1. 1 Options Global Financial Management Campbell R. Harvey Fuqua School of Business Duke University charvey@mail.duke.edu http://www.duke.edu/~charvey

2. 2 Overview l Options: »Uses, definitions, types l Put-Call Parity »Futures and Forwards l Valuation »Binomial »Black Scholes l Applications »Portfolio Insurance »Hedging

3. 3 Definitions Call Option is a right (but not an obligation) to buy an asset at a pre- arranged price (=exercise price) on or until a pre-arranged date (=maturity). Put Option is a right (but not an obligation) to sell an asset at a pre- arranged price (=exercise price) on or until a pre-arranged date (=maturity). European Options can be exercised at maturity only. American Options can be exercised at any time before maturity

4. 4 Examples of Options Securities 4 Equity options Warrants Underwriting Call provisions Convertible bonds Caps Interest rate options Insurance Loan guarantees 4 Risky bonds 4 Equity Real Options Options to expand Abandonment options Options to delay investment Model sequences Options are everywhere!

5. 5 Values of Options at Expiry Buying a Call Payoff Stock Price X 0 Payoff = max[0, S T - X] Buy Call Option

6. 6 Payoff Stock Price X 0 Sell Call Option Payoff = - max[0, S T - X] Values of Options at Expiry Writing a Call

7. 7 Payoff Stock Price X 0 Payoff = max[0, X - S T ] Buy Put Option X Values of Options at Expiry Buying a Put

8. 8 Payoff Stock Price X 0 Sell Put Option Payoff = - max[0, X - S T ] -X Values of Options at Expiry Selling a Put

9. 9 Example l What are the payoffs to the buyer of a call option and a put option if the exercise price is X=$50?

10. 10 Valuation of Options: Put-Call Parity l Principle: »Construct two portfolios »Show they have the same payoffs »Conclude they must cost the same l Portfolio I: Buy a share of stock today for a price of S 0 and simultaneously borrowed an amount of PV(X)=Xe -rT. »How much would your portfolio be worth at the end of T years? –Assume that the stock does not pay a dividend.

11. 11 Payoff of Portfolio I Payoff Stock Price STST -X S T - X 0 Payoff on Stock Payoff on Borrowing Net Payoff X

12. 12 Put-Call Parity l Portfolio II: 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? l Since the two portfolios have the same payoffs at date T, they must have the same price today. l The put-call parity relationship is: l This implies: Call - Put = Stock - Bond C E - P E = S 0 - PV(X)

13. 13 Put-Call Parity Payoff Stock Price -X X S T - X 0 Payoff on short put Payoff on long call Net Payoff

14. 14 Put-Call Parity and Arbitrage l 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? l From Put-Call Parity, the price of the call option should be equal to: »C E = P E + S 0 - Xe -rT =$13.12 l 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 PV($90)= 88.88 for 3 months.

15. 15 Put-Call Parity and Arbitrage l The cash flows for this investment are outlined below: l Hence, realize an arbitrage profit of 1.12 »This is independent of the value of the stock price!

16. 16 Options and Futures l Compare this with a futures contract that specicifies that you buy a stock at X at time T. The futures contract trades today at F 0. »What is the price of the futures if there is no arbitrage? –Construct zero-payoff portfolio: Buy a Put, Write a Call, and buy the futures contract »Hence, the relationship between futures and options is:

17. 17 Options and Futures Payoff Stock Price -X-X X S T - X 0 Payoff on short put Payoff on long call Payoff on Future l Call is right to purchase l Short Put is obligation to sell l Future combines both l When is F 0 =0?

18. 18 Debt and Equity as Options l 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?

19. 19 Debt and Equity l Consider a firm with zero coupon debt outstanding with a face value of F. The debt will come due in exactly one year. l 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. l Similarly, the payoff to the firm ’ s bondholders one year from now will be: Payoff to Bondholders = V - max[0,V-F] l 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.

20. 20 Debt and Equity Payoffs Firm Value0 Equityholders Bondholders F

21. 21 Valuing Options Establish bounds for Options l Upper bound on European call: »Compare to following portfolio: buy one share, borrow PV of exercise price »Consider value at maturity: l Hence, since the call is worth more at maturity, C E >S-PV(X) before maturity

22. 22 Bounds on Option Values l CE>S-PV(X); dominates portfolio of stock and borrowing X. l CE<S, otherwise buy stock straightaway S, C Stock Price C=SC=S-PV(X) PV(X)

23. 23 Example on Option Bounds I l 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 l Since the price is only $1.50, the call is underpriced by at least $0.46.

24. 24 Example on Option Bounds II l 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 l Since the price is only $2.50, the put is underpriced by at least $0.34

25. 25 Valuing Options as Contingent Claims Idea: l Investors attach different values to states in which assets pay off: $1 is worth more in bad times than in good times. l Values depend on preferences for insuring against bad times and discounting (time value of money). l Value of $1 in good times or bad times (or a continuum of states) can be inferred from prices of stocks and bonds. Procedure: »Determine value of $1 in good and in bad state »Use the value to infer the value of the option Stock Price = 100 125 80 High State Low State r=10%

26. 26 Pricing Contingent Claims Step 1: Determine the value of states Method l Break up payment to shareholders into two components: »Shareholders receive at least 80 for sure (in good and bad state). »Shareholders receive an additional 45 if the share price is high, otherwise nothing. Steps: 1.The present value of a safe payment of 80 is simply: 2.The value shareholders attach to the uncertain 45=125-80 must be the difference between the current share price and the value of the safe payment: 100 - 72.73 = 27.27 3.The present value of $1 in the good state is 27.27/45=0.606.

