Abstract
We study the pricing of the American options with fractal transmission system under two-state regime switching models. This pricing problem can be formulated as a free boundary problem of time-fractional partial differential equation (FPDE) system. Firstly, applying Laplace transform to the governing FPDEs with respect to the time variable results in second-order ordinary differential equations (ODEs) with two free boundaries. Then, the solutions of ODEs are expressed in an explicit form. Consequently the early exercise boundaries and the values for the American option are recovered using the Gaver-Stehfest formula. Numerical comparisons of the methods with the finite difference methods are carried out to verify the efficiency of the methods.
Highlights
We have developed Laplace transform methods to solve the time-fractional American option pricing under regime switching models
The value of the American option with regime switching is formulated as the solution to a free boundary problem of time-fractional partial differential equation system
The Laplace transform is executed for the time variables and the resulting system partial differential equations (PDEs) are solved analytically
Summary
Markov regime switching models were first introduced by Hamilton [1] and recently have become popular in financial applications including equity options [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17], bond prices and interest rate derivatives [18,19,20], portfolio selection [21], and trading rules [22,23,24,25,26]. Laplace transform methods have been developed to solve the free boundary problems arising in American option pricing under geometric Brownian motion (GBM) (see Mallier and Alobaidi [27], Zhu [28], and Zhu and Zhang [29]) and constant elasticity of variance (CEV) (see Wong and Zhao [30]). Chen et al [31] studied a predictor-corrector finite difference methods for pricing American options under the FMLS model which is a kind of space-fractional derivative model. The solution of the timefractional PDEs with two free boundaries is more challenging than solving the fractional PDEs with fixed boundary for European option pricing in [33] and much more complex than solving single Black-Scholes-Merton differential equation in [32]. We develop the Laplace transform methods for pricing time-fractional American options under regime switching models. The generated option value and early exercise boundary are compared with the finite difference method which are usually used as the benchmark methods in the area of option pricing
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