Option Valuation Bodie, Kane and Marcus Essentials of Investments 9 th Global Edition 16

16.1 I NTRODUCTION Intrinsic Value Stock price minus exercise price, or profit that could be attained by immediate exercise of in-the-money call option Time Value Difference between option’s price and intrinsic value

16.1 I NTRODUCTION Determinants of Option Value Stock price Exercise price Volatility of price Time to expiration Interest rate Dividend rate

F IGURE 16.1 C ALL O PTION V ALUE BEFORE E XPIRATION

T ABLE 16.1 D ETERMINANTS OF C ALL O PTION V ALUES

16.2 B INOMIAL O PTION P RICING Two-State Option Pricing Assume stock price can take one of two values Increase by u = 1.2, or fall by d =.9

16.2 B INOMIAL O PTION P RICING Two-State Option Pricing Compare payoff to one share of stock, borrowing of $ % interest Outlay is $18.18; payoff is three times call option

16.2 B INOMIAL O PTION P RICING Two-State Option Pricing Portfolio of one share and three call options is perfectly hedged Hedge ratio for two-state option

16.2 B INOMIAL O PTION P RICING Generalizing the Two-State Approach Break up year into two intervals Subdivide further using the same pattern

F IGURE 16.2A P ROBABILITY D ISTRIBUTIONS FOR F INAL S TOCK P RICE, T HREE S UBINTERVALS

F IGURE 16.2B P ROBABILITY D ISTRIBUTIONS FOR F INAL S TOCK P RICE, S IX S UBINTERVALS

F IGURE 16.2C P ROBABILITY D ISTRIBUTIONS FOR F INAL S TOCK P RICE, 20 S UBINTERVALS

16.3 B LACK -S CHOLES O PTION V ALUATION Black-Scholes Pricing Formula Where

16.3 B LACK -S CHOLES O PTION V ALUATION

16.3 B LACK -S CHOLES O PTION V ALUATION Black-Scholes Pricing Formula And where r = Risk-free interest rate T = Time until expiration ln = Natural logarithm σ = Standard deviation of annualized continuously compounded RoR

F IGURE 16.3 S TANDARD N ORMAL P ROBABILITY F UNCTION

16.3 B LACK -S CHOLES O PTION V ALUATION Black-Scholes Pricing Formula Implied volatility Standard deviation of stock returns consistent with option’s market value

S PREADSHEET 16.1 B LACK -S CHOLES C ALL O PTION V ALUES

F IGURE 16.4 F INDING I MPLIED V OLATILITY

F IGURE 16.5 I MPLIED V OLATILITY OF S&P 500

16.3 B LACK -S CHOLES O PTION V ALUATION Put-Call Parity Relationship Represents relationship between put and call prices Extension for European call options on dividend- paying stocks

F IGURE 16.6 P AYOFF -P ATTERN OF L ONG C ALL –S HORT P UT P OSITION

T ABLE 16.3 A RBITRAGE S TRATEGY

16.3 B LACK -S CHOLES O PTION V ALUATION Put Option Valuation Black-Sholes formula for European put option

16.4 U SING THE B LACK -S CHOLES F ORMULA Hedge Ratios and Black-Scholes Formula Hedge ratio or delta Number of shares required to hedge price risk of holding one option Option elasticity Percentage increase in option’s value given 1% increase in value of underlying security

F IGURE 16.7 C ALL O PTION V ALUE AND H EDGE R ATIO

16.4 U SING THE B LACK -S CHOLES F ORMULA Portfolio Insurance Portfolio strategies that limit investment losses while maintaining upside potential Dynamic hedging Constant updating of hedge positions as market conditions change

F IGURE 16.8 P ROFIT ON P ROTECTIVE P UT S TRATEGY

F IGURE 16.9 H EDGE R ATIOS C HANGE AS S TOCK P RICE F LUCTUATES

16.4 U SING THE B LACK -S CHOLES F ORMULA Option Pricing and the Crisis of When banks buy debt with limited liability, they implicitly write put option to borrower

F IGURE R ISKY L OAN

F IGURE V ALUE OF I MPLICIT P UT O PTION ON L OAN G UARANTEE AS % OF F ACE V ALUE OF D EBT

F IGURE I MPLIED V OLATILITY AS F UNCTION OF E XERCISE P RICE