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


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1 Week 11 The BlackScholes Model: Hull, Ch
2 The BlackScholes Model Objective: To show how the BlackScholes formula is derived and how it can be used to value options. 2
3 The BlackScholes 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 nondividend 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 BlackScholes 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 BlackScholes 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 BlackScholes 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 = u1 S e s Dt S ST = S d DS/S = d1 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 BlackScholes 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 BlackScholes 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 riskfree rate. 17
18 The Concept Underlying the BlackScholes 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 BlackScholes formula. 18
19 6. Pricing Formulas 19
20 The BlackScholes 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 BlackScholes Formula Putcall 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 RiskNeutral Valuation Notice that m does not appear in the BlackScholes 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 BlackScholes 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 BlackScholes price equals the market price. The is a onetoone 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 dividendpaying stocks are valued by substituting the stock price less the present value of dividends into the BlackScholes formula. Only dividends with exdividend 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 nondividendpaying 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 exdividend date. 36
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