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