2. 0 Chapters 14/15 – Part 1 Options: Basic Concepts l Options l Call Options l Put Options l Selling Options l Reading The Wall Street Journal l Combinations of Options l Valuing Options l An Option ‑ Pricing Formula l Investment in Real Projects and Options l Summary and Conclusions

3. 1 Options Contracts: Preliminaries l Option Definition. l Calls versus Puts l Call options l Put options. l Exercising the Option l Strike Price or Exercise Price l Expiration Date l European versus American options

4. 2 Options Contracts: Preliminaries l Intrinsic Value l Speculative Value Option Premium = Intrinsic Value Speculative Value +

5. 3 Value of an Option at Expiration Impact of leverage… Stock price is $50. Buy 100 shares Call strike is $50, price is $10. Buy 1 contract. Put strike is $50, price is $10. Buy 1 contract. ===================== C = S – E P = E - S

6. 4 Call Option Payoffs -20 1009080706001020304050 -40 20 0 -60 40 60 Stock price ($) Option payoffs ($) Write a call Buy a call

7. 5 Put Option Payoffs Write a put Buy a put -20 0 -40 20 0 -60 40 60 Option payoffs ($) Stock price ($) 100908070601020304050

8. 6 Call Option Payoffs -20 1009080706001020304050 -40 20 0 -60 40 60 Stock price ($) Option payoffs ($) Buy a call Exercise price = $50

9. 7 Call Option Payoffs -20 1009080706001020304050 -40 20 0 -60 40 60 Stock price ($) Option payoffs ($) Write a call Exercise price = $50

10. 8 Call Option Profits -20 1009080706001020304050 -40 20 0 -60 40 60 Stock price ($) Option profits ($) Write a call Buy a call Exercise price = $50; option premium = $10

11. 9 Put Option Payoffs -20 1009080706001020304050 -40 20 0 -60 40 60 Stock price ($) Option payoffs ($) Buy a put Exercise price = $50

12. 10 Put Option Payoffs -20 1009080706001020304050 -40 20 0 -60 40 60 Option payoffs ($) write a put Exercise price = $50 Stock price ($)

13. 11 Put Option Profits -20 1009080706001020304050 -40 20 0 -60 40 60 Stock price ($) Option profits ($) Buy a put Write a put Exercise price = $50; option premium = $10 10 -10

14. 12 Selling Options – Writing Options l The seller (or writer) of an option has an obligation. l The purchaser of an option has an option. -20 1009080706001020304050 -40 20 0 -60 40 60 Stock price ($) Option profits ($) Buy a put Write a put 10 -10 -20 1009080706001020304050 -40 20 0 -60 40 60 Stock price ($) Option profits ($) Write a call Buy a call

15. 13 Call Option Payoffs at Expiration (Δ exercise) Stock price ($) 1009080706001020304050 Buy a call 20 10 40 30 0 50 60 Option payoffs ($) E=50 E=0

16. 14 Option Pricing Bounds at Expiration l Upper bounds  Call Options  Put Options l Lower Bounds  Call option intrinsic value  = max [0, S - E]  Put option intrinsic value  = max [0, E - S] l In-the-money / Out-of-the-money l Time premium/time decay l At expiration, an American call option is worth the same as a European option with the same characteristics.

17. 15 Reading The Wall Street Journal

18. 16 Valuing Options l The last section concerned itself with the value of an option at expiration. l This section considers the value of an option prior to the expiration date.

19. 17 Option Value Determinants Call Put 1. Exercise price 2. Stock price 3. Interest rate 4. Volatility in the stock price 5. Expiration date The value of a call option C 0 must fall within max (S 0 – E, 0) < C 0 < S 0. The precise position will depend on these factors.

20. 18 Varying Option Input Values l Stock price: l Call: as stock price increases call option price increases l Put: as stock price increases put option price decreases l Strike price: l Call: as strike price increases call option price decreases l Put: as strike price increases put option price increases

21. 19 Varying Option Input Values l Time until expiration: l Call & Put: as time to expiration increases call and put option price increase l Volatility: l Call & Put: as volatility increases call & put value increase l Risk-free rate: l Call: as the risk-free rate increases call option price increases l Put: as the risk-free rate increases put option price decreases

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26. 24 Option Value Determinants Call Put 1. Exercise price– + 2. Stock price+ – 3. Interest rate + – 4. Volatility in the stock price+ + 5. Expiration date+ + The value of a call option C 0 must fall within max (S 0 – E, 0) < C 0 < S 0. The precise position will depend on these factors.

27. 25 Market Value, Time Value and Intrinsic Value for an American Call C aT > Max[S T - E, 0] Profit loss E STST Market Value Intrinsic value S T - E Time value Out-of-the-money In-the-money STST The value of a call option C 0 must fall within max (S 0 – E, 0) < C 0 < S 0.

