The Theoretical Analysis Of American Spread Option Pricing

An American option is a contract which gives the holder the right, but not the obligation, to buy or sell a specific amount of underlying assets at a specified price (the strike price) at any workday before the specified expiry date. And an American spread option is a special American option written on the difference of two underlying assets, known as the spread.When considering American spread option pricing problems, we intend to analyze the influence of various variables, such as assets'prices, dividend, riskless interest, volatility, expiry date and exercise price etc on the option pricing under the assumption of arbitrage-free. Then we can make an optimal strategy for the holder exercise the option. So we should find the optimal exercise boundary of the option, which divide the survival region of the option into two parts, that is, the continuation region and the stopping region.Mathematically the pricing of American spread options can be converted to a free boundary problem in PDE. From this viewpoint we will study the properties of the solution to the corresponding PDE and discuss the influence of solution's various factors to the option price.This thesis is organized as follows. The first section will introduce the basic background and give an overview of the subject, then under the assumption that the underlying asset's price process follows standard geometric Brownian Motion and using theΔ-hedging principle and stochastic analysis theory, we build the mathematical model for American spread options and turn it into the solving of a variational inequation with appropriate transformations. By constructing penalty functions to the option price function, which is the solution to the variational inequation, we turn the variational inequation into a Cauchy problem in PDE in the second section. In the third section, by the convergence of penalty problems and maximum principles, we get a series of estimates to various variables of the option price, such as dividend, riskless interest, volatility, underlying asset's price etc. In the last section, under two different assumptions that the strike price equals zero and is larger than zero, we separately discuss the properties of optimal exercise boundary, stopping region and continuation region of American spread option and study the perpetual case, then we further get some conclusions about their convexity, convergence and monotonicity etc.