,Valuing Options,Principles of Corporate FinanceSeventh Edition Richard A. Brealey Stewart C. Myers,Slides by Matthew Will,Chapter 21,McGraw Hill/Irwin,Copyright 2003 by The McGraw-Hill Companies, Inc. All rights reserved,Topics Covered,Simple Valuation Model Binomial Model Black-Scholes Model Black Scholes vs. Binomial,Probability Up = p = (a - d) Prob Down = 1 - p(u - d)a = erD t d =e-s D t.5 u =es D t.5 Dt = time intervals as % of year,Binomial Pricing,Example Price = 36 s = .40 t = 90/365 D t = 30/365 Strike = 40 r = 10%a = 1.0083 u = 1.1215 d = .8917 Pu = .5075 Pd = .4925,Binomial Pricing,40.3732.10,36,Binomial Pricing,40.3732.10,36,Binomial Pricing,50.78 = price40.3732.1025.52,45.283628.62,40.3732.10,36,Binomial Pricing,50.78 = price 10.78 = intrinsic value40.37 .3732.10 025.52 0,45.283628.62,36,40.3732.10,Binomial Pricing,50.78 = price 10.78 = intrinsic value40.37 .3732.10 025.52 0,45.28 5.603628.62,40.3732.10,36,The greater of,Binomial Pricing,50.78 = price 10.78 = intrinsic value40.37 .3732.10 025.52 0,45.28 5.6036 .1928.62 0,40.37 2.9132.10 .10,36 1.51,Binomial Pricing,50.78 = price 10.78 = intrinsic value40.37 .3732.10 025.52 0,45.28 5.6036 .1928.62 0,40.37 2.9132.10 .10,36 1.51,Binomial Pricing,Option Value,Components of the Option Price 1 - Underlying stock price 2 - Striking or Exercise price 3 - Volatility of the stock returns (standard deviation of annual returns) 4 - Time to option expiration 5 - Time value of money (discount rate),Option Value,Black-Scholes Option Pricing Model,OC = PsN(d1) - SN(d2)e-rt,OC = PsN(d1) - SN(d2)e-rt,OC- Call Option Price Ps - Stock Price N(d1) - Cumulative normal density function of (d1) S - Strike or Exercise price N(d2) - Cumulative normal density function of (d2) r - discount rate (90 day comm paper rate or risk free rate) t - time to maturity of option (as % of year) v - volatility - annualized standard deviation of daily returns,Black-Scholes Option Pricing Model,(d1)=,ln + ( r + ) t,Ps S,v2 2,v t,32 34 36 38 40,N(d1)=,Black-Scholes Option Pricing Model,(d1)=,ln + ( r + ) t,Ps S,v2 2,v t,Cumulative Normal Density Function,(d2) = d1 -,v t,Call Option,Example What is the price of a call option given the following? P = 36 r = 10% v = .40 S = 40 t = 90 days / 365,Call Option,(d1) =,ln + ( r + ) t,Ps S,v2 2,v t,(d1) = - .3070,N(d1) = 1 - .6206 = .3794,Example What is the price of a call option given the following? P = 36 r = 10% v = .40 S = 40 t = 90 days / 365,Call Option,(d2) = - .5056,N(d2) = 1 - .6935 = .3065,(d2) = d1 -,v t,Example What is the price of a call option given the following? P = 36 r = 10% v = .40 S = 40 t = 90 days / 365,Call Option,OC = PsN(d1) - SN(d2)e-rt,OC = 36.3794 - 40.3065e - (.10)(.2466),OC = $ 1.70,Example What is the price of a call option given the following? P = 36 r = 10% v = .40 S = 40 t = 90 days / 365,Put - Call Parity,Put Price = Oc + S - P - Carrying Cost + Div.,Carrying cost = r x S x t,ExampleABC is selling at $41 a share. A six month May 40 Call is selling for $4.00. If a May $ .50 dividend is expected and r=10%, what is the put price?,Put - Call Parity,Expanding the binomial model to allow more possible price changes,1 step 2 steps 4 steps (2 outcomes) (3 outcomes) (5 outcomes)etc. etc.,Binomial vs. Black Scholes,How estimated call price changes as number of binomial steps increases,No. of steps Estimated value1 48.12 41.03 42.15 41.810 41.450 40.3100 40.6 Black-Scholes 40.5,Binomial vs. Black Scholes,