1. 1 Mechanics of Options Markets Chapter 8

2. 2 OPTIONS ARE CONTRACTS Two parties:Seller and buyer A contract:Specifying the rights and obligations of the two parties. An underlying asset: a financial asset, a commodity or a security, that is the basis of the contract.

3. 3 Assets Underlying Exchange-Traded Options (p. 190) Stocks Foreign Currency Stock Indices Futures Options Bonds

4. 4 OPTIONS BASICS A contingent claim: The option’s value is contingent upon the value of the underlying asset Two Types of Options: A Call: THE RIGHT TO BUY THE UNDERLYING ASSET A Put: THE RIGHT TO SELL THE UNDERLYING ASSET

5. 5 CALL Buyer  holder  long. In exchange for making a payment of money, the call premium, the call buyer has the right to BUY a specified quantity of the underlying asset for the exercise (strike) price before the option’s expiration date.

6. 6 PUT Buyer  holder  long. In exchange for making a payment of money, the put premium, the put buyer has the right to SELL a specified quantity of the underlying asset for the exercise (strike) price before the option’s expiration date.

7. 7 Call Seller  writer  short. In exchange for receiving the call’s premium, the Call seller has the obligation to SELL the underlying asset for the predetermined exercise (strike) price upon being served with an exercise notice during the life of the option, I.e., before the option expires.

8. 8 Put Seller  writer  short. In exchange for receiving the put premium, the Put seller has the obligation to BUY the underlying asset for the predetermined exercise (strike) price upon being served with an exercise notice during the life of the option, I.e., before the option expires.

9. 9 The two main types of Options (PUTS and CALLS) American Options exercisable any time before expiration European Options exercisable only on expiration date

10. 10 OPTIONS NOTATIONS: S –The underlying asset’s market price K -The exercise (strike) price t –The current date T –The expiration date T -t The time till expiration c, p European call, put premiums C, P American call, put premiums

11. 11 Options definitions using the above notation: LONG CALL On date t, the BUYER of a call option pays the call’s market price, c t, C t, and holds the right to buy the underlying asset at the strike price, K, before the call expires on date T. (or on T, if the call is European). Thus => the call holder expects the price of the underlying asset, S t, to increase during the life of the option contract.

12. 12 SHORT CALL On date t, the SELLER of a call option receives c t, C t, and must sell the underlying asset for K, if the option is exercised by its holder before the option expires on date T. Thus => expects the price of the underlying asset, S t, to remain below or at the exercise price, K, during the option’s life. This way the writer keeps the premium.

13. 13 LONG PUT On date t, the BUYER of a put option pays p t, P t, and holds the right to sell the underlying asset for K before the put expires on date T. Thus => expects the market price of the underlying asset, S t, to decrease during the life of the put.

14. 14 SHORT PUT On date t, the SELLER of a put receives p t, P t, and must buy the underlying asset for K if the put is exercised by its holder before the put expires on date T. Thus => expects the market price of the underlying asset, S t, to remain at or above K during the life of the put. This way the put writer keeps the premium.

15. 15 A numerical example: LONG CALL C(S= $47.27share;K = $45/share;T-t =.5yrs) C t = $5.78/share On date t, the BUYER of this call pays the call’s market price, $5.78/share, and holds the right to buy the underlying asset at the strike price, K = $45/share, before the call expires at T, half a year from now (at T, if the call is European). Thus => the call holder expects the price of the underlying asset, S t = $47.27/share, to increase during the life of the option contract.

16. 16 A numerical example: SHORT CALL C( S=$47.27share;K =$45/share; T-t =.5yrs) C t = $5.78/share On date t, the SELLER of this call receives $5.78/share and must sell the underlying asset for K = $45/share, if the option is exercised by its holder before the option expires at T, half a year from now. Thus => hopes the price of the underlying asset, currently S t = $47.27/share, remains below or at the exercise price, K = $45/share, during the option’s life of half a year and hence, keep the premium c t = $5.78/share

17. 17 A numerical example: LONG PUT p(S= $47.27share;K = $45/share;T-t =.5yrs) p t = $2.25/share On date t, the BUYER of this put pays the market price of p t = $2.25/share and holds the right to sell the underlying asset for K = $45/share before the put expires half a year from now, at T. Thus => expects the market price of the underlying asset, S t = $47.27/share, to decrease during the half a year life span of the put.

