Introduction

• No-arbitrage argument –
• Risk-neutral valuation –
• Consider a three-month call option on the stock with a strike price of USD 52. If the stock price turns out to be USD 60, the payoff for the option will be USD 8. If it turns out to be USD 40, the payoff will be zero.

• Consider a portfolio consisting of a long position in ∆ shares of the stock and a short position in one call option ( ∆ is the Greek capital letter for delta).

Generalization

• Suppose that the price of a non-dividend paying stock is currently S, and that during a time T it will either move up to (providing a return of u – 1) or down to (providing a return of d – 1). Consider a derivative that provides a payoff of if the stock price increases, and a payoff of if the stock price decreases.

• Now a portfolio is formed consisting of:
  • A short position in one unit of the derivative; and
  • A position of ∆ in the stock

where:

The value of the portfolio at time T is

if the stock price increases, and

if the stock price decreases.

• Put-call parity states that when a European put option and a European call option on a non-dividend paying stock have the same strike price and time to maturity, then

Call Price + PV of Strike Price = Put Price + Stock Price

Multi-Step Trees

• Of course, it is quite unrealistic to model stock price changes using a tree with a single step. To create a more realistic valuation model, the life of an option is divided into many steps.

• Let’s define the length of a tree step as ∆t. Hence

where:

• The parameters u and d should be chosen to reflect the volatility of the stock price. If annual volatility is σ then appropriate values for the parameters are

where ∆t is measured in years.

A Two-Step Tree For European Call Option

American Options

• European options can only be exercised at maturity. The calculations change while dealing with American options (i.e., options that can be exercised at any time).

• At each node, two calculations need to be carried out to determine:

1. How much the option is worth if it is exercised at the node, and
2. How much the option is worth if it is not exercised.

The value at the node is the greater of these two.

Since ______________________, the option should _______________

at node C, and thus the value at this node is _______________.

Since ______________________, the option should _______________

at node B, and thus the value at this node is _______________.

Multi-Step Trees

Delta

• Delta was calculated as

Other Assets – Options On Dividend Paying Stocks

• A dividend yield of q implies that the formulas that have been presented so far must be adjusted slightly. The total return in a risk-neutral world is r. Dividends provide a return of q. The expected growth of the stock price must therefore be r – q. In the case where dividends are paid at rate q,

so that:

Everything else about the tree, including the calculation of u and d and the roll back procedure, is the same as before.

Other Assets – Call On Stock Index

• For an option on a stock index, it is usually assumed the index provides a dividend yield

• EXAMPLE –

Consider a stock index standing at 2,500. Suppose the dividend yield on the index is 2% while the risk-free rate is 3%. Suppose further that the volatility of the index is 15% per annum. If a three step-tree is used  to value a European call option with a strike price of 2,500 and a time to maturity of six months, then

 In this case:

and the value of the option given by the three-step tree is 119.579.

• This figure shows a three-step tree for valuing a European call option on an index with a strike price of 2,500. The upper number at each node is the stock price, and the lower number is the option price.

Other Assets – Options On Currency

• A currency can be considered as an asset providing a yield at the foreign risk-free rate. Therefore, the analysis that has been presented for a stock paying a continuous dividend yield applies, with q equal to the foreign risk-free rate (r_x). This means that:

• The rest of the analysis is as explained earlier in this chapter for non-dividend paying stocks.

Other Assets – Call On Currency

EXAMPLE

As an example, consider a four-step tree for a one-year American option to buy a foreign currency for 0.8000 when the current exchange rate is 0.7800. The volatility of the exchange rate is 12%, while the domestic and foreign risk-free rates are 2% and 6% (respectively). In this case:

• The tree is shown in this figure. (The option is exercised early at nodes A and B. The value of the option to buy one unit of the foreign currency is 0.0188.)

Other Assets – Options On Futures

• Because it costs nothing to enter into a futures contract, the return on a futures contract in a risk-neutral world must be zero. This means that a futures contract can be treated like a stock, paying a continuous dividend yield equal to r. This is because when q = r, the expected growth rate of the stock is zero.

Hence

• The rest of the analysis is as explained earlier in this chapter for non-dividend paying stocks.

Other Assets – Put On Futures

EXAMPLE

Consider a three-step tree to value an American nine-month put option on a futures contract when the current futures price is 38, and the strike price is 40. assume the volatility to be 20%, and the risk-free rate to be 4%. In this case:

The value of the option is 3.828.

• This figure shows a three-step tree for valuing an American put option on a futures contract with a strike price of 40. The upper number at each node is the futures price, and the lower number is the option price.