1. - 1 - Topic 4: Derivative Securities in Global Financial Markets Purpose: Provide background on the basics of Option Pricing Theory (OPT) Examine some recent applications of derivatives in international finance

2. - 2 - Derivatives Derivative = obligation to accomplish a transaction in the future Forward Contract = basic derivative from which all others have evolved Repurchase Agreements –Reverse Repurchase Agreements Futures Contracts

3. - 3 - Derivatives Swaps –Futures and Forward Contracts On Swaps Options –Currency Options –Swaptions –Options On Futures –Futures On Options

4. - 4 - What are options? Options are financial contracts whose value is contingent upon the value of some underlying asset Such arrangements are also known as contingent claims –because equilibrium market value of an option moves in direct association with the market value of its underlying asset. OPT measures this linkage

5. - 5 - The basics of options Calls and puts defined Call: privilege of buying the underlying asset at a specified price and time Put: privilege of selling the underlying asset at a specified price and time

6. - 6 - The basics of options Regional differences American options can be exercised anytime before expiration date European options can be exercised only on the expiration date Asian options are settled based on average price of underlying asset

7. - 7 - The basics of options Options may be allowed to expire without exercising them Options game has a long history –at least as old as the “premium game” of 17th century Amsterdam –developed from an even older “time game” which evolved into modern futures markets and spawned modern central banks

8. - 8 - Binomial Approach

9. - 9 -

10. - 10 - DCF only Augmented

11. - 11 -

12. - 12 - As the binomial change process runs faster and faster, it approaches something known as Brownian Motion Let’s have a sneak preview of the Black-Scholes model, using a similar example

13. - 13 - Illustration using Black-Scholes Value of 1st year’s option = $1135.45 Value of 2nd year’s option = $1287.59 NPV = –2000 + 1135.40 + 1287.59 = $423.04

14. - 14 - Put-Call Parity Consider two portfolios Portfolio A contains a call and a bond: C(S,X,t) + B(X,t) Portfolio B contains stock plus put: S + P(S,X,t)

15. - 15 - Put-Call Parity Consider two portfolios Portfolio A contains a call and a bond: C(S,X,t) + B(X,t) Portfolio B contains stock plus put: S + P(S,X,t)

16. - 16 - Put-Call Parity C(S,X,t) + B(X,t) = S + P(S,X,t) News leaks about negative event Informed traders sell calls and buy puts

17. - 17 - Put-Call Parity News leaks about negative event Informed traders sell calls and buy puts Arbitrage traders buy the low side and sell the high side C(S,X,t) + B(X,t) = S + P(S,X,t)

18. - 18 - Put-Call Parity News leaks about negative event Informed traders sell calls and buy puts Arbitrage traders buy the low side and sell the high side Stock price falls — “the tail wags the dog” C(S,X,t) + B(X,t) = S + P(S,X,t)

19. - 19 - C(S,X,t) + B(X,t) = S + P(S,X,t) Boundaries on call values C(S,X,t) + B(X,t) = S + P(S,X,t) Upper Bound: C(S,X,t) < S Stock Call

20. - 20 - C(S,X,t) + B(X,t) = S + P(S,X,t) Boundaries on call values C(S,X,t) + B(X,t) = S + P(S,X,t) Upper Bound: C(S,X,t) < S Lower bound: C(S,X,t) ≥ S – B(X,t) Stock Call B(X,t)

21. - 21 - C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

22. - 22 - C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

23. - 23 - C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

24. - 24 - C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

25. - 25 - C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

26. - 26 - C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

27. - 27 - C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

28. - 28 - C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

29. - 29 - C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

30. - 30 - C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

31. - 31 - C(S,X,t) = S - B(X,t) + P(S,X,t) Call values C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t)

32. - 32 - C(S,X,t) = S - B(X,t) + P(S,X,t) Keys for using OPT as an analytical tool C(S,X,t) = S - B(X,t) + P(S,X,t) Stock Call B(X,t) Stock Call B(X,t) S C X C t C C  R C P P P P P

33. - 33 -

34. - 34 -

35. - 35 -

36. - 36 -

37. - 37 -

38. - 38 -

39. - 39 -

40. - 40 -

41. - 41 -

42. - 42 -

43. - 43 - Impact of Limited Liability C(V,D,t) = V - B(D,t) + P(V,D,t) B(D,t) V Equity Equity = C(V,D,t) Debt = V - C(V,D,t)

44. - 44 - Basic Option Strategies Long Call Long Put Short Call Short Put Long Straddle Short Straddle Box Spread

45. - 45 - Long Call S $ 0 - C X X+C

46. - 46 - Short Call S $ 0 - C X X+C Long Call X S $ 0 X+C C

47. - 47 - Long Put S $ 0 - C X X+C X S $ 0 C Long Call Short Call S $ 0 X - P X-P

48. - 48 - Short Put S $ 0 - C X X+C X S $ 0 C Long Put Long Call Short Call S $ 0 X - P X-P S $ 0 P X

