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 But the historical information might be outdated.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