《金融期货与期权》（英文版） Chapter 12 The blackscholes Model

12.1 The black-scholes Model Chapter 12 Options, Futures, and Other Derivatives, Sth Edition C 2002 by John C. Hull

12.1 Options, Futures, and Other Derivatives, 5th Edition © 2002 by John C. Hull 1 The Black-Scholes Model Chapter 12

12.2 The stock Price Assumption o Consider a stock whose price is S ° In a short period of time of lengthδt,the return on the stock is normally distributed OS ≈u6t,o√δt where u is expected return and o is volatility Options, Futures, and Other Derivatives, Sth Edition C 2002 by John C. Hull

12.2 Options, Futures, and Other Derivatives, 5th Edition © 2002 by John C. Hull 2 The Stock Price Assumption • Consider a stock whose price is S • In a short period of time of length dt, the return on the stock is normally distributed: where m is expected return and s is volatility ( t t) S S   md s d d

The lognormal Property 12.3 (Equations 12.2 and 12.3, page 235) o It follows from this assumption that InpTInso 2 or nSr≈dlnS+| o Since the logarithm of s is normal, sr is lognormally distributed Options, Futures, and Other Derivatives, Sth Edition C 2002 by John C. Hull

12.3 Options, Futures, and Other Derivatives, 5th Edition © 2002 by John C. Hull 3 The Lognormal Property (Equations 12.2 and 12.3, page 235) • It follows from this assumption that • Since the logarithm of ST is normal, ST is lognormally distributed ln ln , ln ln , S S T T S S T T T T −  −              + −             0 2 0 2 2 2  m s s  m s s or

12.4 The lognormal distribution E(ST)=S var(S)=Se(e°-1) Options, Futures, and Other Derivatives, Sth Edition C 2002 by John C. Hull

12.4 Options, Futures, and Other Derivatives, 5th Edition © 2002 by John C. Hull 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 s

12.5 Continuously Compounded Return, n(Equations 12.6 and 12.7), page 236) T 0e7 or T n In or 2 o 2 Options, Futures, and Other Derivatives, Sth Edition C 2002 by John C. Hull

12.5 Options, Futures, and Other Derivatives, 5th Edition © 2002 by John C. Hull 5 Continuously Compounded Return, h (Equations 12.6 and 12.7), page 236) S S e T S S T T T T =  −       0 0 1 2 or = or 2 h h h  m s s ln

12.6 The Expected Return The expected value of the stock price is T e The expected return on the stock is T 儿-2/2 hnE(S/S)]=u Options, Futures, and Other Derivatives, Sth Edition C 2002 by John C. Hull 6

12.6 Options, Futures, and Other Derivatives, 5th Edition © 2002 by John C. Hull 6 The Expected Return • The expected value of the stock price is S0 e mT • The expected return on the stock is m – s 2 /2    = m = m −s ln ( / ) ln( / ) / 2 0 2 0 E S S E S S T T

127 The volatility The volatility of an asset is the standard deviation of the continuously compounded rate of return in 1 year As an approximation it is the standard deviation of the percentage change in the asset price in 1 year Options, Futures, and Other Derivatives, Sth Edition C 2002 by John C. Hull

12.7 Options, Futures, and Other Derivatives, 5th Edition © 2002 by John C. Hull 7 The Volatility • The volatility of an asset is the standard deviation of the continuously compounded rate of return in 1 year • As an approximation it is the standard deviation of the percentage change in the asset price in 1 year

128 Estimating volatility from Historical Data (page 239-41) Take observations So, Sy on at intervals ofτ years 2. Calculate the continuously compounded return in each interval as u =n 3. Calculate the standard deviation s of the u. s y The historical volatility estimate is: 0= Options, Futures, and Other Derivatives, Sth Edition C 2002 by John C. Hull

12.8 Options, Futures, and Other Derivatives, 5th Edition © 2002 by John C. Hull 8 Estimating Volatility from Historical Data (page 239-41) 1. Take observations S0 , S1 , . . . , Sn at intervals of t years 2. Calculate the continuously compounded return in each interval as: 3. Calculate the standard deviation, s , of the ui ´ s 4. The historical volatility estimate is: u S S i i i =       − ln 1 t s = s ˆ

129 The Concepts underlying Black-scholes 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 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, Sth Edition C 2002 by John C. Hull

12.9 Options, Futures, and Other Derivatives, 5th Edition © 2002 by John C. Hull 9 The Concepts Underlying Black-Scholes • 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 • The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate • This leads to the Black-Scholes differential equation

12.10 The derivation of the black scholes Differential equation δS=μSt+oSz δf= 0f,。,0f,10 +2 f-2c2 of St+=oS 8z dt aS C W e set up a portfolio consisting of 1: derivative +- shares aS Options, Futures, and Other Derivatives, Sth Edition C 2002 by John C. Hull

12.10 Options, Futures, and Other Derivatives, 5th Edition © 2002 by John C. Hull 10 The Derivation of the Black-Scholes Differential Equation : shares ƒ + : derivative W e set up a portfolio consisting of ƒ ƒ ½ ƒ ƒ ƒ S S z S S t t S S S S S t S z   − s d   d +         s   +   m +   d = d = m d + s d 1 2 2 2 2