PhD Oral Exam – Ali Boloor Foroosh

Abstract

In this thesis we present three papers in empirical and theoretical derivative pricing.

In the first essay, which is an empirical study on equity options, we use simultaneous data from equity, index and option markets in order to estimate a single factor market model in which idiosyncratic volatility is allowed to be priced. We model the index dynamics' P-distribution as a mean-reverting stochastic volatility model as in Heston (1993), and the equity returns as single factor models with stochastic idiosyncratic volatility terms. We derive theoretically the underlying assets' Q-distributions and estimate the parameters of both P- and Q-distributions using a joint likelihood function. We document the existence of a common factor structure in option implied idiosyncratic variances. We show that the average idiosyncratic variance, which proxies for the common factor, is priced in the cross section of equity returns, and that it reduces the pricing error when added to the Fama- French model. We find that the idiosyncratic volatilities differ under P- and Q-measures, and we estimate the price of this idiosyncratic volatility risk, which turns out to be always significantly different from zero for all the stocks in our sample. Further, we show that the idiosyncratic volatility risk premiums are not explained by the usually equity risk factors. Finally, we explore the implications of our results for the estimates of the conditional equity betas.

In the next two essays, we present a theoretical methodology for the pricing of catastrophe derivatives. In the second essay, we present a new approach to the pricing of catastrophe event derivatives that does not assume a fully diversifiable event risk. Instead, we assume that the event occurrence and intensity affect the return of the market portfolio of an agent that trades in the event derivatives. Based on this approach, we derive values for a CAT option and a reinsurance contract on an insurers assets using recent results from the option pricing literature. We show that the assumption of unsystematic event risk seriously underprices the CAT option.

Last, we present numerical results for our derivatives using real data from hurricane landings in Florida. In the last essay, we extend the methodology developed in the second essay by relaxing several of its assumptions, and apply it to the valuation of a reinsurance contract given the value of a futures contract indexed on the CAT event. Since the payoff of the reinsurance contract has the form of a vertical spread, our methodology is also applicable to the valuation of derivatives with non-convex payoffs in other markets. Our approach recognizes the fundamental incompleteness of financial markets arising from the occurrence of rare events. Moreover, our approach does not rely on the existence of a representative agent and his/her risk preferences, and we do not assume knowledge of the martingale probability measure beyond the futures price. Using stochastic dominance methodology, we derive bounds for the value of the reinsurance contract. The derived bounds represent reservation write and purchase prices for the reinsurance contract, violation of which result in second degree stochastically dominating strategies that increase the expected utility of any risk averse investor.

Our method is general in nature and is independent of distributional assumptions on the CAT event amplitude. It can be generalized without reformulation to any Markovian process that may include dependence between the amplitude distribution and the frequency of occurrence of the event.