49. - 49 - Long Straddle S $ 0 - C X X+C X S $ 0 C Long Put Long Call Short Call S $ 0 P X X-P S $ 0 X - P X-P Short Put S $ 0 X -(P+C) X-P-C X+P+C

50. - 50 - Short Straddle S $ 0 - C X X+C X S $ 0 C Long Put Long Call Short Call S $ 0 P X X-P S $ 0 X - P X-P Short Put S $ 0 X -(P+C) X-P-C X+P+C Long Straddle $ 0 X P+C X-P-C X+P+C S

51. - 51 - Box Spread Long call, short put, exercise = X Same as buying a futures contract at X S X $ 0

52. - 52 - Box Spread Long call, short put, exercise = X Short call, long put, exercise = Z S X $ 0 Z

53. - 53 - Box Spread You have bought a futures contract at X And sold a futures contract at Z S X $ 0 Z

54. - 54 - Box Spread Regardless of stock price at expiration –you’ll buy for X, sell for Z –net outcome is Z - X S X $ 0 Z Z - X

55. - 55 - Box Spread How much did you receive at the outset? + C(S,Z,t) - P(S,Z,t) - C(S,X,t) + P(S,X,t) S X $ 0 Z Z - X

56. - 56 - Box Spread Because of Put/Call Parity, we know: C(S,Z,t) - P(S,Z,t) = S - B(Z,t) - C(S,X,t) + P(S,X,t) = B(X,t) - S S X $ 0 Z Z - X

57. - 57 - Box Spread So, building the box brings you S - B(Z,t) + B(X,t) - S = B(X,t) - B(Z,t) S X $ 0 Z Z - X

58. - 58 - Assessment of the Box Spread At time zero, receive PV of X-Z At expiration, pay Z-X You have borrowed at the T-bill rate. S X $ 0 Z Z - X

59. - 59 - Currency Options Options to exchange one currency for another

60. - 60 - The basics of currency options Call: privilege of buying the underlying currency at a specified exchange rate and time –A call option written on the U.S. Dollar in London, for example, gives the holder the privilege (but not the obligation) of buying U.S. Dollars in exchange for British Pounds at a specified exchange rate Put: privilege of selling the underlying currency at a specified exchange rate and time –a put option written on the Pound in New York conveys the privilege of selling Pounds in exchange for Dollars at a specified rate

61. - 61 - The basics of currency options Call: privilege of buying the underlying currency at a specified exchange rate and time Put: privilege of selling the underlying currency at a specified exchange rate and time The twist is that the put option written in New York is the same thing as the call option written in London, when both have the same expiration date –Any disparity in prices would present a lucrative but short-lived arbitrage opportunity

62. - 62 - Example of Parity in Currency Options New York $5 buys a put to sell £60 in exchange for $100 (exchange at the forward rate) London Find equilibrium price for a call to buy $100 in exchange for £60 (exchange at the forward rate) Answer: $5 *.62 = £3.10 $1 = £0.62 spot $1 = £0.60 forward

63. - 63 - Example of Parity in Currency Options New York $5 buys a put to sell £60 in exchange for $100 (exchange at the forward rate) London Find equilibrium price for a call to buy $200 in exchange for £120 (exchange at the forward rate) Answer: $10 *.62 = £6.20 $1 = £0.62 spot $1 = £0.60 forward

64. - 64 - Using Scale to Compare Options The previous example gives us insight into the ability to adjust the scale of options: The New York involves half as much money as the London option ($100 and £60, compared with $200 and £120) We should scale the price of the option accordingly, and find the value of $10 translated into Pounds at the spot exchange rate

65. - 65 - Standardizing Options The ability to adjust the scale of options also makes it possible to standardize options for improved comparability in the search for potential arbitrage opportunities: Divide the underlying asset’s price by the exercise price, creating an option with an exercise price of one –Then the value of the underlying asset is adjusted to S/X Time and volatility stay the same The price of the standardized option will be the old price divided by the exercise price

66. - 66 - Example of a London Dollar Call Expiration is in 180 days Use the “Margrabe” tab on the option calculator spreadsheet

67. - 67 - Options on Futures A Solution for Some Hedgers Frustrated with the Practical Difficulties of Hedging

68. - 68 - Options on Futures Futures options may expire before the maturity of the futures contract Notation to be used: –T is the maturity of the futures contract –Expiration of the option is t

69. - 69 - Options on Futures A call conveys the right (but not the obligation) to take a long position in a specified futures contract at a specified price Example: –Buy a September Swiss Franc 60 call in July (quantity for delivery is CHF 100, so price per franc is $0.6000) –On option expiration, the current futures price for the September Swiss Franc contract is $62 –Upon exercise you would receive $2, the difference between the $62 futures price and the $60 exercise price

70. - 70 - Options on Futures A put conveys the right (but not the obligation) to take a short position in a specified futures contract at a specified price Example: –Buy a September Swiss Franc 60 put in July (quantity for delivery is CHF 100, so price per franc is $0.6000) –On option expiration, the current futures price for the September Swiss Franc contract is $58 –Upon exercise you would receive $2, the difference between the $60 strike price and the $58 futures price

