Analytical and Numerical Approaches to Pricing the Path-Dependent Options with Stochastic Volatility

TL;DR

This paper introduces new analytical and numerical methods for pricing European path-dependent options under stochastic volatility, enhancing accuracy and flexibility in valuation by generalizing existing approaches and studying implied volatility behavior.

Contribution

It extends the path integral method to path-dependent options with stochastic volatility, allowing for more accurate and versatile option valuation.

Findings

Improved path integral method for path-dependent options

Implied volatility can be accurately modeled with proper parameters

Methods applicable to options with any payoff function

Abstract

In this paper new analytical and numerical approaches to valuating path-dependent options of European type have been developed. The model of stochastic volatility as a basic model has been chosen. For European options we could improve the path integral method, proposed B. Baaquie, and generalized it to the case of path-dependent options, where the payoff function depends on the history of changes in the underlying asset. The dependence of the implied volatility on the parameters of the stochastic volatility model has been studied. It is shown that with proper choice of model parameters one can accurately reproduce the actual behavior of implied volatility. As a consequence, it can assess more accurately the value of options. It should be noted that the methods developed here allow evaluating options with any payoff function.