Three essays in mathematical finance: A risk-neutral stochastic volatility model: Analytical and statistical studies. E-ARCH model for term-structure of implied vol of foreign exchange options. A numerical scheme for pricing Asian options

The dissertation consists of three essays. The first essay consists of the first two chapters. It addresses issues in stochastic volatility model. The first chapter deals with the risk-neutral stochastic volatility model. This is done based on Black-Scholes assumption and no-arbitrage pricing principle. The asymptotic behavior of the implied volatility "smile" curve is derived, with implication of skewness for positive and negative correlations between the underlying asset and its volatility. The near-money property of "smile" curve is studied in the second chapter. We use Monte Carlo simulation to study the implied volatility behavior in terms of different model parameters.; The second essay deals with the implied volatility term structure issue, continuing on the same topic as the first two chapters of stochastic volatility model, but with different philosophy as well as different approach. We construct a statistical model for term-structure of implied volatilities of currency options based on daily historical data of Foreign Exchange market. We examine the joint evolution of 1 month, 2 month, 3 month, 6 month and 1 year (50 {dollar}Delta{dollar}) options in all the currency pairs. We show that there exist three uncorrelated state variables (principal components) which account for the parallel movement, slope oscillation, and curvature of the term structure and which explain, on average, the movements of the term structure of volatility to more than 95% in all cases. We test and construct an exponential ARCH, or E-ARCH, model for each state variable. One of the applications of this model is to produce confidence bands for the term structure of volatility.; The last chapter is devoted to the two-dimensional tree model for a class of path-dependent options with averaging aspect, known as "Asian options". A detail numerical algorithm is presented and the result is compared with Monte Carlo simulation. The computational time is generally shorter than published results.