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

Black-Scholes Formula must satisfy the following PDE: and

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 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 But the historical information might be outdated.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: Prices of Actively Traded Options Volatility Pricing 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