Computational Finance

Lecture 6

Black-Scholes Formula

Agenda

How to use the B-S formula in Excel;

Some possible extensions: Stocks with dividends; Options on foreign currencies

Implied volatility and historical volatility

Black-Scholes Formula

Stock price process:

Drift: Volatility: Risk free interest rate:

Black-Scholes Formula

Option prices: Call option: Strike price Time to maturity Put option: Strike price Time to maturity

Black-Scholes Formula

must satisfy the following PDE:

and

Black-Scholes Formula

Black-Scholes formula: European call:

European put:

where

Example

What is the price of a European call option on a non-dividend-paying stock when the stock price is $52, the strike price is $50, the risk-free interest rate is 12% per annum and the volatility is 30% per annum and the time to maturity is three months?

Put-Call Parity Revisited

Suppose that a call and a put with the same strike price, the same time to maturity and on the same underlying stock. Then,

Some Extensions:Options on Dividend Stocks

Options on dividend stocks: Consider a 6-month European call

option on a stock when there are two dividend payments expected in two months and five months.

The dividend of each payment is expected to be $0.5.

Current stock price $40, volatility 30% per annum, risk free interest: 9% per annum;

Strike price $40

Some Extensions:Options on Dividend Stocks

Usually we can view the whole stock prices as the sum of two parts: Riskless component that corresponds

to the known dividend during the life of the option;

Risky component. Reset to be the current stock

price minus the present value of dividends. Then we can use the B-S formula.

Some Extensions:Currency Options

Options on foreign currencies: Consider a four-month European

call option traded in the US market on the British pound. Current exchange rate US$1.9/pound; Strike price: US$1.95 Risk free interest rates: 8% in US,

11% in UK Exchange rate volatility: 20%

Some Extensions:Currency Options

The duplication argument will lead to

Some Extensions:Currency Options

Black-Scholes formula for foreign currency options: Call option:

Put option:

where

Implied Volatility

Recall: European call:

European put:

where

Implied Volatility

In the B-S formula, only one thing is unobservable: stock’s volatility.

One way: Use the historical volatility to price

options. But the historical information might be outdated.

Implied Volatility

More commonly, traders use the following way:

Implied volatility

Prices of Actively Traded Options

VolatilityPricing Non-Actively

Traded Options

Implied Volatility

Objective: Note that

where

Knowing or , solving for

Implied Volatility

Example: Call option with strike price $30. Two stocks, A and B. A is more volatile and B is more placid.

A: Price at maturity $10 $20 $30 $40 $50

Payoffs $0 $0 $0 $10 $20

B: Price at maturity $20 $25 $30 $35 $40

Payoffs $0 $0 $0 $5 $10

Implied Volatility

Mathematically, option prices and are both increasing functions of . Then we can use the so called bisection method.

Implied Volatility

Pseudo code:

Do while ( );

Let ;

If , then

;

else

;

End If

End Loop

Implied Volatility

European call option: Price: $1.875 Underlying stock price: $21 Strike price: $20 Interest rate: 10% Time to maturity: 0.25