Understanding Options

Tuesday, February 26, 2019

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

Monday, January 21, 2019

The One-Period Binomial Model

First, let us define what we mean by a one period world. An option has a defined life. Assume the option's life is one unit of time, typically expressed in days. This time period can be as short or as long as necessary.

The model is called a binomial model. It allows the stock price to either up or down, possibly at different rates. A binomial probability distribution is a distribution in which there are two outcomes or states. The probability of an up or down movement is governed by the binomial probability distribution. Because of this, the model is also called a two-state model.

Consider a world in which there is a stock price at S on which call options are available. The call has one period remaining before it expires. The beginning oldie period is today and is referred to as time 0. The end of the period is referred to as time 1. When the call expires, the stock can take one of two values: It can go up by a factor of u or down by a factor of d. If it goes up, the stock price will be S_u_.If it goes up, the stock price will be S_d.

Alternatively, suppose the stock price is currently $10 and can go either up to $13 or up to $11. Thus, u = 1.30 and d = 1.1. The variable u and d, therefore, are 1.0 plus the rate of return on the stock. When the call expires, the stock will be either S_u, = 10(1.3) = $13 or S_d 10(1.1) = $11.

Consider a call option on the stock with an exercise price of X and a current price of C. When the option expires, it will be worth either C_u or C_d. Because at expiration the call price is its intrinsic value:

C_u = Max[0, S_u – X]

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.

V_u = hS_u- C_u

V_d = hS_d- C_d

If the same outcome is achieved regardless of what the stock price does, the position is riskless. We can choose a value of h that will make this happen. We simply set V_u = V_d so that:

hS_u- C_u = hS_d- C_d

Solving for h gives:

Moreover, if the portfolio's current value grows at the risk free rate, its value at the option's expiration will be

(hS - C)(1+r).

The two values of the portfolio at expiration, V_u and V_d, are equal, so we can choose either one. Choosing Vu and setting it equal to the original value of the portfolio compounded at the risk free rate gives:

V(1 + r) = V_u

(hS - C)(1+r) = hS_u- C_u

Substituting the formula for h and solving the equation for C gives the option pricing formula:

where p is defined as

The variables affecting the call option price are the current stock price, S, the exercise price, X, the risk free rate, r, and the parameters, u and d, which define the possible future stock prices at expiration.

Tuesday, October 31, 2017

Put-Call Parity

The prices of European puts and calls on the same stock with identical exercise prices and expiration dates have a special relationship. The put price, call price, stock price, exercise price, time to expiration, and risk-free rate are all related by a formula called put-call parity. Let us see how this formula is derived.

Imagine a portfolio, called portfolio A, consisting of one share of stock and one European put. This portfolio will require an investment of S₀+ P_e(S₀, T, X). Now consider a second portfolio, called portfolio B, consisting of a European call with the same exercise price and risk-free pure discount bonds with a face value off. That portfolio will require an investment of C_e(S₀, T, X) + X(1 +r)^-T. Now let us look at what happens at expiration. Table 2.1 presents the outcomes.

Table 2.1

				Payoffs from Portfolio
Portfolio		Current Value	S_T ≤ X	S_T > X
A	Long put	P_e(S₀, T, X)	S_T – X	0
	Stock	S₀	S_T	S_T
			X	S_T
B	Long call	C_e(S₀, T, X)	0	S_T – X
	Long bond	X(1 +r)^-T	X	X
				S_T

The stock is worth S_T regardless of whether S_T is more or less than X. Likewise the risk-free bonds are worth X regardless of the outcome. If S_T exceeds X, the call expires in the money and is worth S_T – X and the put expires worthless. If S_T is less than or equal to X, the put expires in-the-money worth S_T – X and the call expires worthless.

The total values of portfolios A and B are equal regardless of the outcome of the stock price. However, given the Law of One Price, the current value of the two portfolios must be equal. Thus we require that:

S₀+ P_e(S₀, T, X) = C_e(S₀, T, X) + X(1 +r)^-T

This statement is referred to as put-call-parity and it is probably one of the most important results in understanding options. It says that a share of stock plus a put is equivalent to a call plus risk-free bonds. It shows the relationship between the call and put prices, the stock price, the time to expiration, and the exercise price.

Suppose the combination of the put and the stock is worth less than the combination of the call and the bonds. Then you can create an arbitrage portfolio by buying the put and stock and short selling the call and the bonds. Selling short a call just means writing the call, and selling short the bonds simply means to borrow the present value of X and promise to pay back X at the option's expiration. Since everyone would start doing this transaction, the prices would be forced back in line with the put-call parity equation.

By observing the signs in front of each term, we can easily determine which combinations replicate others. If the sign is positive, we should buy the option, stock or bond. If the sign is negative, we should sell. For example, suppose we isolate the call price:

C_e(S₀, T, X) = S₀+ P_e(S₀, T, X) - X(1 +r)^-T

Then owning a call is equivalent to owning a put, owning the stock, and selling the bonds (borrowing). If the stock pay dividends, once again, we simply insert

which is the stock price minus the present value of the dividends.

While put-call parity is an extremely important and useful result, it does not hold so neatly if the options are American. The put-call parity rule for American options must be stated as inequalities:

where S’₀ is, again, the stock price minus the present value of the dividends.

The Effect of Stock Volatility

The effect on a pat's price is the same as that for a call: Higher volatility increases the possible gains for a put holder. This is because greater volatility increases the gains on the put if the stock price increases, because the stock price can drop below the exercise price by a larger amount. On the other hand, greater volatility means that if the stock price goes up, it can also be much higher than the exercise price. To a put holder, however, this does not matter because the potential loss is limited, it is said to be truncated at the exercise price.

The Effect of Interest Rates

In contrast to call option prices, which vary directly with interest rates, put option prices vary inversely with interest rates. Purchasing a put is like deferring the sale of the stock. When you finally sell the stock by exercising the put, you receive X dollars. If interest rates increase, the X dollars will have a lower present value. Thus, a put holder forgoes higher interest while waiting to exercise the option and receive the exercise price. Higher interest rates make puts less attractive to investors.

The Early Exercise of American Puts

Let us suppose there are no dividends. Suppose you hold an American put and the stock goes bankrupt, meaning that the stock price goes to zero. You are holding an option to sell it for X dollars. There is no reason to wait until expiration to exercise it and obtain your X dollars. You might as well exercise it now. Thus, bankruptcy is one obvious situation in which an American put would be exercised early. However, bankruptcy is not required to justify early exercise. If the stock price falls to a critical level - and thus cannot fall much further - an American option might be exercised early and the funds reinvested.

If the stock pays dividends, it might still be worthwhile to exercise it early, but because dividends drive the stock price down, they may make American puts less likely to be exercised early. In fact, if the dividends are sufficiently large, it can sometimes be shown that the put would never be exercised early, thus making it effectively a European put.