27. 27 l Consider the following option: Maturity:1 year Exercise price:110 Type:European How does the option value develop? l The present value of $1 in the good state is $0.606, hence the option value is: Option value = $0.606*15=$9.09 Pricing Contingent Claims Step 2: Value an Option Option Value = ? 15 0 High State Low State

28. 28 Why does this work? Contingent Claim Pricing and Arbitrage l Compare two portfolios: Portfolio 1: 1 Call option Portfolio 2: 1/3 share; 1 loan which pays off 80/3 at the end l Value of call option =Share price - Loan value = l You can make an option through buying a share and borrowing.

29. 29 Arbitrage: The General Idea General Rule: l Use arbitrage principle by constructing portfolio with same payoffs as option (this is called replication). l Portfolio has delta shares and loan which pays exactly the lowest value of the delta shares. delta is called the option delta: l If portfolio replicates option, then it must have the same value as the option. Implications: l Options can be valued by replicating their payoffs through forming portfolios of other assets. l Having an option is similar to buying stock and borrowing.

30. 30 Options with Many States l Suppose there are more than two possible states at the end of the period. Then: subdivide period. Example: 3 states at the end of the period: Divide movement into two periods with two-states in each. Solution: l Value the option for each of the mid-period nodes and then fold it backwards into the first node. l Repeat this for ever smaller intervals to cover larger numbers of states.

31. 31 The Black-Scholes Formula Alternative Solution: l Repeat the above process until infinity; Continuum of different states. l Use mathematical theory to determine result of this process. Black-Scholes Formula: Option value=[delta x share price]-[bank loan] N(d 1 ) x P - N(d 2 ) x PV(X) where:

32. 32 Call Option Sensitivities l The Option Pricing formula gives the following sensitivies for a call option:

33. 33 Intuition for Black-Scholes

34. 34 Black-Scholes Put Option Formula l We can use the put-call parity relationship to derive the Black-Scholes put option formula: l Use Put-Call Parity and the fact that the normal distribution is symmetric around the mean: P E = C E - S + Xe -rT P E = -SN(-d 1 ) + Xe -rT N(-d 2 )

35. 35 Put Option Sensitivities l The Option Pricing formula gives the following sensitivies for a put option:

36. 36 Example l On February 2, 1996, Microsoft stock closed at a price of $93 per share. »Annual standard deviation is about 32%. »The one-year T-bill rate is 4.82%. l 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?

37. 37 How to Use Black-Scholes l The inputs for the Black-Scholes formula are: »S = $93.00  r = 4.82% »X = $100.00   = 32% »T = 77/365 l This gives: d 1 = -0.351 d 2 = -0.498. l The cumulative normal density for these values are N(d 1 ) = 0.3628 N(d 2 ) = 0.3103. l Plugging these values into the Black-Scholes formula gives: c = $3.02 p = $9.02.

38. 38 How to Use Black-Scholes l Microsoft Put and Call Options

39. 39 Implied Volatility

40. 40 Implied Volatilities l It is common for traders to quote prices in terms of implied volatilities. l This is the volatility (  ) that sets the Black-Scholes price equal to the market price. l This can be computed using SOLVER in EXCEL.

41. 41 Applications of Options I: Volatility Bets l Suppose you have no information about the return of the stock, but you believe that the market underrates the volatility of the stock: »Give an example! –How can you trade? l Buy Straddle: »Buy a call and a put on the same stock –same exercise price –same time to maturity..

42. 42 Option Trading Strategies: The Straddle Payoff Stock Price X 0 X Put Payoff Call Payoff Straddle Payoff

43. 43 Hedging with Options l Initial investment (option premium) is required l You eliminate downside risks, while retaining upside potential Example »It is the end of August and we will receive 1m DM at the end of October. »At this point, we will sell DM, converting them back into dollars. »We are concerned about the price at which we will be able to sell DM. »We can lock in a minimum sale price by buying put options. –Since the total exposure is for 1m DM and each contract is for 62,500 DM we buy 16 put option contracts. –Suppose we choose the puts struck at 0.66 - locking in a lower bound of 0.66 $/DM.

44. 44 Heding with Currency Options Scenario I: Deutschmark falls to $0.30 l We have the right to sell 1m DM for $0.66 each by exercising the put options. l Since DM ’ s are only worth $0.30 each we do choose to exercise. l Our cash inflow is therefore $660,000 Scenario II: Deutschemark rises to $0.90 l We have the right to sell 1m DM for $0.66 each by exercising the put options. l Since DM ’ s are worth $0.90 each we do not choose to exercise. l We sell the DM on the open market for $0.90 each. l Our cash inflow is therefore $900,000

45. 45 Portfolio Insurance l Reconsider the case of a fund manager who wishes to insure his portfolio

46. 46 Summary l Options are derivative securities: »Replicate payoffs with combinations of underlying assets l Put and Call prices are linked l Valuation as contingent claims »Use Black-Scholes as approximation l Value of option increases with volatility of underlying assets l Use options for »Volatility bets »Portfolio Insurance »Hedging