# Option Pricing for a General Stock Model in Discrete Time

August 2014

Thesis

## Degree Name

Master of Science

## Department

Mathematics

Richard H. Stockbridge

## Committee Members

Richard H. Stockbridge, Eric Key, Bruce Wade

## Abstract

{As there are no arbitrage opportunities in an efficient market, the seller of an option must find a risk neutral price. This thesis examines different characterizations of this option price. In the first characterization, the seller forms a hedging portfolio of shares of the stock and units of the bond at the time of the option's sale so as to reduce his risk of losing money. Before the option matures, the present value stock price fluctuates in discrete time and, based on those changes, the seller alters the content of the portfolio at the end of each time period. The primal linear program captures the seller's hedging activities. We use linear programming to explore the pricing of options for both the Trinomial Asset Pricing Model and the General Asset Pricing Model, allowing us to consider the pricing of any style of option.

We first look at the Trinomial Asset Pricing Model. This model yields a finite-dimensional linear program and is included to motivate the results for the General Asset Pricing Model. We use the strong duality results for finite-dimensional linear programs to characterize the solution to the primal linear program through the solution of the dual linear program. The dual program can be interpreted as minimizing the expected present value of the contingent claim with respect to measures under which the present value stock price process is a martingale relative to its natural filtration. The dual program provides a second characterization of the option price.

We then present a general asset pricing model in which the present value stock price is a random process. The thesis examines the dual linear program corresponding to the primal linear program arising from the seller's hedging portfolio. The optimal values of the two linear programs are related by weak duality in the general case. In the interpretation of the dual linear program, this paper examines expectations and conditional expectations of stock prices over time. It is here that the use of measure theory in combination with the definition of conditional expectation reveals that, even for this general model, our dual optimization problem minimizes the expected present value of the contingent claim over measures under which the present value stock price process is a martingale. The validity of strong duality between the primal and dual linear programs is not addressed in this thesis.

At the end, we present possibilities for further work on this model.

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