l1.11 The Assumptions Underlying Black-Scholes 1. The stock follows a Brownian motion with constant u and o 2. Short selling of securities with full use of proceeds is permitted 3. No transaction cost or taxes 4. Securities are perfectly divisible 5. No dividends paid during the life of the option 6. There are no arbitrage opportunities 7. Security trading is continuous 8. The risk-free rate of interest,r is constant and is the same for all maturities Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University
11.11 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University The Assumptions Underlying Black-Scholes 1. The stock follows a Brownian motion with constant m and 2. Short selling of securities with full use of proceeds is permitted 3. No transaction cost or taxes 4. Securities are perfectly divisible 5. No dividends paid during the life of the option 6. There are no arbitrage opportunities 7. Security trading is continuous 8. The risk-free rate of interest, r, is constant and is the same for all maturities
1.12 I of 3: The derivation of the Black-Scholes Differential Equation △S=μS△t+oS△ f 了f,102f △f=pS++ f S2△t+aS△z at aS We set up a portfolio consisting of 1: derivative f shares Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University
11.12 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 1 of 3: The Derivation of the Black-Scholes Differential Equation D D D D D D S S t S z S S t S S t S S z S = + = + + + − m m ƒ ƒ ƒ ½ ƒ ƒ We set up a portfolio consisting of : derivative + ƒ : shares 2 2 2 2 1
11.13 2 of3: The derivation of the Black-Scholes Differential Equation The value of the portfolio li is given by ∏=-f+ of aS The change in its value in time At is given by △I=4/O aS Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University
11.13 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University The value of the portfolio is given by ƒ ƒ The change in its value in time is given by ƒ ƒ D D D D = − + = − + S S t S S 2 of 3: The Derivation of the Black-Scholes Differential Equation
11.14 3 of 3: The derivation of the Black-Scholes Differential Equation The return on the portfolio must be the risk-free rate. Hence △I=r∏△t We substitute for Af and As in these equations to get the Black-Scholes differential equation d+.02o28f=rf +rS=+ Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University
11.14 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 3 of 3: The Derivation of the Black-Scholes Differential Equation 2 2 2 The return on the portfolio must be the risk-free rate. Hence We substitute for and in these equations to get the Black-Scholes differential equation: 1 2 f f S f f rS S t S r t S + = + D D D D 2 = rf