016228 1.97 641 321 0.007861 2.06 1281 641 0.003457 2.27 2561 1281 0.001170 2.96

Table 6.2.  Computed rates of convergence for the put option with $a = 0.01$ and $b=0.07$

 $M$ $N$ $\|\cdot\|_{h,2}$ Ratio$(\|\cdot\|_{h,2})$ 21 11 0.454596 41 21 0.438884 1.04 81 41 0.390547 1.12 161 81 0.327934 1.19 321 161 0.259319 1.26 641 321 0.189478 1.37 1281 641 0.121703 1.56 2561 1281 0.058168 2.09
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