| Monte Carlo simulation is an important tool in the pricing and hedging of financial instruments. In the dissertation, which consists of three essays, we demonstrate various applications of Monte Carlo simulation in derivative securities pricing.; In the first essay, we consider the pricing of American-style options by parameterizing the early exercise boundary and optimizing with respect to the parameters using a gradient-based simulation algorithm. First, we consider an American call option on a single underlying asset paying multiple discrete dividends and provide the perturbation analysis gradient estimators. Then a different asset price model for the dynamics of the underlying stock price process is introduced and the corresponding gradient estimators are derived. We conclude with a discussion of extensions of the estimator to American-style options with continuous dividend rate.; Next, we consider discrete American-Asian call options in a general setting. First we derive structural properties of the optimal exercise policy. We show that the optimal policy is a threshold policy: the option should be exercised as soon as the average asset price reaches a characterized threshold, which can be written as a function of the asset price at that time. Furthermore, we prove that the threshold level is a nondecreasing and unbounded function of the asset price at that time, and for a large class of models the threshold level is also convex. Then, parameterizing the exercise boundary with a piecewise linear function, we derive gradient estimators with respect to the parameters via perturbation analysis. Using an iterative stochastic approximation algorithm based on the perturbation analysis estimators, we obtain an estimate for the price of the American-Asian option. Numerical experiments carried out indicate that the algorithm performs extremely well.; In the second essay, we consider the valuation of barrier options using stochastic volatility models and local volatility models. The stochastic volatility models are based on time changing homogeneous Levy jump processes. We discuss in detail how to simulate these stochastic volatility models and how to construct a local volatility surface. Barrier option prices from all of the models are compared empirically using a common S&P 500 data set. We observe that local volatility models tend to yield higher down-and-in call option prices and lower up-and-out call option prices. The price differences within stochastic volatility or local volatility models are relatively small. In the third essay, we generalize the trading strategy used on a log contract for replicating the payoff of a variance swap to general convex payoff contracts. Following this generalized trading strategy, the holder of the contract can always receive a positive cashflow at maturity. Monte Carlo simulation is used to investigate the profit and loss of the trading strategy. Under the statistical measure, we consider both geometric diffusion and variance gamma (VG) models, with parameters estimated using actual S&P 500 historical data, whereas the contract value is estimated under the risk-neutral measure. We find that other convex contracts can yield better returns than the log contract, and that the skewness and kurtosis of returns differ somewhat between the geometric diffusion and VG models. |
