A Numerical Scheme Based on Semi-Static Hedging Strategy

In the present paper, we introduce a numerical scheme for the price of a barrier option when the price of the underlying follows a diffusion process. The numerical scheme is based on an extension of a static hedging formula of barrier options. For getting the static hedging formula, the underlying process needs to have a symmetry. We introduce a way to "symmetrize" a given diffusion process. Then the pricing of a barrier option is reduced to that of plain options under the symmetrized process. To show how our symmetrization scheme works, we will present some numerical results applying (path-independent) Euler-Maruyama approximation to our scheme, comparing them with the path-dependent Euler-Maruyama scheme when the model is of the Black-Scholes, CEV, Heston, and $ (\lambda) $-SABR, respectively. The results show the effectiveness of our scheme.