18. 18 A numerical example: SHROT PUT p(S =$47.27share;K = $45/share;T-t =.5yrs) p t = $2.25/share On date t, the SELLER of this put receives the market premium p t =$2.25/share and must buy the underlying asset for K = $45 if the put is exercised by its holder before the put expires half a year from now at T. Thus => expects the market price of the underlying asset, S t = $47.27/share to remain at or above K = $45 during the life span of the put and to keep the premium, p t = $2.25/share.

19. 19 $ 47.27 K = 45 t = now T =.5yrS

20. 20 More terminology Premium =The option Market Price Premium = [Intrinsic value + extrinsic value] Intrinsic value: CallsMax{0, S t - K) ≥ 0 PutsMax{0, K - S t ) ≥ 0 Extrinsic value (time value): Premium – Intrinsic value

21. 21 At-the-money S t = K In this case the intrinsic value for both calls and puts is zero: S t - K = K - S t = 0 and the premium consists of the Extrinsic (time) value only. PREMIUM = 0 + extrinsic value

22. 22 In-the-money CallsPuts S t > K S t < K or: S t – K > 0 K – S t > 0 The Intrinsic value of an option that is in-the money is positive.

23. 23 Out-of-the-money CallsPuts S t K or S t - K< 0 K – S t < 0 In this case the intrinsic value is zero and the premium consists of the extrinsic (time) value only. PREMIUM = 0 + extrinsic value

24. 24 The next table shows the market prices (premiums) of calls and puts on IBM On Friday NOV 30 2007 = t When IBM was trading at S t = $105/share. Notice that there where options traded for several expiration dates and for a wide range of strike prices. Blanks mean that the option did not trade on NOV 30 2007 OR did not exist.

25. S=105CALLSPUTS KDEC07JAN08APR08JUL08JAN09JAN10DEC07JAN08APR08JUL08JAN09JAN10 5548.90 6043.4048.30.051.05 6539.0040.1052.60.10.30.501.75 7033.7032.2037.6048.20.15.55.701.90 7533.2830.8032.3033.50.20.601.102.40 8023.3025.6028.6028.8032.38.25.751.853.97 8518.9020.2024.4024.9025.70.05.301.402.454.38 9017.3618.0018.4020.4022.9027.00.10.601.953.405.708.70 9511.4011.6014.7018.1018.25.301.303.205.906.80 1006.308.2011.0013.6018.0021.50.902.304.806.108.9011.83 1052.904.808.2010.5513.602.354.106.908.0011.00 110.952.705.907.7011.8016.805.506.909.109.8012.8018.50 115.201.313.956.7510.309.8010.6010.8015.0015.60 120.05.602.605.207.1012.5016.3015.0014.5018.0018.5023.50 125.251.653.7010.3016.9018.8024.80 130.151.151.804.309.6022.6022.2027.5031.20 135.10.801.7533.10 140.37.953.107.2032.7038.70 145.30.80 150.151.654.70 1601.204.00 170.50

26. 26 Options Markets 1.OTC options: Over the counter (OTC) Meaning Not on an organized exchange. 2.Exchange traded options: An organized exchange Options clearing corporation (OCC)

27. 27 WHEN OPTIONS ARE TRADED ON THE OTC TRADERS BEAR Credit risk Operational risk Liquidity risk

28. 28 Credit Risk: Does the other party have the means to pay? Operational Risk: Will the other party deliver the commodity? Will the other party pay?

29. 29 Liquidity Risk. Liquidity = the speed (ease) with which investors can buy or sell securities (commodities) in the market. In case either party wishes to get out of its side of the contract, what are the obstacles? How to find another counterparty? It may not be easy to do that. Even if you find someone who is willing to take your side of the contract, the other party may not agree.

