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- 1. Options Finance 100 Prof. Michael R. Roberts Copyright © Michael R. Roberts 1 Topic Overview Options: » Uses, definitions, types Put-Call Parity Valuation » Black Scholes Applications » Portfolio Insurance » Hedging » Speculation and arbitrage Copyright © Michael R. Roberts 2 1 1
- 2. Definitions Call Option A standard call option is an option giving the buyer the right to buy from the seller an underlying asset for a fixed price (strike/exercise price) at any time on or before a fixed date (expiration date) Put Option A standard put option is an option giving the buyer the right to sell to the seller an underlying asset for a fixed price (strike/exercise price) at any time on or before a fixed date (expiration date) Exercise Styles: » European: can be exercised at maturity only. » American: can be exercised at any time before maturity » Bermudan: can be exercised only at some predefined times (e.g. employee stock options) » Atlantic: can be exercised at times dependent on underlying asset (e.g. “cap” and “barrier” options) Copyright © Michael R. Roberts 3 Options Markets Exchange traded options » Stock options (CBOE, Philli, AMEX, NYSE), ForEx options (Philli), Index options (CBOE), Futures options (CBOE) OTC options Specification of an Option Contract » Expiration Date (typically third Friday of the month) » Strike Price » Class & Series » OTM, ATM, ITM (Deep) » Splits & Dividends Trading » No margin investing! Options Clearing Corporation (OCC) » Similar to clearinghouse for futures Regulation » OCC » SEC (options on stocks, stock indices, currencies & bonds) » CFTC (options on futures) Warrants & ESOPs Copyright © Michael R. Roberts 4 2 2
- 3. Options Quotes for Amazon.com Stock Copyright © Michael R. Roberts 5 Values of Options at Expiration Buying a Call 50 40 Net Payoff at Maturity 30 20 10 0 0 20 40 60 80 100 -10 Stock Price at Maturity This is the payoff (at maturity) to the buyer of a call option: » PayoffT = max(0,ST – K) - C Copyright © Michael R. Roberts 6 3 3
- 4. Values of Options at Expiration Writing a Call 10 0 0 20 40 60 80 100 Net Payoff at Maturity -10 -20 -30 -40 -50 Stock Price at Maturity This is the payoff (at maturity) to the seller (or writer) of a call option: » PayoffT = -[max(0,ST – K) – C] = min(0,K – ST) + P Copyright © Michael R. Roberts 7 Values of Options at Expiration Buying a Put 50 40 Net Payoff at Maturity 30 20 10 0 0 20 40 60 80 100 -10 Stock Price at Maturity This is the payoff (at maturity) to the buyer of a put option: » PayoffT = max(0,K - ST) – P Copyright © Michael R. Roberts 8 4 4
- 5. Values of Options at Expiration Selling a Put 10 0 0 20 40 60 80 100 Net Payoff at Maturity -10 -20 -30 -40 -50 Stock Price at Maturity This is the payoff (at maturity) to the seller (or writer) of a put option: » PayoffT = -[max(0,K - ST) – P] = min(0,ST – K) + P Copyright © Michael R. Roberts 9 Sample Payoffs What are the gross payoffs (ignoring the price of the contract) to the buyer of a call option and a put option if the exercise price is K=$50? Stock Buy Write Buy Write Price Call Call Put Put max(0,ST - K) min(0,K - ST) max(0,K - ST) min(0,ST - K) 20 0 0 30 -30 40 0 0 10 -10 60 10 -10 0 0 80 30 -30 0 0 Copyright © Michael R. Roberts 10 5 5
- 6. Put-Call Parity Example What is the relationship between a put and a call option that are otherwise identical. Consider: » stock whose current price is $90 and pays no dividends » a risk-free rate of 5%, » One call and an otherwise identical put option each with one year to expiration and strike price of 100. Can we replicate the call option & what does no arbitrage imply? Copyright © Michael R. Roberts 11 Put-Call Parity Example (Cont.) No arbitrage implies: Copyright © Michael R. Roberts 12 6 6
