A Kernel Method for Pricing and Hedging American Path-Dependent Options

An American option is a contract which provides its holder the right to buy or sell a certain amount of underlying asset at a pre-specified price at any time up to the maturity date. In financial markets, most of the financial derivatives such as exchangetraded stock options are American-style, and thus the pricing and hedging of these options is of tremendous practical interest. Compared to European options, the early exercise feature of American options poses a significant challenge in American option pricing and hedging, and makes it one of the most challenging problems in financial engineering.Among the American options, some of them are path-dependent in the sense that their payoffs depend on path history of the underlying assets. These path-dependent options are actively traded in the market. While the pricing of ordinary American option is already challenging, the presence of path-dependent feature makes the problem even more complicated. Typically no closed-form solutions are available for American path-dependent options, and thus one has to resort to numerical methods. Among various numerical methods, Monte Carlo simulation is a popular one in pricing complex American options, which allows for general model settings for the underlying asset.This project aims to propose an efficient simulation method for pricing and hedging American path-dependent options. We call it a kernel method because it stems from the kernel regression estimation in nonparametric statistics. While a direct implementation of the kernel method may be time-consuming, we propose an approximate algorithm which enables a fast implementation. In its nature, the proposed kernel method, along with two other existing methods, namely, the traditional stochastic mesh method and the least-squares method, can be viewed as members in the family of generalized stochastic mesh methods. Compared to the traditional stochastic mesh method, the kernel method does not require any density information, and can be implemented much faster. Compared to the least-squares method which might be biased, the kernel method is expected to be asymptotically unbiased, leading to a convergent estimator as the sample size increases.In addition to option price, we are also interested in its sensitivities with respect to some market parameters, which play important roles in constructing hedging strategies. It is expected that the kernel method may also be applied in estimating these price sensitivities, with little extra computational effort.