Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 11. The Black-Scholes Model: Hull, Ch. 13.

Transcription

1 Week 11 The Black-Scholes Model: Hull, Ch

2 The Black-Scholes Model Objective: To show how the Black-Scholes formula is derived and how it can be used to value options. 2

3 The Black-Scholes Model 1. Introduction. 2. Black and Scholes Price Dynamics. 3. Volatility. 4. Assumptions Underlying the Black and Scholes Model. 5. The Concept Behind the Black and Scholes Model. 6. Pricing Formulas. 7. Properties of the Black and Scholes Formula. 8. Risk Neutral Valuation. 9. Implied Volatilities. 10. Dividends. 11. American Options. 3

4 1. Introduction 4

5 Introduction Option Pricing Model named after the two economists who discovered it in Goal: Value a European option on a non-dividend paying stock. Steps: Assumptions on the dynamics of stock price, financial markets and interest rates. Analysis Pricing Formula. Applications. Extensions. 5 Jorge Cruz Lopez - Bus 316: Derivative Securities

6 2. Black and Scholes Price Dynamics 6

7 Black-Scholes Price Dynamics In a short period of time of length Dt the percentage change in the stock price (i.e., return) DS/S is assumed to be normal with mean mdt and variance σ 2 Dt Therefore, DS S ~ N m Dt,s Dt where m is the expected return and s is the volatility. 7 Jorge Cruz Lopez - Bus 316: Derivative Securities

8 Black-Scholes Price Dynamics The previous equation indicates that stock prices follow a lognormal distribution (i.e. the log of the prices is normally distributed). Therefore, And D D t T N S S t T S N S T T s s m s s m, 2 ~ ln, 2 ln ~ ln T T e S S E m 0 ) ( 1 ) var( T T T e e S S s m 8 Jorge Cruz Lopez - Bus 316: Derivative Securities

9 Black-Scholes Price Dynamics The Geometric Brownian Motion (GBM): Discrete approximation: 9

10 Binomial Price Dynamics S is the price of the stock. DS/S the percentage change in the price of the stock (i.e. the stock return). ST = S u DS/S = u-1 S e s Dt S ST = S d DS/S = d-1 S e s Dt 10

11 3. Volatility 11

12 Volatility The volatility is the standard deviation of the continuously compounded rate of return in one year. It is a measure of risk, a measure of uncertainty about future returns. The standard deviation of the return in during Dt is: s Dt 12

13 Estimating Volatility from Historical Data 1. Take observations S 0, S 1,..., S n at intervals of t years. 2. Define the continuously compounded return as: u i Si ln S i1 3. Calculate the standard deviation s of the u i. 4. The historical volatility estimate is: (see Appendix 1). sˆ s t 13 Jorge Cruz Lopez - Bus 316: Derivative Securities

14 4. Assumptions Underlying the Black and Scholes Model 14

15 Assumptions Underlying the Black-Scholes Model Stock returns are normally distributed. No transaction costs, no taxes. No dividends. No arbitrage opportunities. Continuous trading. Borrow and lend at r. r is constant through time. 15

16 5. The Concept Behind the Black and Scholes Model 16

17 The Concept Underlying the Black-Scholes Model The option price and the stock price depend on the same underlying source of uncertainty. We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty (i.e. we can create a riskless portfolio). Since the portfolio is instantaneously riskless it must instantaneously earn the risk-free rate. 17

18 The Concept Underlying the Black-Scholes Model Matematically: S risky c(s) risky too V(c,S) = - c(s) + D S dv(c,s) = - dc(s) + D ds where ds denotes the change in S dv(c,s) / V(c,S) = r dt Solve for c and get the Black-Scholes formula. 18

19 6. Pricing Formulas 19

20 The Black-Scholes Formulas c S rt 0 d N( d1) K e N( 2) p K e rt N( d2) S0 N( d1) ln( S0 where d 1 / K) ( r s 2 / 2) T s T d 2 ln( S0 / K) ( r s 2 / 2) T d1 s T s T 20

21 The N(x) Function N(x) is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x. See tables at the end of the textbook. 21

22 7. Properties of the Black and Scholes Formula 22

23 Properties of the Black-Scholes Formula Put-call parity holds Option Lower bounds hold: c max(s 0 Ke rt, 0) p max(ke rt S 0, 0) As S 0 becomes very large c tends to its lower limit, S 0 Ke -rt, and p tends to zero (see Appendix). As S 0 becomes very small c tends to zero and p tends to its lower limit, Ke -rt S 0. 23

24 8. Risk Neutral Valuation 24

25 Risk-Neutral Valuation Notice that m does not appear in the Black-Scholes equation. The equation is independent of all variables affected by risk preference. This is consistent with the riskneutral valuation principle. 25

26 Application of the Black-Scholes Pricing Formula Pricing a Call: (see Normal Distribution Tables) S 0 = 52 K = 50 r = 0.12 s = 0.3 T = 0.25 Pricing a Put: (see Normal Distribution Tables) S 0 = 69 K = 70 r = 0.05 s = 0.35 T = 0.5 Pricing a Call and a Put using Excel. 26

27 9. Implied Volatilities 27

28 Implied Volatility The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price. The is a one-to-one correspondence between prices and implied volatilities. Traders and brokers often quote implied volatilities rather than dollar prices. If the assumptions hold, then the implied volatility is the expected level of volatility during the remaining life of the option. 28

29 1/2/1990 8/14/1990 3/27/ /6/1991 6/19/1992 2/1/1993 9/14/1993 4/26/ /7/1994 7/21/1995 3/4/ /14/1996 5/29/1997 1/12/1998 8/25/1998 4/9/ /18/1999 7/3/2000 2/14/ /3/2001 5/17/ /30/2002 8/13/2003 3/24/ /4/2004 6/20/2005 VIX Index (CBOE) January 1990 to July Gulf War Russian Default Asian Currency Crisis Bursting Dotcom Bubble 9/11 Gulf War II Source: 29 Jorge Cruz Lopez - Bus 316: Derivative Securities

30 INTEL Implied Volatility (CBOE) January 1996 to December 2002 Source: Dubinsky and Johannes, 2005, Columbia University 30

31 10. Dividends 31

32 Dividends European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into the Black-Scholes formula. Only dividends with ex-dividend dates during life of option should be included. The dividend should be the expected reduction in the stock price. 32

33 Dividends (Continued) When the dividend is paid continuously at rate q, we can value European options by reducing the stock price to S 0 e q T and then pricing the option as if there were no dividend payments. 33

34 11. American Call Options 34

35 American Calls An American call on a non-dividendpaying stock should never be exercised early. An American call on a dividendpaying stock should only ever be exercised immediately prior to an exdividend date. 35

36 Black s Approximation This is Black s approach to dealing with dividends in American Call Options. Set the American price equal to the maximum of two European prices: 1. The 1st European price is for an option maturing at the same time as the American option. 2. The 2nd European price is for an option maturing just before the final ex-dividend date. 36