- 7. Put-Call Parity Example 200 Call Price 10 Strike 100 Put Price 15.24 Spot Price 90 IR 5% Combined Position 150 Future Stock Buy Buy Buy Borrow Combined Own Put 100 Price Call Put Asset PV(Strike) Position Payoff at Expiration 0 -10 84.76 -90 -4.76 -10 50 20 -10 64.76 -70 -4.76 -10 40 -10 44.76 -50 -4.76 -10 0 60 -10 24.76 -30 -4.76 -10 0 50 100 150 200 80 -10 4.76 -10 -4.76 -10 -50 100 -10 -15.24 10 -4.76 -10 -100 120 10 -15.24 30 -4.76 10 Own Asset 140 30 -15.24 50 -4.76 30 -150 Borrowing 160 50 -15.24 70 -4.76 50 Future Stock Price 180 70 -15.24 90 -4.76 70 200 90 -15.24 110 -4.76 90 We see the combined position in the put, asset and cash exactly replicates the payoffs to the call. Copyright © Michael R. Roberts 13 Put-Call Parity General Expression More generally, consider: » An underlying asset with current price = S0, price at expiry = ST » A put and a call option with T years to maturity and identical strike price = K » An annualized risk-free return and dividend yield equal to r and d, respectively. Today Expiration Date ST < K ST > K Buy Call -C 0 ST-K Buy Put -P K-ST 0 -T Buy 1 unit of Asset -S0(1+d) ST ST -T Borrow PV(Strike Price) K(1+r) -K -K -T -T Total -P-S(1+d) +K(1+r) 0 ST-K No arbitrage implies: C = P + S 0 (1 + d ) − T − K (1 + r ) −T Copyright © Michael R. Roberts 14 7 7
- 8. Put-Call Parity General Expression Continuous Compounding More generally, consider: » An underlying asset with current price = S0, price at expiry = ST » A put and a call option with T years to maturity and identical strike price = X » Continuously compounded risk-free return and dividend yield equal to r and d, respectively. Today Expiration Date ST < K ST > K Buy Call -C 0 ST-K Buy Put -P K-ST 0 -dT Buy 1 unit of Asset -S0e ST ST -rT Borrow PV(Strike Price) Ke -K -K -dT -rT Total -P-Se +Ke 0 ST-K No arbitrage implies: C = P + S 0 e − dT − Ke − rT Copyright © Michael R. Roberts 15 Put-Call Parity Implications By rearranging the put-call parity relation, we find numerous implications » How can we replicate borrowing with options and the underlying asset ? Ke − rT = − C + P + S 0 e − dT » How can we replicate shorting the underlying asset with options ? S 0 e − dT = C − P + Ke − rT » What is the implies risk-free return ? ⎛ − C + P + S 0 e − dT ⎞1 r = − ln ⎜ ⎜ ⎟ ⎟T ⎝ K ⎠ » Protective Put = Fiduciary Call ? S 0 e − dT + P = C + Ke − rT Copyright © Michael R. Roberts 16 8 8
- 9. Put-Call Parity and Arbitrage Example A non-dividend paying stock is currently selling for $100. » A call option with an exercise price of $90 and maturity of 3 months has a price of $12. » A put option with an exercise price of $90 and maturity of 3 months has a price of $2. » The one-year T-bill rate is 5.0%. What does PCP imply? What is the first step of your arbitrage strategy? Copyright © Michael R. Roberts 17 Put-Call Parity and Arbitrage Example (Cont.) Copyright © Michael R. Roberts 18 9 9
- 10. Call Option Valuation Black-Scholes Black Scholes price of a call option on a non-dividend-paying stock, C C = S × N (d1 ) − PV (K ) × N (d 2 ) S is current price of the stock K is the exercise price N(x) is the Standard Normal CDF, Pr(X<x) ln[S / PV (K )] σ T d1 = + and d 2 = d1 − σ T σ T 2 σ is the annual volatility T is the number of years left to expiration Copyright © Michael R. Roberts 19 Valuing a Call Option with Black-Scholes Copyright © Michael R. Roberts 20 10 10
- 11. Valuing a Call Option with Black-Scholes The BS parameters: » S = (12.58+12.59)/2 = $12.585 » T = 45/365 » rf = 4.38% » σ = 25% » K = $12.50 PV ( K ) = 12.50 / (1.0438 ) 45/365 = $12.434 ln[S / PV (K )] σ T ln (12.585 / 12.434 ) 0.25 45 / 365 d1 = + = + = 0.181 σ T 2 0.25 45 / 365 2 d 2 = d1 − σ T = 0.181 − 0.25 45 / 365 = 0.094 C = S × N (d1 ) − PV (K ) × N (d 2 ) = 12.585 ( 0.572 ) − 12.434 ( 0.537 ) = $0.52 Copyright © Michael R. Roberts 21 Black-Scholes Call Option Values Future Payoff Value of Call before Maturity Value of Call at Maturity Future Stock Price Copyright © Michael R. Roberts 22 11 11
