1. Class 5 Option Contracts

2. Options n A call option is a contract that gives the buyer the right, but not the obligation, to buy the underlying security at a prespecified price (called the strike or exercise price) within a prespecified period of time. n A put option is a contract that gives the buyer the right, but not the obligation, to sell the underlying security at a prespecified price (called the strike or exercise price) within a prespecified period of time.

3. Options n European options (both calls and puts) may only be exercised at the expiration date of the option. n American options (both calls and puts) may be exercised at any time prior to the expiration date of the option.

4. Call Option: Payoff Diagram Payoff Stock Price X 0 Payoff = max[0, S T - X] Buy Call Option

5. Call Option: Payoff Diagram Payoff Stock Price X 0 Sell Call Option Payoff = - max[0, S T - X]

6. Put Option: Payoff Diagram Payoff Stock Price X 0 Payoff = max[0, X - S T ] Buy Put Option X

7. Put Option: Payoff Diagram Payoff Stock Price X 0 Sell Put Option Payoff = - max[0, X - S T ] -X

8. Example n What are the payoffs on a call option and a put option if the exercise price is X=$50?

9. Option Trading Strategies: The Straddle n Buy a call and a put on the same stock with the same exercise price and time to maturity. n Appropriate when you believe the stock price will change a lot, but you are unsure of the direction.

10. Option Trading Strategies: The Straddle Payoff Stock Price X 0 X Put Payoff Call Payoff Straddle Payoff

11. Option Trading Strategies: The Spread n Buy a call and sell another call with a higher strike price on the same stock with the same time to maturity. n Appropriate when you believe the stock price will increase and you are willing to trade off some upside potential to reduce the cost of your investment.

12. Option Trading Strategies: The Spread Payoff Stock Price X 1 0 Short Call Payoff Long Call Payoff Spread Payoff X 2 X 2 -X 1

13. Valuation of Options: Put-Call Parity n Suppose you bought a share of stock today for a price of S 0 and simultaneously borrowed an amount of Xe -rT. How much would your portfolio be worth at the end of T years? Assume that the stock does not pay a dividend.

14. Put-Call Parity Payoff Stock Price STST -X S T - X 0 Payoff on Stock Payoff on Borrowing Net Payoff X

15. Put-Call Parity n Now assume you buy a call option and sell a put option with a maturity date of T and an exercise price of X. How much will your options be worth at the end of T years?

16. Put-Call Parity Payoff Stock Price -X X S T - X 0 Payoff on short put Payoff on long call Net Payoff

17. Put-Call Parity n Since the two portfolios have the same payoffs at date T, they must have the same price today. n The put-call parity relationship is: C E - P E = S 0 - Xe -rT

18. Example n A 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%. Is there an arbitrage opportunity available in these prices?

19. Example n From Put-Call Parity, the price of the call option should be equal to:  C E = P E + S 0 - Xe -rT  C E = 2.00 +100.00 -90.00 e -(0.05)0.25  C E = 13.12 n Since the market price of the call is $12, it is underpriced by $1.12. We would want to buy the call, sell the put, sell the stock, and invest $90e -(0.05)0.25 for 3 months.

20. Example n The cash flows for this investment are outlined below:

21. Lower Bounds for European Option Prices n Since both put options and call options must have non-negative prices, the put-call parity relationship establishes the following lower bounds for European option prices: C E > max[ 0, S 0 - Xe -rT ] P E > max[ 0, Xe -rT - S 0 ]

22. Example n Suppose a stock is selling for $50 per share. The riskfree interest rate is 8%. A call option with an exercise price of $50 and 6 months to maturity is selling for $1.50. Is there an arbitrage opportunity available?  C E > max[ 0, S 0 - Xe -rT ]  C E > max[ 0, 50 - 50e -(0.08)0.5 ] = 1.96 n Since the price is only $1.50, the call is underpriced by at least $0.46.

23. Example n The arbitrage involves the following cash flows.

24. Example n Now suppose you observe a put option with an exercise price of $55 and 6 months to maturity selling for $2.50. Does this represent an arbitrage opportunity?  P E > max[ 0, Xe -rT - S 0 ]  P E > max[ 0, 55e -(0.08)0.5 - 50] = 2.84 n Since the price is only $2.50, the put is underpriced by at least $0.34

25. Example n The arbitrage involves the following cash flows:

26. American vs. European Options n Recall that American options allow the holder of the option to exercise at any time prior to maturity, whereas a European option only permits the holder to exercise at maturity. n Because the option to exercise early cannot have a negative value, American options must be more valuable than European options.

