A Numerical Scheme Based on Semi-Static Hedging Strategy

TL;DR

This paper introduces a numerical scheme for barrier option pricing based on symmetrizing the underlying diffusion process, simplifying the problem to plain option pricing, and demonstrates its effectiveness through numerical experiments.

Contribution

It proposes a novel symmetrization method for diffusion processes to facilitate barrier option pricing, extending static hedging formulas to more general models.

Findings

The scheme effectively prices barrier options under various models.

Numerical results show improved accuracy and efficiency.

The method applies to models like Black-Scholes, CEV, Heston, and SABR.

Abstract

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.