Exponential functionals of Levy processes and variable annuity guaranteed benefits

TL;DR

This paper models equity returns using exponential Levy processes instead of geometric Brownian motion to better capture market features and explicitly computes the distribution of exponential functionals relevant to variable annuity guarantees.

Contribution

It introduces a Levy process-based model for equity returns and derives explicit distributions of exponential functionals for valuing annuity guarantees.

Findings

Explicit distribution formulas for exponential functionals of Levy processes

Improved modeling of equity returns over classical Brownian motion

Enhanced valuation methods for variable annuity guarantees

Abstract

Exponential functionals of Brownian motion have been extensively studied in financial and insurance mathematics due to their broad applications, for example, in the pricing of Asian options. The Black-Scholes model is appealing because of mathematical tractability, yet empirical evidence shows that geometric Brownian motion does not adequately capture features of market equity returns. One popular alternative for modeling equity returns consists in replacing the geometric Brownian motion by an exponential of a Levy process. In this paper we use this latter model to study variable annuity guaranteed benefits and to compute explicitly the distribution of certain exponential functionals.