27. American Put Options n The possibility to exercise American options at any time prior to maturity allows us to derive a tighter lower bound for the price of an American put option: P A > max[ 0,X-S 0 ]

28. Example n Consider the previous example where the stock price is $50. What is the lower bound for the price of an American put option with an exercise price of $55?  P A > max[ 0, X - S 0 ]  P A > max [ 0, 55 - 50 ] = $5.00 n Note that $5.00 is the minimum price for an American put, regardless of the time to maturity.

29. American Call Options n Because of the possibility of early exercise, the price of an American call option is always at least as high as the price of its European counterpart. Hence, C A > C E > max [ 0, S 0 - Xe -rT ]

30. American Call Options n For stocks that do not pay dividends, C A = C E. n The exercise value of an American call option is S 0 -X. n The unexercised value of an American call option is at least: C A > max [ 0, S 0 - Xe -rT ] n Since the unexercised value is higher than the exercised value, it is never optimal to exercise early for non-dividend-paying stocks.

31. Black-Scholes Option Pricing Formula n The Black-Scholes option pricing formula prices European options on non-dividend-paying stocks. n Black-Scholes Call Option Formula: N(d 1 ) = cumulative normal probability distribution, or NORMSDIST(.) in EXCEL. C E = S N(d 1 ) - Xe -rT N(d 2 )

32. Call Option Sensitivities

33. Intuition for Black-Scholes

38. Black-Scholes Put Option Formula n We can use the put-call parity relationship to derive the Black-Scholes put option formula: n We have used the fact that 1-N(d 1 ) = N(-d 1 ) and 1-N(d 2 ) = N(-d 2 ). P E = C E - S + Xe -rT P E = -SN(-d 1 ) + Xe -rT N(-d 2 )

39. Put Option Sensitivities

40. Example n On February 2, 1996, Microsoft stock closed at a price of $93 per share. Microsoft’s annual standard deviation is about 32%. The one-year T-bill rate is 4.82%. What are the Black-Scholes prices for both calls and puts with an exercise price of $100 and a maturity of April 1996 (77 days)? How do these prices compare to the actual market prices of these options?

41. Example n The inputs for the Black-Scholes formula are:  S = $93.00  r = 4.82%  X = $100.00   = 32%  T = 77/365 n This gives d 1 = -0.351 and d 2 = -0.498. n The cumulative normal density for these values are N(d 1 ) = 0.3628 and N(d 2 ) = 0.3103. n Plugging these values into the Black-Scholes formula gives: c = $3.02 and p = $9.02.

42. Example n Microsoft Put and Call Options

43. Implied Volatilities n It is common for traders to quote prices in terms of implied volatilities. n This is the volatility (  ) that sets the Black- Scholes price equal to the market price. n This can be computed using SOLVER in EXCEL.

44. Hedging with Options n Initial investment (option premium) is required n You eliminate downside risks, while retaining upside potential

45. Option Hedging Example n It is the end of August and we will receive 1m DM at the end of October. n At this point, we will sell DM, converting them back into dollars. n We are concerned about the price at which we will be able to sell DM. n We can lock in a minimum sale price by buying put options.

46. Option Hedging Example n Since the total exposure is for 1m DM and each contract is for 62,500 DM we buy 16 put option contracts. n Suppose we choose the puts struck at 0.66 - locking in a lower bound of 0.66 $/DM.

47. Deutschemark Falls to $0.30 n We have the right to sell 1m DM for $0.66 each by exercising the put options. n Since DM’s are only worth $0.30 each we do choose to exercise. n Our cash inflow is therefore $660,000

48. Deutschemark Rises to $0.90 n We have the right to sell 1m DM for $0.66 each by exercising the put options. n Since DM’s are worth $0.90 each we do not choose to exercise. n We sell the DM on the open market for $0.90 each. n Our cash inflow is therefore $900,000

49. Debt and Equity n Consider a firm with zero coupon debt outstanding with a face value of F. The debt will come due in exactly one year. n 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.

50. Debt and Equity n Similarly, the payoff to the firm’s bondholders one year from now will be: Payoff to Bondholders = V - max[0,V-F] n Equity has a payoff like that on a call option. Risky debt has a payoff that is equal to the total value of the firm, less the payoff on a call option.

51. Debt and Equity Payoffs Firm Value0 Equityholders Bondholders F

52. Debt and Equity n Since bondholders have essentially sold a call option on the value of the firm’s assets to equityholders, conflicts of interest can arise.  Payout policy.  Asset substitution problem.  Underinvestment problem. n These problems can be resolved to some extent with debt covenants, conversion features, callability features, and putability features.