71. - 71 - Put-Call Parity for Options on Futures (T is time until maturity of futures, t is time until expiration of the options) Consider two portfolios Portfolio A contains a call and a bond: Sept SF 60 call + e -rt (60-59) Portfolio B contains future plus put: f t (T) - 59 + Sept SF 60 put Today’s futures price for Sept SF is 59

72. - 72 - Put-Call Parity at time 0 (European Options) Call e + e -rt (60 - 59) = Future + Put e Lower bound of European Call: Call e ≥ Max [0, e -rt (f 0 (T) - X)] In the example of the September Swiss Franc 60 options, the lower bound is zero

73. - 73 - Put-Call Parity at time 0 (European Options) Future = Call e – Put e – e -rt (60 - 59) = 0

74. - 74 - Put-Call Parity at time 0 (European Options) Call e = Put e – e -rt (60 – 59) In the example of the September Swiss Franc 60 options, the call has lower value than the put

75. - 75 - Lower Bound (American Options on Futures) Lower bound of American Call on a future: Call a ≥ Max [0, (f 0 (T) - X)] –There is a futures price at which the option premium for an in- the-money American call equals its intrinsic value. Early exercise would occur if the futures price rose above that point Lower bound of American Put on a future: Put a ≥ Max [0, (X - f 0 (T))] –There is a futures price at which the option premium for an in- the-money American put equals its intrinsic value. Early exercise would occur if the futures price fell below that point

76. - 76 -

77. - 77 -

78. - 78 - Swaps

79. - 79 - Floating-Fixed Swaps Fixed If net is positive, underwriter pays party. If net is negative, party pays underwriter. Illustration of a Floating/Fixed Swap Party Underwriter Counterparty Variable Fixed Variable

80. - 80 - Floating to Floating Swaps LIBOR If net is positive, underwriter pays party. If net is negative, party pays underwriter. Illustration of a Floating/Floating Swap Party Underwriter Counterparty T-Bill LIBOR T-Bill

81. - 81 - Parallel Loan United States Germany Loan guarantees Debt service in $ Illustration of a parallel loan German Parent U.S. subsidiary of German Firm U.S. Parent German subsidiary of U.S. Firm Principal in $ Debt service in Euro Principal in Euro

82. - 82 - Currency Swap German rate x €1,000,000 € 1,000,000 2 2 U.S. rate x $1,500,000 German rate x €1,000,000 U.S. rate x $1,500,000 1 1 € 1,000,000 $1,500,000 € 1,000,000 3 3 $1,500,000 € 1,000,000 $1,500,000 Illustration of a straight currency swap Step 1 is notional Steps 2 & 3 are net Borrow in US, invest in Europe Borrow in Europe, invest in US

83. - 83 - Swaps Investor Underwriter Libor ± Spread Equity Index Return* *Equity index return includes dividends, paid quarterly or reinvested Illustration of an Equity Return Swap

84. - 84 - Swaps Investor Underwriter Foreign Equity Index Return* A Illustration of an Equity Asset Allocation Swap *Equity index return includes dividends, paid quarterly or reinvested Foreign Equity Index Return* B

85. - 85 - Equity Call Swap Investor Underwriter Illustration of an Equity Call Swap Equity Index Price Appreciation* * No depreciation—settlement at maturity Libor ± Spread

86. - 86 - Equity Asset Swap Underwriter Equity Index Return* * Equity index return includes dividends, paid quarterly or reinvested Income Stream Investor Income Stream Asset

87. - 87 - Bringing these innovations to the retail level

88. - 88 - PENs SCPERS BT Counterpary PEFCO $5 mm $5mm + Appreciation 1% Coupon Fixed Undisclosed Flow Appreciation

89. - 89 - Equity Call Swap Investor Underwriter Illustration of an Equity Call Swap Equity Index Price Appreciation* * No depreciation—settlement at maturity Libor ± Spread

90. - 90 -

91. - 91 - Box Spread Because of Put/Call Parity, we know: C(S,Z,t) + B(Z,t) = S + P(S,Z,t) S X $ 0 Z Z - X

92. - 92 - Box Spread C(S,Z,t) + B(Z,t) = S + P(S,Z,t) Now, let’s subtract the bond from each side: C(S,Z,t) = S + P(S,Z,t) - B(Z,t) S X $ 0 Z Z - X

93. - 93 - Box Spread C(S,Z,t) = S + P(S,Z,t) - B(Z,t) Next, let’s subtract the put from each side: C(S,Z,t) - P(S,Z,t) = S - B(Z,t) S X $ 0 Z Z - X

94. - 94 - Box Spread C(S,Z,t) - P(S,Z,t) = S - B(Z,t) Given this, we also know: - C(S,X,t) +P(S,X,t) = - S + B(X,t) S X $ 0 Z Z - X

95. - 95 - Box Spread So, because of Put/Call Parity, we know: C(S,Z,t) - P(S,Z,t) = S - B(Z,t) S X $ 0 Z Z - X