Pricing and hedging derivative securities using non-Gaussian distributions

The essays in my dissertation study the problem of pricing and hedging interest rate derivative securities. Two assumptions that underlie the basic no-arbitrage methodology of pricing derivatives are relaxed. The aim of studying the effects of relaxing these assumptions is that they will allow us to build models that are more realistic and those that result in prices closer to the market price.; The assumptions that are relaxed are the ones that require that--(1) the volatility of the price of the underlying asset be constant/deterministic; and (2) price paths be continuous. Models/interest rate process specifications that result in option prices that fit observed market prices better are considered. Current research on the subject has found that simple diffusion processes result in option pricing errors that have statistically and economically significant biases related to option maturity and moneyness. Processes with jumps and stochastic volatility are used to see if these biases can be reduced or eliminated.; I find that models with stochastic volatility specifications result in option prices that better fit observed market prices. Stochastic volatility specifications cause the distribution of future interest rates to be more fat-tailed than the normal distribution, capturing the 'volatility smile' related effects. Further improvement in model performance is obtained if the volatility is allowed to depend on the level of the short-term interest rates. This allows the model to generate the skewed volatility smiles observed in market prices.; Stochastic volatility alone is insufficient to capture all the 'volatility smile' related effects observed in interest rate options that are close to expiration. Jumps/other dis-continuous interest rate processes are needed to fit observed prices of these options. The second part of my thesis develops a general option pricing formula when the uncertainty is driven by a dis-continuous process. This is done in an equilibrium framework where the jump risk is systematic. The option pricing formula obtained is similar in form to Merton's jump-diffusion option pricing formula. Using examples I show how this formula can be specialized to the case of the jump-diffusion process and the gamma process.