30. 30 THE Option Clearing Corporation (OCC) (p. 198) The exchanges understood that there will exist no efficient options markets without contracts standardization and an absolute guarantee to the options’ holders – that the market is default-free, so they have created the: OPTIONS CLEARING CORPORATION (OCC) The OCC is a nonprofit corporation

31. 31 CLEARING MEMBERS NONCLEARING MEMEBRS EXCHANGE CORPORATION OPTIONS CLEARING CORPORATION BROKERSCLIENTES THE OPTION CLEARING CORPORATION PLACE IN THE MARKET OCC MEMBER

32. 32 The OCC’s absolute guarantee The holders of calls and puts will always be able to exercise their options if they so wish to do!!!

33. 33 The absolute guarantee The OCC’s absolute guarantee provides traders with a default-free market. Thus, any investor who wishes to engage in options buying knows that there will be no operational default.

34. 34 The OCC Also, clears all options trading. Maintains the list of all long and short positions. Matches all long positions with short positions. Hence, the total sum of all options traders positions must be ZERO at all times.

35. 35 The OCC Maintains the accounting books of all trades. Charges fees to cover costs Assigns Exercise notices Given the OCC’s guarantee, the market is anonymous and traders only have to offset their positions in order to come out of the market. The OCC has no control over the market prices. These are determined by trader’s supply and demand.

36. 36 The OCC The OCC’s absolute guarantee together with matching all short and long trading makes the market very liquid. 1 – traders are not afraid to enter the market 2 – traders can quit the market at any point in time by OFFSETTING their original position.

37. 37 CLIENT A BUY ORDER INFO: TRADERS LONG SHORT OPTIONS PRICE OPEN INTEREST VOLUME BROKER OCC MEMBER PRICE THE TRADING FLOOR TRADE THE OCC PRICE OCC MEMBERMARGINS BROKERMARGINS SELL ORDER CLIENT B MARGINS

38. 38 OFFSETTING POSITIONS A trader with a LONG position who wishes to get out of the market MAY: a)Exercise, or b)open a SHORT position with equal number of the same options. Example: Suppose LONG 5, SEP, $85, IBM puts; p 0 = $4/share This position must be offset by SHORT 5, SEP, $85, IBM puts; p 1 = $3/share Cash flows: -$2,000 + $1,500 = -$500.

39. 39 OFFSETTING POSITIONS A trader with a SHORT position who wishes to get out of the market MUST open a LONG position with equal number of the same options. Example:Suppose SHORT 25, JAN, $75, BA calls; c = $7/share This position must be offset by LONG 25, JAN, $75, BA calls; c = $5/share Cash flows: $17,500 - $12,500 = $5,000.

40. 40 THE OCC Standardization: Contract size:the number of units of the underlying asset covered in one option. Exercise prices:Mostly, increments of $2.5, $5.00 and $10.00. Exercise notice and assignment procedures Delivery sequence.

41. 41 THE OCC Standardization: Expiration dates:Saturday, immediately following the third Friday of the expiration month. The basic expiration cycles: 1.[JAN  APR  JUL  OCT] 2.[FEB  MAY  AUG  NOV] 3.[MAR  JUN  SEP  DEC]

42. 42 A Review of Some Financial Economics Principles Arbitrage: A market situation whereby an investor can make a profit with: no equity and no risk. Efficiency: A market is said to be efficient if prices are such that there exist no arbitrage opportunities. Alternatively, a market is said to be inefficient if prices present arbitrage opportunities for investors in this market.

43. 43 Valuation: The current market value (price) of any project or investment is the net present value of all the future expected cash flows from the project. One-Price Law: Any two projects whose cash flows are equal in every possible state of the world have the same market value. Domination: Let two projects have equal cash flows in all possible states of the world but one. The project with the higher cash flow in that particular state of the world has a higher current market value and thus, is said to dominate the other project.

44. 44 The Holding Period Rate of Return (HPRR): Buy shares of a stock on date t and sell them later on date T. While holding the shares, the stock has paid a cash dividend in the amount of $D/share. The Holding Period Rate of Return HPRR is:

45. 45 Example: S t = $50/share S T = $51.5/share D T-t = $1/share T = t + 73days.

