First-Order Asymptotics of Path-Dependent Derivatives in Multiscale Stochastic Volatility Environment

TL;DR

This paper extends first-order asymptotic analysis to path-dependent derivatives in multiscale stochastic volatility models, showing that vanilla option calibrations can price exotic derivatives effectively, with some cases allowing closed-form solutions.

Contribution

It introduces a method to apply first-order asymptotics to path-dependent derivatives using Dupire's calculus, enabling practical pricing based on vanilla option calibration.

Findings

Market parameters calibrated to vanilla options can price exotic derivatives to the same order.

Conditional expectation representations facilitate numerical evaluation of first-order conditions.

Closed-form solutions are obtainable for certain path-dependent derivatives like Asian options.

Abstract

In this paper, we extend the first-order asymptotics analysis of Fouque et al. to general path-dependent financial derivatives using Dupire's functional Ito calculus. The main conclusion is that the market group parameters calibrated to vanilla options can be used to price to the same order exotic, path-dependent derivatives as well. Under general conditions, the first-order condition is represented by a conditional expectation that could be numerically evaluated. Moreover, if the path-dependence is not too severe, we are able to find path-dependent closed-form solutions equivalent to the fist-order approximation of path-independent options derived in Fouque et al. Additionally, we exemplify the results with Asian options and options on quadratic variation.