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 BlackScholes ➢ 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