Lecture 16

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

O C - Call Option Price P - Stock Price N(d 1 ) - Cumulative normal density function of (d 1 ) PV(EX) - Present Value of Strike or Exercise price N(d 2 ) - Cumulative normal density function of (d 2 ) 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

N(d 1 )= Black-Scholes Option Pricing Model

Cumulative Normal Density Function

Call Option Example - Genentech What is the price of a call option given the following? P = 80r = 5%v =.4068 EX = 80t = 180 days / 365

Call Option Example - Genentech What is the price of a call option given the following? P = 80r = 5%v =.4068 EX = 80t = 180 days / 365

Call Option Example - Genentech What is the price of a call option given the following? P = 80r = 5%v =.4068 EX = 80t = 180 days / 365

Call Option Example What is the price of a call option given the following? P = 36r = 10%v =.40 EX = 40t = 90 days / 365

.3070=.3 =.00 =.007

Call Option Example What is the price of a call option given the following? P = 36r = 10%v =.40 EX = 40t = 90 days / 365

Call Option Example What is the price of a call option given the following? P = 36r = 10%v =.40 EX = 40t = 90 days / 365

Example What is the price of a call option given the following? P = 36r = 10%v =.40 EX = 40t = 90 days / 365

(d 1 ) = ln + (.1 + ) 30/ /365 (d 1 ) =.3335N(d 1 ) =.6306 Example What is the price of a call option given the following? P = 41r = 10%v =.42 EX = 40t = 30 days / 365

(d 1 ) = ln + (.1 + ) 30/ /365 (d 1 ) =.3335N(d 1 ) =.6306 Example What is the price of a call option given the following? P = 41r = 10%v =.42 EX = 40t = 30 days / 365

(d 2 ) =.2131 N(d 2 ) =.5844 (d 2 ) = d 1 -v t = (.0907) Example What is the price of a call option given the following? P = 41r = 10%v =.42 EX = 40t = 30 days / 365

O C = P s [N(d 1 )] - S[N(d 2 )]e -rt O C = 41[.6306] - 40[.5844]e - (.10)(.0822) O C = $ 2.67 Example What is the price of a call option given the following? P = 41r = 10%v =.42 EX = 40t = 30 days / 365

Example What is the price of a call option given the following? P = 41r = 10%v =.42 EX = 40t = 30 days / 365

 Intrinsic Value = = 1  Time Premium = = 1.67  Profit to Date = =.94  Due to price shifting faster than decay in time premium Example What is the price of a call option given the following? P = 41r = 10%v =.42 EX = 40t = 30 days / 365

 Q: How do we lock in a profit?  A: Sell the Call

 Q: How do we lock in a profit?  A: Sell the Call

 Q: How do we lock in a profit?  A: Sell the Call

 Q: How do we lock in a profit?  A: Sell the Call

Black-Scholes O p = EX[N(-d 2 )]e -rt - P s [N(-d 1 )] Put-Call Parity (general concept) Put Price = Oc + EX - P - Carrying Cost + D Carrying cost = r x EX x t Call + EXe -rt = Put + P s Put = Call + EXe -rt - P s

N(-d 1 ) =.3694 N(-d 2 )=.4156 Black-Scholes O p = EX[N(-d 2 )]e -rt - P s [N(-d 1 )] O p = 40[.4156]e -.10(.0822) - 41[.3694] O p = 1.34 Example What is the price of a call option given the following? P = 41r = 10%v =.42 EX = 40t = 30 days / 365

Put-Call Parity Put = Call + EXe -rt - P s Put = e -.10(.0822) - 41 Put = = 1.34 Example What is the price of a call option given the following? P = 41r = 10%v =.42 EX = 40t = 30 days / 365

Put-Call Parity & American Puts P s - EX < Call - Put < P s - EXe -rt Call + EX - P s > Put > EXe -rt - P s + call Example - American Call > Put > e -.10(.0822) > Put > 1.34 With Dividends, simply add the PV of dividends