Equilibrium market prices of risks and risk aversion in a complete stochastic volatility model with habit formation: Empirical risk aversion from S&P 500 index options

Considering a pure exchange economy with habit formation utility, the theoretical part of this dissertation explores the equilibrium relationships between the market pricing kernel, the market prices of risks and the market risk aversion under a continuous time stochastic volatility model completed by liquidly traded put options. We demonstrate with these equilibrium relations that the risk neutral pricing partial differential equation is a restricted version of the fundamental pricing equation provided in Garman (1976). We also show that in this completed market stochastic volatility cannot explain the documented empirical pricing kernel puzzle (Jackwerth (2000)). Instead, a habit formation utility offers a possible explanation of the puzzle. The derived quantitative relation between the market prices of risks and the market risk aversion also provides a new way to extract empirical market risk aversion. Based upon this theoretical relation between market prices of risks and the market risk aversion in a Heston model, we empirically extract the market prices of risks and risk aversion from the options market using cross-sectional fitting. Specifically we consider a restricted model where only the volatility risk is allowed to freely change and an unrestricted model where all model parameters are allowed to freely change. For the restricted model, we determine other parameters by Efficient Method of Moments (EMM). Using European call options data, we find an implied risk aversion smile, indicating that individual groups of investors trading options with different strike prices have different risk aversions. We also extracted an average or aggregated market risk aversion by minimizing the mean squared pricing error across all strikes. This represents the risk aversion level for the whole market in the sense of "averaging". None of these risk aversions are negative across moneyness, hence indicating that adding stochastic volatility to the model will not reproduce the documented pricing kernel puzzle. In addition, the market price of volatility risk is small in values compared with the market price of asset risk, implying that the major driving factor of market risk aversion and pricing kernel is the asset risk. This is consistent with the sensitivity analysis conducted on the option prices with respect to the market prices of risks. For the unrestricted model, we observe similar behavior for the two market prices of risks using a different data set, S&P500 index futures options. We find that the asset risk and volatility risk premium generally move opposite across the strikes. The variation of volatility risk decreases and the absolute values converge to zero with longer time to maturity. So the asset risk dominates the pricing more for options with longer maturities.