1. Introduction to options TIP If you do not understand something, ask me! Basic and advanced concepts

2. 2 Today’s plan Introduction of options yDefinition of options yPosition diagrams yNo arbitrage argument yPut-call parity yApplication of put-call parity yHow parameter values affect option values?

3. 3 Introduction to options What is an option? zAn option is a right to do something at a specified price or cost on or before some specified date. zAn option, is a contract, and is therefore “written” – just means it exists zOptions are everywhere. yAOL offers its CEO a bonus (stock options) if its stock price exceeds $65 per share yYou have the option to come to my office hours at the cost of walking several extra steps.

4. 4 Brief History zOptions are a form of insurance, so in that sense they have been around for quite some time. zThe first organized exchange on which options were traded was opened in Chicago in 1973. Before that, options were traded over-the-counter.

5. 5 Brief History (cont’d) zIn the same year, the Black-Scholes formulae for option prices was published. The prices predicted by the formulae turned out to be extremely close to actual option prices. zThe popularity of options skyrocketed. They are arguably the most successful derivative security ever!

6. 6 Financial Options vs. Real Options zFinancial Options yOptions written on financial asset are called financial options, or simply “options” x(ex: option written on IBM or Dell) zReal options yOptions written on real assets are called real options yFor example, the option to set up a factory or discontinue a division is called real option

7. 7 Now we focus on two types of (financial) options… zCall yAn option to buy an underlying security (for example, a stock) for a fixed price (that is, the strike or exercise price) on or before a certain date (expiration date or maturity date). zPut yAn option to sell the underlying security (for example, a stock) for a fixed price (that is the strike or exercise price) on or before a certain date (expiration date or maturity date).

8. 8 Option Terms zExercising the Option yEnforcing the contract, i.e., buy or selling the underlying asset using the option zStriking, Strike, or Exercise Price yThe fixed price specified in the option contract for which the holder can buy or sell the underlying asset. zExpiration Date yThe last date on which the contract is still valid. After this date the contract no longer exists.

9. 9 Option terminology zIn-the-money call – a call option whose exercise price is less than the current price of the underlying stock. zOut-of-the-money call – a call option whose exercise price exceeds the current stock price. zAnother way to remember whether an option is in the money: if you can make money by immediately exercising your option, the option is in the money. (You may not be able to exercise it, though.)

10. 10 European vs. American Options zEuropean yA European option can only be exercised on the exercise date. zAmerican yAn American option can be exercised on any date up to the exercise date.

11. 11 Option Obligations zOptions are rights (to the buyer), and are obligations (to the seller) zThis means that: ythe buyer of an option may or may not exercise the option. yHowever, the seller of the option must sell or buy the underlying assets if the buyer decides to exercise the option.

12. 12 What is a short position in an option? yIn this case the other party has the option. yIs a long position in a call the same as a short position in a put?

13. 13 Payoff or cash flows from options at expiration date zThe payoff of a call option with a strike price K at the expiration date T is yWhere S(T) is the stock price at time T zThe payoff of a put option with a strike price K at the expiration date T is yWhere S(T) is the stock price at time T

14. 14 Example on payoffs Suppose that you have bought one European put and an European call on AOL with the same strike price of $55. The payoffs of your options certainly depend on the price of AOL on expiration

15. 15 Option payoff at expiration Call option value (graphic) given a $55 exercise price. Share Price Call option $ payoff 55 75 $20

16. 16 Option payoff Put option value (graphic) given a $55 exercise price. Share Price Put option value 50 55 $5

17. 17 Option payoff Call option payoff (to seller) given a $55 exercise price. Share Price Call option $ payoff 55

18. 18 Option payoff Put option payoff (to seller) given a $55 exercise price. Share Price Put option $ payoff 55

19. 19 Let's do some examples. zGoing short, selling an option you do not own, or writing an option are all the same thing. zYou have written a call with a strike of $50 on GM stock. What is your position if, on the expiration date, GM closes at $55 $45 zWho has the option in this case?

20. 20 Value of the position at expiration Stock Price

21. 21 Shorting Puts zYou have written a put with a strike of $50 on GM stock. What is your position if, on the expiration date, GM closes at $55 $45 zWho has the option in this case?

22. 22 Value of the position at expiration Stock Price

23. 23 What is the payoff if you go long a call and short a put, both with a strike of $50? zSay I add $50, what is another name for this position? Stock Price

24. 24 Some examples zPlease draw position diagrams for the following investment: yBuy a call and put with the same strike price and maturity (straddle)

25. 25 Option payoff Straddle - Long call and long put Share Price Position Value Straddle

26. 26 More examples yBuy a stock and a put (protective put)

27. 27 Option payoff Protective Put - Long stock and long put Share Price Position Value Protective Put

28. 28 Valuation of options zAt expiration an option must be worth its exercise value or zero. zAn American option's value is as least as large as its immediate exercise value (why?) and since it gives an extra right (which can always be ignored) is always at least as valuable as its European counterpart.

29. 29 Valuation of options zAn American call's (put's) value can never exceed the value of the stock (strike price) yWhy? zDoes this principle hold for European options? zYes.

