Equilibruim approach of asset pricing under Lévy process

Abstract

This work considers the equilibrium approach of asset pricing for Lévy process. It derives the equity premium and pricing kernel analytically for the stock price process, obtains an equilibrium option pricing formula, and explains some empirical evidence such as the negative variance risk premium, implied volatility smirk, and negative skewness risk premium by comparing the physical and risk-neutral distributions of the log return. Different from most of the current studies in equilibrium pricing under jump diffusion models, this work models the underlying asset price as the exponential of a Lévy process and thus allows nearly an arbitrage distribution of the jump component.

Equilibrium approach of asset and option pricing under Lévy process and stochastic volatility

“…In the general equilibrium framework, it is assumed that there is a representative investor who wants to maximize an objective function in a rational expectation economy where there are one risk-free asset and one risky asset. In this work, we model the risky asset as an exponential of a Lévy process with stochastic volatility, which extends the work of Santa-Clara and Yan (2004), Fu and Yang (2012) and Ruan et al (2013). Then we develop exact expressions of the equity premium and pricing kernel in a general equilibrium economy.…”

“…x ( ) , and B t S and B t V are one-dimensional Brownian motions with dB dB dt t S t V = ρ . Then based on the work of Fu (2012) for the Lévy process with constant volatility, we construct the following Lévy process with stochastic volatility for our model…”

“…The above model includes various existing models as special cases. For example, if the volatility for the diffusion process in equation 3is taken as a constant σ , then the model reduces to Fu and Yang (2012). From (2) and (3), by using the Ito formula, we obtain…”

“…1 which is the result of Fu (2012). We now establish the pricing kernel which acts like the physical and risk-neutral densities in the economy and is the key to obtain the PDE for option price.…”

“…Zhang et al (2012) removes these restrictions, and uses a compound Poisson distribution for the jump component. Moreover, Fu and Yang (2012) also relaxes those restrictions and uses a general Lévy process, but a constant volatility is assumed. Ruan et al (2013) considers stochastic volatility, but only with Poisson distribution for the jump component.…”