华东师范大学：《金融工程》英文版 Chapter 11 The Black-Scholes Model

The black-Scholes Model Chapter 11 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

11.1 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University The Black-Scholes Model Chapter 11

11.2 The Stock Price Assumption >Consider a stock whose price is s >In a short period of time of length At the change in then stock price S is assumed to be normal with mean uSat and standard deviation oS√△t, that is, S follows geometric Brownian motion ds=u Sdt+oSdz Then dInS=( )at t odi u is expected return and o is volatility Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

11.2 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University The Stock Price Assumption ➢Consider a stock whose price is S ➢In a short period of time of length Dt the change in then stock price S is assumed to be normal with mean mSdt and standard deviation , that is, S follows geometric Brownian motion ds=m Sdt+Sdz. Then ➢m is expected return and  is volatility S Dt

11.3 The Lognormal property >It follows from this assumption that lnS-lnS≈d|!y √T or In so √T Since the logarithm of Sr is normal, ST is lognormally distributed Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

11.3 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University The Lognormal Property ➢It follows from this assumption that ➢Since the logarithm of ST is normal, ST is lognormally distributed 2 0 2 0 ln ln , 2 or ln ln , 2 T T S S T T S S T T   m    m       + −             −  −        

11.4 Modeling stock Prices in Finance >In finance, frequently we model the evolution of stock prices as a generalized Wiener Process ds=usdt +osd Also, assume prices are distributed lognormal and returns are distributed normal Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

11.4 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Modeling Stock Prices in Finance ➢In finance, frequently we model the evolution of stock prices as a generalized Wiener Process Also, assume prices are distributed lognormal and returns are distributed normal dS = mSdt +Sdz

11.5 The Lognormal distribution E(ST) var(Sr) T Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

11.5 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University The Lognormal Distribution E S S e S S e e T T T T T ( ) ( ) ( ) = = − 0 0 2 2 2 1 var m m 

11.6 Continuously compounded rate of Return, n(Equation(11.7) mT T or n n or n≈中 √T Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

11.6 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Continuously Compounded Rate of Return, h (Equation (11.7)) S S e T S S T T T T =  −       0 0 1 2 or = or 2 h h h  m   ln

The Expected return The expected value of the stock price is E(ST) The expected continuously compounded return on the stock is E(m=u-o/2(the geometric average) u is the the arithmetic average of the returns Note that E[n(ssl is not equal to InE(ST) In[E(STI-In So+u T, EIn(ST) =In So+(u-0/2)T Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

11.7 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University The Expected Return ➢ The expected value of the stock price is E(ST )=S0e mT ➢ The expected continuously compounded return on the stock is E(h)=m – 2 /2 (the geometric average) ➢ m is the the arithmetic average of the returns ➢ Note that E[ln(ST )] is not equal to ln[E(ST )] ln[E(ST )]=ln S0+m T, E[ln(ST )] =ln S0+(m-2 /2)T

11.8 The Expected Return Example ake the following 5 annual returns: 10%, 12%, 8%, 9%, and 11% The arithmetic average is x=∑x=(010+012+098+009+01)=y*050=010 However, the geometric average is ∏I(+x)-1=(1.10*112108*109*1113-1=0099 Thus, the arithmetic average overstates the geometric average The geometric is the actual return that one would have earned The approximation for the geometric return is -a2/2=010-0015811)2=0098 This differs from g as the returns are not normally distributed Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

11.8 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University The Expected Return Example Take the following 5 annual returns: 10%, 12%, 8%, 9%, and 11% The arithmetic average is However, the geometric average is Thus, the arithmetic average overstates the geometric average. The geometric is the actual return that one would have earned. The approximation for the geometric return is This differs from g as the returns are not normally distributed. (0.10 0.12 0.08 0.09 0.11) 5 * 0.50 0.10 1 1 5 1 1 _ =  = + + + + = = = n i n i x x (1 ) 1 (1.10*1.12*1.08*1.09*1.11) 5 1 0.09991 1 1 1 − = − =         =  + = n n i i g x / 2 0.10 (0.015811 )/ 2 0.09988 2 2 m − = − =

11.9 Is Normality realistic? >If returns are normal and thus prices are lognormal and assuming that volatility is at 20%(about the historical average) On 10/19/87 the 2 month s&P 500 Futures dropped 29% This was a-27 sigma event with a probability of occurring of once in every 10160 days On 10/13/89. the s&P 500 index lost about 6% This was a-5 sigma event with a probability of 0.00000027 or once every 14, 756 years Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

11.9 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Is Normality Realistic? ➢ If returns are normal and thus prices are lognormal and assuming that volatility is at 20% (about the historical average) – On 10/19/87, the 2 month S&P 500 Futures dropped 29% • This was a -27 sigma event with a probability of occurring of once in every 10160 days – On 10/13/89, the S&P 500 index lost about 6% • This was a -5 sigma event with a probability of 0.00000027 or once every 14,756 years

11.10 The Concepts Underlying Black Se choles The option price 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 The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate This leads to the Black-Scholes differential equation Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

11.10 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University The Concepts Underlying Black￾Scholes ➢ The option price & 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 ➢ The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate ➢ This leads to the Black-Scholes differential equation