30. 30 Valuation of options zEverything else equal, the longer maturity for An American option, the more valuable. yWhy? zDoes this principle hold for European options?

31. 31 Valuation of options zAn American call (put) with a higher exercise price will be worth less (more). zWhy? zDoes this principle apply to European options? zYes.

32. 32 Put-Call Parity zLet P(K,T) and C(K,T) be the prices of a European put and a call with strike prices of K and maturity of T. S0 is current stock price. Then we have or Where

33. 33 No arbitrage concept zIf two securities have the exactly the same payoff or cash flows in every state of each future period, these two securities should have the same price; otherwise there is an arbitrage opportunity or money making opportunity.

34. 34 Let’s show put-call parity zWe can first use position diagrams to show put-call parity zThis exercise is a good way of getting used to the ideas of the single price rule or no arbitrage argument.

35. 35 Position diagram Payoff of investing PV(K) in risk-free security and buying a call Share Price Position Value K

36. 36 Position diagram Payoff of long stock and long put Share Price Position Value K

37. 37 The conclusion zSince both portfolios in the previous two slides give you exactly the same payoff, they must have the same price. That is,

38. 38 In the above we have assumed that the stock will not pay any dividend. zConsider dividend payment D before expiration date. For European options:

39. 39 Things to note about Put- Call parity zOnly works for European options. zBased on arbitrage so it works exactly. This is how brokers created puts out of calls when options were traded over the counter.

40. 40 European vrs American Calls zIt turns out that you would never want to exercise an American call on a non- dividend paying stock early. zWhy might you wish to exercise an American call early when the stocks pays dividend?

41. 41 European vs American Puts zThere are times when you will want to exercise an American put on a non- dividend paying stock early. zWhy?

42. 42 Applications of option concepts and put-call parity zOne important application of option concepts and put-call parity is the valuation of corporate bonds. zFor example, suppose that a firm has issued $K million zero-coupon bonds maturing at time T. Let the market value of the firm asset at time t be V(t).

43. 43 Applications of option concepts and put-call parity (continue) Payoff of equity Market value of asset Position Value K

44. 44 Applications of option concepts and put-call parity (continue) zSo based on the position payoff diagram in the previous slide, we can see that the value of equity is just the value of a call option with strike price K. zThen bond value =Asset value –equity value (value of call: C(K,T) zUsing the put-call parity, we have zBond value=V(A)-(V(A)+P(K,T)-PV(K))=PV(K)- P(K,T) (value of put )

45. 45 Applications of option concepts and put-call parity (continue) zWhat does this result mean? zThe value of risky corporate bonds is equal to the value of the safe corporate bonds minus the cost of default. zWhen will the firm default? yAt time T, if the value of asset is less than K, the firm will default. P(K,T) is the cost of this default to bond holders.

46. 46 Some bounds about option values zSince an option is a right to buy or sell securities, its price is always non-negative. zSince at expiration, we have payoff Max(S(T)-K,0) for a call with a price C(K,T) at time 0 Max(K-S(T),0) for a put with a price P(K,T) at time 0 zThen

47. 47 Some bounds about option values (continue) zFrom put-call parity, we have zThus

48. 48 The impact of volatility of the stock price on the call option zConsider the following two call options written on stocks A and B with the same strike price of $50 and same maturity, respectively: current price S A =S B =$40 and stock A is much more volatile than stock B. Then At maturity, stock A has a much larger chance that the stock price is larger than $50 than stock B. Thus, the payoff from the option on stock A is expected to be larger than from the option A. Thus the option on stock A is more valuable than the option on stock B.

49. 49 Volatility and option values. zFor call options, the larger the volatility of the underlying asset, the larger the value of the option. zSuppose a firm has both debt and equity. yIf the managers are to take riskier projects than bond holders expect, should the bond holders or equity holders benefit from this?

50. 50 How option values are affected by variables? z If this variable increases The value of an American or European call The value of an European put The value of an American put Stock price (S)IncreaseDecrease Exercise price (K)DecreaseIncrease Volatility (σ)Increase? Time to expiration (T) Increase? Interest rate (rf)IncreaseDecrease Dividend payoutDecreaseIncrease

51. 51 The Black-Scholes formula for a call option zThe Black-Scholes formula for a European call is zWhere

52. 52 The Black-Scholes formula for a put option zThe Black-Scholes formula for a European put is zWhere

53. 53 Of course, if you already know a call with same maturity and expiration… zYou can get the put price by put-call parity.

54. 54 Intuition for the Black-Scholes formula zOne way to understand the Black-Scholes formula is to find the present value of the payoff of the call option if you are sure that you can exercise the option at maturity, i.e., S - exp(-rt)K. zComparing this present value of this payoff to the Black-Scholes formula, we know that N(d 1 ) can be regarded as the probability that the option will be exercised at maturity

55. 55 An example zMicrosoft sells for $50 per share. Its return volatility is 20% annually. What is the value of a call option on Microsoft with a strike price of $70 and maturing two years from now suppose that the risk-free rate is 8%? zWhat is the value of a put option on Microsoft with a strike price of $70 and maturing in two years?

56. 56 Solution zThe parameter values are zThen