This thesis is composed of two parts. The first parts deals with a technique for pricing American-style contingent options. The second part details a statistical arbitrage model using statistical process control approaches. We propose a novel simulation approach for pricing American-style contingent claims. We develop an adaptive policy search algorithm for obtaining the optimal policy in exercising an American-style option. The option price is first obtained by estimating the optimal option exercising policy and then evaluating the option with the estimated policy through simulation. Both high-biased and low-biased estimators of the option price are obtained. We show that the proposed algorithm leads to convergence to the true optimal policy with probability one. This policy search algorithm requires little knowledge about the structure of the optimal policy and can be naturally implemented using parallel computing methods. As illustrative examples, computational results on pricing regular American options and American-Asian options are reported and they indicate that our algorithm is faster than certain alternative American option pricing algorithms reported in the literature. Secondly, we investigate arbitrage opportunities arising from continuous monitoring of the price difference of highly correlated assets. By differentiating between two assets, we can separate common macroeconomic factors that influence the asset price movements from an idiosyncratic condition that can be monitored very closely by itself. Since price movements are in line with macroeconomic conditions such as interest rates and economic cycles, we can easily see out of the normal behaviors on the price changes. We apply a statistical process control approach for monitoring time series with the serially correlated data. We use various variance estimators to set up and establish trading strategy thresholds.

In this thesis we mainly focused on the usage of the Conditional Value-at-Risk (CVaR) in risk management and on the pricing of the arithmetic average basket and Asian options in the Black-Scholes framework via a new log-normal sum approximation method. Firstly, we worked on the linearization procedure of the CVaR proposed by Rockafellar and Uryasev. We constructed an optimization problem with the objective of maximizing the expected return under a CVaR constraint. Due to possible intermediate payments we assumed, we had to deal with a re-investment problem which turned the originally one-period problem into a multiperiod one. For solving this multi-period problem, we used the linearization procedure of CVaR and developed an iterative scheme based on linear optimization. Our numerical results obtained from the solution of this problem uncovered some surprising weaknesses of the use of Value-at-Risk (VaR) and CVaR as a risk measure. In the next step, we extended the problem by including the liabilities and the quantile hedging to obtain a reasonable problem construction for managing the liquidity risk. In this problem construction the objective of the investor was assumed to be the maximization of the probability of liquid assets minus liabilities bigger than a threshold level, which is a type of quantile hedging. Since the quantile hedging is not a perfect hedge, a non-zero probability of having a liability value higher than the asset value exists. To control the amount of the probable deficient amount we used a CVaR constraint. In the Black-Scholes framework, the solution of this problem necessitates to deal with the sum of the log-normal distributions. It is known that sum of the log-normal distributions has no closed-form representation. We introduced a new, simple and highly efficient method to approximate the sum of the log-normal distributions using shifted log-normal distributions. The method is based on a limiting approximation of the arithmetic mean by the geometric mean. Using our new approximation method we reduced the quantile hedging problem to a simpler optimization problem. Our new log-normal sum approximation method could also be used to price some options in the Black-Scholes model. With the help of our approximation method we derived closed-form approximation formulas for the prices of the basket and Asian options based on the arithmetic averages. Using our approximation methodology combined with the new analytical pricing formulas for the arithmetic average options, we obtained a very efficient performance for Monte Carlo pricing in a control variate setting. Our numerical results show that our control variate method outperforms the well-known methods from the literature in some cases.