From characteristic functions to implied volatility expansions

TL;DR

This paper introduces an explicit polynomial expansion for implied volatility based on characteristic functions, applicable to various martingale models, and useful for model calibration of implied volatility surfaces.

Contribution

It provides a novel, integral-free polynomial expansion for implied volatility derived from characteristic functions, applicable across different models and aiding in model calibration.

Findings

Effective approximation of implied volatility in Le9vy and stochastic volatility models

No integrals needed, only polynomial calculations

Facilitates model-free calibration of implied volatility surfaces

Abstract

For any strictly positive martingale $S = \exp(X)$ for which $X$ has a characteristic function, we provide an expansion for the implied volatility. This expansion is explicit in the sense that it involves no integrals, but only polynomials in the log strike. We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three well-known martingale models: one finite activity exponential L\'evy model (Merton), one infinite activity exponential L\'evy model (Variance Gamma), and one stochastic volatility model (Heston). Finally, we illustrate how our expansion can be used to perform a model-free calibration of the empirically observed implied volatility surface.