- 12. Put Option Valuation Black-Scholes Black Scholes price of a call option on a non-dividend-paying stock, P P = PV (K )[1 − N (d 2 )] − S[1 − N (d1 )] S is current price of the stock K is the exercise price N(x) is the Standard Normal CDF, Pr(X<x) ln[S / PV (K )] σ T d1 = + and d 2 = d1 − σ T σ T 2 σ is the annual volatility T is the number of years left to expiration Copyright © Michael R. Roberts 23 Stock Option Valuation with Dividends Black-Scholes Very simple adjustment to the previous formulas: Define the ex-dividend stock price: S x = S − PV (Div) Substitute Sx for S in the formulas » A special case is when the stock will pay a dividend that is proportional to its stock price at the time the dividend is paid. If q is the stock’s (compounded) dividend yield until the expiration date, then: S x = S / (1 + q) Copyright © Michael R. Roberts 24 12 12
- 13. Call Option Sensitivities The Option Pricing formula gives the following sensitivities for a call option: Effect on Increase in… Call Price Intuition S up More likely to finish ITM σ up Asymmetric Payoff T up Asymmetric Payoff r up Time Value of $ K down Less likely to finish ITM Copyright © Michael R. Roberts 25 Compaq Options: Using Black-Scholes (cont.) Compaq (Ok…) Black -Scholes prices and quoted prices: Calls Puts K BS Price Quoted Price BS Price Quoted Price 60 18.854 20.250 1.029 NA 65 14.942 15.500 2.027 2.500 70 11.539 12.875 3.534 3.000 75 8.691 8.750 5.597 6.125 80 6.394 6.000 8.211 8.250 85 4.604 4.375 11.331 NA Copyright © Michael R. Roberts 26 13 13
- 14. Compaq Call Options Market prices and Black-Scholes Prices 30.000 BS Price 25.000 Quoted Price 20.000 Option valu 15.000 10.000 5.000 0.000 50 55 60 65 70 75 80 85 90 95 Strike price Copyright © Michael R. Roberts 27 Debt and Equity as Options Suppose a firm has debt with a face value of $1m outstanding that matures at the end of the year. What is the value of debt and equity at the end of the year? Payoff to Payoff to Asset Value Shareholders Debtholders 0.3 0.0 0.3 0.6 0.0 0.6 0.9 0.0 1.0 1.2 0.2 1.0 1.5 0.5 1.0 (Amounts are in millions of dollars) Copyright © Michael R. Roberts 28 14 14
- 15. Debt and Equity as Options An Illustration Net Payoffs Firm Bondholders Equityholders Face Value 0 Future Firm Value of Debt Copyright © Michael R. Roberts 29 Debt and Equity as Options Consider a firm with zero coupon debt outstanding with a face value of F. The debt will come due in exactly one year. The payoff to the equityholders of this firm one year from now will be the following: Payoff to Equity = max[0, V-F] where V is the total value of the firm’s assets one year from now. Similarly, the payoff to the firm’s bondholders one year from now will be: Payoff to Bondholders = V - max[0,V-F] Equity has a payoff like that of a call option. Risky debt has a payoff that is equal to the total value of the firm, less the payoff of a call option. Copyright © Michael R. Roberts 30 15 15
- 16. Junior vs. Senior Debt 150 125 Senior Debt 100 Payoff 75 Junior Debt 50 25 Equity 0 0 50 100 150 200 250 300 Asset Value Copyright © Michael R. Roberts 31 Hedging with Foreign Currency Options Initial investment (option premium) is required You eliminate downside risks, while retaining upside potential Example: Recall the American firm selling 20 machines to a German company at 50,000ECU per machine. » What’s our exposure ? » What options position should we take to hedge the risk ? (Assume that the puts are struck at $1.57/ECU) Copyright © Michael R. Roberts 32 16 16
- 17. Hedging with Foreign Currency Options (Cont.) Scenario 1: Exchange rate falls to $1.00/ECU » Profits from options position = ? » Profits from sale of machines = ? » Total profit in $US = ? Scenario 2: Exchange rate rises to $2.00/ECU » Profits from futures position = ? » Profits from sale of machines = ? » Total profit in $US =? Punch line: Copyright © Michael R. Roberts 33 Hedging with Options vs. Futures Payoffs Payoff with Options Payoff with Futures 0 Exchange Rate Copyright © Michael R. Roberts 34 17 17
- 18. Summary Options are derivative securities Put-Call Parity Valuation: use Black-Scholes Value of option increases with volatility of underlying assets Use options for » Volatility bets » Portfolio Insurance » Hedging Copyright © Michael R. Roberts 35 18 18

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