Since the volatility smile had been found in the real financial market, many literatures attempted to better explain and predict the behavior of option prices across moneyness and maturity. Merton (1976) assumes that the stock return is discontinuous and has a diffusion-jump process instead of the geometric brownian motion. Heston (1993) relax the assumption of constant volatility and assume that the volatility it self follows a square root process. Bates (1996, 2000) and Bakshi et al. (1997, 2000) incorporate both stochastic volatility and Poisson jump, and find a better fitting for the real empirical data.
Huang and Wu (2004) and Carr and Wu (2004) further introduce a time-changed Lévy process that can be used to generate a wide class of jump-diffusion stochastic volatility models such as the variance-gamma jump model of Madan et al. (1998) and the log stable model of Carr and Wu (2003). In the presence of jumps, additional sources of uncertainty including the random jump size, jump frequency, and jump timing make the market incomplete. Consequently, the state price density and the pricing kernel are not unique. Incorporating additional risk components in the underlying process, Pan (2002) and Santa-Clara and Yan (2008) price different market risks in terms of a candidate pricing kernel.
Employing the stochastic volatility and jumps, previous studies significantly reduce the pricing and hedging errors of the Black-Scholes model. All these studies are build on the theorem of no arbitrage and assume a risk neutral preference for individual agents. In this framework, the economy setting is a partial equilibrium. In contrast to no arbitrage method, an alternative pricing approach is based on a fully-stated general equilibrium and incorporates an investor’s risk preference.
In this strand of literature, Naik and Lee (1990) work within the general equilibrium framework to price options written on the market portfolio with discontinuous returns in a representative agent economy of Lucas (1978). They show that the risk premium is equal to the covariance of option payoff with the change in the marginal utility of equilibrium aggregate wealth. Ma (2006) further generalizes the model by assuming the recursive utility of Epstein and Zin (1989) and adopting the Lévy process to model jumps. His option pricing formula is expressed via the inverse Laplace transformation of a complex function through which preference parameters and other aggregate risk factors in the economy affecting option prices are explicitly modeled. Liu et al. (2005) consider a jump-diffusion process of endowment and gauge the ability of an equilibrium model to explain post-crash S&P 500 option prices.
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PDF A Comparison of Option Pricing Models Between General Equilibrium and No Arbitrage Method
