Essays on equilibrium valuation of options, theorem and empirical estimates

grantor: University of TorontoThis thesis consists of three essays which study the valuation of options in an equilibrium framework. The first essay uses a general equilibrium model to study the valuation of options on the market portfolio with predictable returns and stochastic volatility in a complete market. In a closed endowment economy where aggregate dividend is the only source of uncertainty, I investigate why the stock return exhibits certain predictable features. I also examine the equilibrium relationship between the price of the market portfolio and its volatility, as well as the relationship between the spot interest rate and the market volatility. Equilibrium conditions imply that the predictable feature of the market portfolio is induced by the mean-reverting of the rate of dividend growth. It is shown that there is strong interdependence between the stock price process and its volatility process. Using the Euler equation, I derive equilibrium pricing formulas for options on the market portfolio which incorporate both stochastic volatility and stochastic interest rates. Since there is only one source of uncertainty, this model preserves the completeness feature for hedging and risk management purposes. With realistic parameter values, numerical examples show that stochastic volatility and stochastic interest rates are both necessary for correcting the Black-Scholes pricing biases. The second essay focuses on the currency options in an incomplete market where the economy is subject to shocks in aggregate dividend and money supply. The key feature is that the exchange rate exhibits systematic jump risks which should be priced in the currency options. The closed-endowment equilibrium model in the first essay is extended to a small open monetary economy with stochastic jump-diffusion processes for both the money supply and aggregate dividend. It is shown that the exchange rate is affected by both government monetary policies and aggregate dividends. Since the jump in the exchange rate is correlated with aggregate consumption, the jump risk in the exchange rate derived from aggregate consumption must be priced by means of utility maximization. I further derive the foreign agents' risk-neutral valuation of the European currency option and provide restrictions that ensure the law of one price in currency option pricing. In general, these restrictions depend on the agent's risk preference. The objective of the third essay is to empirically study the existence of systematic jump risks in exchange rates and analyze their importance for currency option pricing. The empirical study is based on the theoretical model studied in the second essay, which argues that exchange rates are inherently correlated with the market and so must exhibit systematic jump risks. The third essay uses the maximum-likelihood method to estimate the joint distribution of exchange rates and the price of the market portfolio. Empirical results show that it is important to incorporate both systematic and non-systematic jump components in exchange rates in order to correctly price currency options.Ph.D