Efficient qricing and numerical methods for derivatives

Abstract

This thesis focuses on numerical methods of computing derivative prices. The approximation methods can be divided into two categories. One is based on the analytic approximation and the other is based on the discretization of variables. First, we study analytic approximations to the prices of American style options. For standard American options, we develop new upper and lower bounds on option prices that improve the bounds by Broadie and Detemple (1996). The main idea is the consideration of doubly capped call options that have two cap prices. We present a new option price approximation based on the two upper bounds. On average, our upper bound extrapolation (named UBE) has an average accuracy better than a 1,000 time-step binomial tree with a computation speed comparable to a 100 time-step binomial tree. We also provide a new method of approximating the optimal exercise boundaries of American options. Price approximation methods for American capped call options and American floating-strike lookback options both of which do not have integral form analytic solutions are also introduced. Second, we propose lattice methods for path-dependent options, reset options and Asian options. A reset option is a path-dependent contingent claim whose strike price can be adjusted in favor of its holders at the predetermined reset dates. A standard reset option uses the underlying stock prices at reset dates as "trigger prices", and an arithmetic average reset option uses the arithmetic averages of the stock prices. Compared with standard reset options, arithmetic average reset options have the advantages of preventing price manipulations and of reducing the hedging problem caused by sudden changes in option delta near the reset dates. We develops a lattice method to numerically compute the value of arithmetic average reset options. We also consider the pricing of discretely sampled Asian and average reset options. Using a change of numeraire under the CRR (Cox, Ross, and Rub...