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