The Smile of certain L\'evy-type Models

TL;DR

This paper develops a series expansion method for pricing European options and deriving implied volatility in a class of Le9vy-type models with local dependencies and default risk, providing explicit formulas and practical applications.

Contribution

It introduces a novel series expansion technique for option pricing and implied volatility in Le9vy-type models with local features, including a new class of CEV-like models.

Findings

Series expansion converges to exact option prices under certain conditions.

Explicit implied volatility formulas within the model class.

CEV-like Le9vy models fit S&P 500 implied volatility surface well.

Abstract

We consider a class of assets whose risk-neutral pricing dynamics are described by an exponential L\'evy-type process subject to default. The class of processes we consider features locally-dependent drift, diffusion and default-intensity as well as a locally-dependent L\'evy measure. Using techniques from regular perturbation theory and Fourier analysis, we derive a series expansion for the price of a European-style option. We also provide precise conditions under which this series expansion converges to the exact price. Additionally, for a certain subclass of assets in our modeling framework, we derive an expansion for the implied volatility induced by our option pricing formula. The implied volatility expansion is exact within its radius of convergence. As an example of our framework, we propose a class of CEV-like L\'evy-type models. Within this class, approximate option prices can be computed by a single Fourier integral and approximate implied volatilities are explicit (i.e., no integration is required). Furthermore, the class of CEV-like L\'evy-type models is shown to provide a tight fit to the implied volatility surface of S{&}P500 index options.