Two applications of Monte Carlo techniques to finance

This dissertation consists of two papers related to Monte Carlo techniques: the first paper is on the efficiency of a Monte Carlo technique and the second paper uses Monte Carlo techniques as a tool to examine a particular model. In the first paper, we describe a technique that can greatly reduce the error variance resulting from using a Monte Carlo simulation to value a derivative security. The technique, known as importance sampling, involves making a change of probability measure under which more sample paths are generated in regions that contribute more to the value of an option. We concentrate on measures under which an asset price has higher expected returns, although we consider a variety of other measures. Instead of taking the sample mean of the discounted payoffs, we must weight each sample by its Radon-Nikodym derivative. We apply the technique to an ordinary call, an Asian option, and an option when the stock price follows a jump diffusion. For an Asian option, we show that importance sampling can significantly outperform the two commonly used variance reduction techniques, the control variate and the antithetic variate methods. The second paper analyzes optimal consumption and portfolio choice when consumers receive service flows from holding durable goods and face transaction costs for adjusting their level of durables in order to determine whether transaction costs can explain the equity premium puzzle. We simulate the optimal behavior of consumers with a given coefficient of relative risk aversion in an economy with transaction costs and then use the resulting covariance between aggregate consumption growth and equity returns to estimate the risk aversion coefficient assuming (incorrectly) that the economy is frictionless. We find that when consumption is derived solely from durables, the frictionless estimates of the level of risk aversion are substantially greater than the true level. However, when a large portion of consumption is in the form of nondurables, the frictionless estimates are close to the true level of risk aversion.