28. 26 Combinations of Options l Puts and calls can serve as the building blocks for more complex option contracts. l If you understand this, you can become a financial engineer, tailoring the risk-return profile to meet your client’s needs.

29. 27 Protective Put Strategy: Buy a Put and Buy the Underlying Stock: Payoffs at Expiration Buy a put with an exercise price of $50 Buy the stock Protective Put strategy has downside protection and upside potential $50 $0 $50 Value at expiration Value of stock at expiration

30. 28 Protective Put Strategy Profits Buy a put with exercise price of $50 for $10 Buy the stock at $40 $40 Protective Put strategy has downside protection and upside potential $40 $0 -$40 $50 Value at expiration Value of stock at expiration

31. 29 Covered Call Strategy Sell a call with exercise price of $50 for $10 Buy the stock at $40 $40 Covered call $40 $0 -$40 $10 -$30 $30$50 Value of stock at expiration Value at expiration

32. 30 Long Straddle: Buy a Call and a Put Buy a put with an exercise price of $50 for $10 $40 A Long Straddle only makes money if the stock price moves $20 away from $50. $40 $0 -$20 $50 Buy a call with an exercise price of $50 for $10 -$10 $30 $60$30$70 Value of stock at expiration Value at expiration

33. 31 Short Straddle: Sell a Call and a Put Sell a put with exercise price of $50 for $10 $40 A Short Straddle only loses money if the stock price moves $20 away from $50. -$40 $0 -$30 $50 Sell a call with an exercise price of $50 for $10 $10 $20 $60$30$70 Value of stock at expiration Value at expiration

34. 32 Put-Call Parity Buy the stock, buy a put, and write a call; the sum of which equals the strike price discounted at the risk-free rate C = Call option priceP = Put option price S = Current stock priceE = Option strike price r = Risk-free rateT = Time until option expiration

35. 33 Put-Call Parity Buy Stock & Buy Put Share Price Position Value Combination: Long Stock & Long Put Long Put Long Stock

36. 34 Put-Call Parity Buy Call & Buy Zero Coupon Risk-Free Bond @ Exercise Price Long Bond Share Price Position Value Combination: Long Stock & Long Bond Long Call

37. 35 Put-Call Parity Share Price Position Value Combination: Long Stock & Long Put Long Put Long Stock Share Price Position Value Combination: Long Stock & Long Bond Long Call Long Bond In market equilibrium, it must be the case that option prices are set such that: Otherwise, riskless portfolios with positive payoffs exist.

38. 36 The Black-Scholes Model Value of a stock option is a function of 6 input factors: 1. Current price of underlying stock. 2. Strike price specified in the option contract. 3. Risk-free interest rate over the life of the contract. 4. Time remaining until the option contract expires. 5. Price volatility of the underlying stock. The price of a call option equals:

39. 37 Black-Scholes Model Where the inputs are: S = Current stock price E = Option strike price r = Risk-free interest rate T = Time remaining until option expiration  = Sigma, representing stock price volatility, standard deviation

40. 38 Black-Scholes Model Where d 1 and d 2 equal:

41. 39 Black-Scholes Models Also, remember at expiration: Remembering put-call parity, the value of a put, given the value of a call equals:

42. 40 The Black-Scholes Model Find the value of a six-month call option on the Microsoft with an exercise price of $150 The current value of a share of Microsoft is $160 The interest rate available in the U.S. is r = 5%. The option maturity is 6 months (half of a year). The standard deviation of the underlying asset is 30% per annum. Before we start, note that the intrinsic value of the option is $10—our answer must be at least that amount.

43. 41 The Black-Scholes Model Then d 2, First calculate d 1 and d 2 Assume S = $160, X = $150, T = 6 months, r = 5%, and  = 30%, calculate the value of a call.

44. 42 The Black-Scholes Model N(d 1 ) = N(0.52815) = 0.7013 N(d 2 ) = N(0.31602) = 0.62401

45. 43 Assume S = $50, X = $45, T = 6 months, r = 10%, and  = 28%, calculate the value of a call and a put. From a standard normal probability table, look up N(d 1 ) = 0.812 and N(d 2 ) = 0.754 (or use Excel’s “normsdist” function) Another Black-Scholes Example

46. 44 Real Options l Real estate developer buys 70 acres in a rural area. He plans on building a subdivision when the population from the city expands this direction. If growth is less than anticipated, the developer thinks he can sell the land to a country club to build a golf course on the property. l The development option is a ______ option. l The golf course option is a _______ option. l How would these real options change the standard NPV analysis?

47. 45 Collar: Buy a Put, Buy the Stock, Sell the Call Buy a put with exercise price of $50 for $0.67 Buy the stock at $80 $80 $49.33 $0 -$80 $50 Value at expiration Value of stock at expiration $120 Sell a call with exercise price of $120 for $2.76 $2.76 -$27.91 $42.11 Collar $0.67 NTS