  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.

Computational Finance;Pricing of Securities