46. 46 Risk-Free Asset: is a security of investment whose return carries no risk. Thus, the return on this security is known and guaranteed in advance. Risk-Free Borrowing And Landing: By purchasing the risk-free asset, investors lend their capital and by selling the risk-free asset, investors borrow capita at the risk-free rate.

47. 47 The One-Price Law: There exists only one risk-free rate in an efficient economy. Proof: By contradiction. Suppose two risk-free rates exist in a market and R > r. Since both are free of risk, ALL investors will try to borrow at r and invest the money borrowed in R, thus assuring themselves the difference. BUT, the excess demand For borrowing at r and excess supply of lending (investing) at R will change them. Supply = demand only when R = r.

48. 48 Compounded Interest (p. 76) Any principal amount, P, invested at an annual interest rate, R, compounded annually, for n years would grow to: A n = P(1 + R) n. If compounded Quarterly: A n = P(1 +R/4) 4n.

49. 49 In general: Invest P dollars in an account which pays an annual interest rate R with m compounding periods every year. The rate in every period is R/m. The number of compounding periods is nm. Thus, P grows to: A n = P(1 +R/m) mn.

50. 50 A n = P(1 +R/m) mn. Monthly compounding becomes: A n = P(1 +R/12) 12n and daily compounding yields: A n = P(1 +R/365) 365n.

51. 51 EXAMPLES: n =10 years;R =12%; P = $100 1.Simple compounding, m = 1, yields: A 10 = $100(1+.12) 10 = $310.5848 2.Monthly compounding, m = 12, yields: A 10 = $100(1 +.12/12) 120 = $330.0387 3.Daily compounding, m = 365, yields: A 10 = $100(1 +.12/365) 3,650 = $331.9462.

52. 52 Notations: The annual rate R will be stated as R m in order to make clear how many times a year it is compounded. For the annual rate is 10% with quarterly compounding, the corresponding formula is: A n = P(1 +.10/4) 4n For the same annual rate with monthly compounding the corresponding formula is: A n = P(1 +.10/12) 12n

53. 53 DISCOUNTING The Present Value today, date t, of a future cash flow, FV T, on a future date T, is given by DISCOUNTING:

54. 54 DISCOUNTING: the general case: Let c ji, j = 1,2,3,…m, be a sequence of m cash flows paid in year i, i = 1,2,3,…,n. Let R m be the annual rate during these years. DISCOUNTING these cash flows yields the Present Value:

55. 55 CONTINUOUS COMPOUNDING In the early 1970s, banks came up with the following economic reasoning: Since the bank has depositors money all the time, this money should be working for the depositor all the time! This idea, of course, leads to the concept of continuous compounding. We want to apply this idea to the formula:

56. 56 CONTINUOUS COMPOUNDING As m increases the time span of every compounding period diminishes CompoundingmTime span Yearly11 year Daily3651 day Hourly87601 hour Every second3,153,600One second Continuously∞Infinitesimally small

57. 57 CONTINUOUS COMPOUNDING This reasoning implies that in order to impose the concept of continuous time on the above compounding expression, we need to solve: This expression may be rewritten as:

58. 58 Recall that the number “e” is: Xe 12 1002.70481382 10,0002.71814592 1,000,0002.71828046 In the limit 2.718281828…..

59. 59 Recall that in our example: n = 10 years. R = 12% P=$100. So, P = $100 invested at a 12% annual rate, continuously compounded for ten years will grow to:

60. 60 Continuous compounding yields the highest return: CompoundingmFactor Simple13.105848208 Quarterly43.262037792 Monthly123.300386895 Daily3653.319462164 Continuously ∞3.320116923

61. 61 This expression may be rewritten as: Continuous Discounting (p. 77)

62. 62 This expression may be rewritten as: Continuous Discounting

63. 63 Recall that in our example: P = $100; n = 10 years and R = 12% Thus, $100 invested at an annual rate of 12%, continuously compounded for ten years will grow to: $332.0117. Therefore, we can write the continuously discounted value of $320.0117:

64. 64 Equivalent Interest Rates (p.77) R m = The annual rate with m compounding periods every year.

