1. Computational Finance
Lecture 6
Black-Scholes Formula

2. 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

3. Black-Scholes Formula
Stock price process:
Drift:
Volatility:
Risk free interest rate:

4. Black-Scholes Formula
Option prices:
Call option: Strike price
Time to maturity
Put option: Strike price

5. Black-Scholes Formula
must satisfy the following PDE:
and

6. Black-Scholes Formula
European call:
European put:
where

7. 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?

8. 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,

9. Some Extensions: 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

10. 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.

11. 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%

12. Some Extensions: Currency Options
The duplication argument will lead to

13. Some Extensions: Currency Options
Black-Scholes formula for foreign currency options:
Call option:
Put option:
where

14. Implied Volatility
Recall:
European call:
European put:
where

15. 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 But the historical information might be outdated.

16. Implied Volatility More commonly, traders use the following way:
Prices of Actively
Traded Options
Volatility
Pricing Non-Actively
Traded Options

17. Implied Volatility
Objective:
Note that
where
Knowing or , solving for

18. 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 $10 $20 B:
Price at maturity $20 $25 $30 $35 $40
Payoffs $ $ $0 $5 $10

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

20. Implied Volatility Pseudo code: Do while ( ); Let ; If , then ; else
End If
End Loop

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