Semi-analytic path integral solution of SABR and Heston equations: pricing Vanilla and Asian options

TL;DR

This paper introduces a semi-analytical path integral method combining analytical and Monte Carlo techniques for efficiently pricing vanilla and Asian options under SABR, Heston, and Bates models with time-dependent parameters.

Contribution

It presents a practical semi-analytical approach using path integrals and Conditional Monte-Carlo for pricing options in complex stochastic models, including Asian options in SABR.

Findings

Efficient pricing of vanilla options in SABR, Heston, Bates models.

Application to Asian options in the SABR model with beta=0.

Compact expressions for three correlated stochastic variables.

Abstract

We discuss a semi-analytical method for solving SABR-type equations based on path integrals. In this approach, one set of variables is integrated analytically while the second set is integrated numerically via Monte-Carlo. This method, known in the literature as Conditional Monte-Carlo, leads to compact expressions functional on three correlated stochastic variables. The methodology is practical and efficient when solving Vanilla pricing in the SABR, Heston and Bates models with time depending parameters. Further, it can also be practically applied to pricing Asian options in the $\beta=0$ SABR model and to other $\beta=0$ type models.