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. To get 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
applied to our
scheme, comparing them with the path-dependent Euler–Maruyama scheme when the model is of the type Black–Scholes,
CEV, Heston, and (λ)-SABR, respectively.
The results show the effectiveness of our scheme.