3. The Binomial Option Pricing Model

Introduction

This chapter examines the first of two general types of option pricing models. A model is a simplified representation of reality that uses certain inputs to produce an output, or result. An option pricing model is a mathematical formula or computational procedure that uses factors determining the option's price as inputs. The output is a theoretical fair value of the option. If the model performs as it should, the option's market price will equal the theoretical fair value. Obtaining the theoretical fair value is a process call option pricing. We begin with a simple model called the binomial option pricing model, which is more of a computational procedure than a formula.

Learning Objectives

o   understand the one-period Binomial Model,

o   understand the two-period Binomial Model,

o   understand the early exercise of American options,

o   perform numerical computation of the Binomial Models.

                    C_u = Max[0, S_d – X]

The following figure illustrates the paths of both the stock and the call price movements. This diagram is simple but will become more complex when we introduce the two-period model.

                                                         $13                                                         $1

                                  $10                                           X = $12                 $0.09

                                                         $11                                                         $0

                                                                                                                               C_u=Max (O, S_u-X)

                                                         S_u

                                 S_o                                                                                   C₀

                                                                                                                               C_d=Max (O, S_d-X)

                                                         S_d

The risk-free rate falls between the rate of return if the stock goes up and the rate of return if the stock goes down. Thus d < I+r < u. We shall assume that investors can borrow and lend at the risk free rate.

The formula for C is developed by constructing a riskless portfolio of stock and options. A riskless portfolio should earn the risk free rate. Given the stock's values and the riskless return of the portfolio, the call's values can be inferred from the other variables. This riskless portfolio is called the hedge portfolio and consists of h shares of stock and a single written call. The model provides the hedge ratio, h. The current value of the portfolio is the value of h shares minus the value of the short call. We subtract the cell's value because the shares are assets and the short call is a liability.

Thus, the portfolio value is assets minus liabilities, or simply net worth. The current portfolio is denoted as V, where V = hS - C. At expiration, the portfolio will have value V_u, if the stock goes up and V_d if the stock goes down. Thus:

                 V_u = hS_u- C_u

                V_d = hS_d- C_d

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