General Theory of Geometric L\'evy Models for Dynamic Asset Pricing

TL;DR

This paper develops a comprehensive theory for geometric Lévy models in asset pricing, extending the classical geometric Brownian motion framework and analyzing the effects of volatility and risk aversion on excess returns.

Contribution

It introduces a generalized geometric Lévy model with a pricing kernel approach, clarifies the nonlinear relationship between excess returns, volatility, and risk aversion, and explores implications for foreign exchange models.

Findings

Excess return is positive and increases with volatility and risk aversion.

In foreign exchange, models with volatility exceeding risk aversion can produce positive excess returns for both exchange rate and inverse.

The nonlinear risk-return relationship differs from the classical market price of risk in GBM models.

Abstract

The geometric L\'evy model (GLM) is a natural generalisation of the geometric Brownian motion model (GBM) used in the derivation of the Black-Scholes formula. The theory of such models simplifies considerably if one takes a pricing kernel approach. In one dimension, once the underlying L\'evy process has been specified, the GLM has four parameters: the initial price, the interest rate, the volatility, and the risk aversion. The pricing kernel is the product of a discount factor and a risk aversion martingale. For GBM, the risk aversion parameter is the market price of risk. For a GLM, this interpretation is not valid: the excess rate of return is a nonlinear function of the volatility and the risk aversion. It is shown that for positive volatility and risk aversion the excess rate of return above the interest rate is positive, and is increasing with respect to these variables. In the case of foreign exchange, Siegel's paradox implies that one can construct foreign exchange models for which the excess rate of return is positive both for the exchange rate and the inverse exchange rate. This condition is shown to hold for any geometric L\'evy model for foreign exchange in which volatility exceeds risk aversion.