65. 65 Equivalent Interest Rates (p.77) r c =The annual rate with continuous compounding

66. 66 Equivalent Interest Rates (p.77) R m = The annual rate with m compounding periods every year. r c = The annual rate with continuous compounding. Definition: R m and r c are said to be equivalent if:

67. 67 Equivalent Interest Rates (p.77)

68. 68 Equivalent Interest Rates (p.77) The same method applies to any two rates with different periods of compounding. Thus, if we have R m1 and another R m2 then the relationship between the two rates is:

69. 69 Risk-free lending and borrowing Treasury bills: are zero-coupon bonds, or pure discount bonds, issued by the Treasury. A T-bill is a promissory paper which promises its holder the payment of the bond’s Face Value (Par- Value) on a specific future maturity date. The purchase of a T-bill is, therefore, an investment that pays no cash flow between the purchase date and the bill’s maturity. Hence, its current market price is the NPV of the bill’s Face Value: P t = NPV{the T-bill Face-Value} We will only use continuous compounding

70. 70 Risk-free lending and borrowing Risk-Free Asset: is a security whose return is a known constant and it carries no risk. T-bills are risk-free LENDING assets. Investors lend money to the Government by purchasing T-bills (and other Treasury notes and bonds) We will assume that investors also can borrow money at the risk-free rate. I.e., investors may write IOU notes, promising the risk-free rate to their buyers, thereby, raising capital at the risk- free rate.

71. 71 Risk-free lending and borrowing LENDING: By purchasing the risk-free asset, investors lend capital. BORROWING: By selling the risk-free asset, investors borrow capital. Both activities are at the risk-free rate.

72. 72 We are now ready to calculate the current value of a T-Bill. P t = NPV{the T-bill Face-Value}. Thus: the current time, t, T-bill price, P t, which pays FV upon its maturity on date T, is: P t = [FV]e -r(T-t) r is the risk-free rate in the economy.

73. 73 EXAMPLE: Consider a T-bill that promises its holder FV = $1,000 when it matures in 276 days, with a risk-free yield of 5%: Inputs for the formula: FV = $1,000; r =.05; T-t= 276/365yrs P t = [FV]e -r(T-t) P t = [$1,000]e -(.05)276/365 P t = $962.90.

74. 74 EXAMPLE: Calculate the yield-to -maturity of a bond which sells for $965 and matures in 100 days, with FV = $1,000. P t = $965; FV = $1,000; T-t= 100/365yrs. Solving for r: P t = [FV]e -r(T-t)

75. 75 SHORT SELLING STOCKS (p. 97) An Investor may call a broker and ask to “sell a particular stock short.” This means that the investor does not own shares of the stock, but wishes to sell it anyway. The investor speculates that the stock’s share price will fall and money will be made upon buying the shares back at a lower price. Alas, the investor does not own shares of the stock. The broker will lend the investor shares from the broker’s or a client’s account and sell it in the investor’s name. The investor’s obligation is to hand over the shares some time in the future, or upon the broker’s request.

76. 76 SHORT SELLING STOCKS Other conditions: The proceeds from the short sale cannot be used by the short seller. Instead, they are deposited in an escrow account in the investor’s name until the investor makes good on the promise to bring the shares back. Moreover, the investor must deposit an additional amount of at least 50% of the short sale’s proceeds in the escrow account. This additional amount guarantees that there is enough capital to buy back the borrowed shares and hand them over back to the broker, in case the shares price increases.

77. 77 SHORT SELLING STOCKS There are more details associated with short selling stocks. For example, if the stock pays dividend, the short seller must pay the dividend to the broker. Moreover, the short seller does not gain interest on the amount deposited in the escrow account, etc. We will use stock short sales in many of strategies associated with options trading. In all of these strategies, we will assume that no cash flow occurs from the time the strategy is opened with the stock short sale until the time the strategy terminates and the stock is repurchased.

78. 78 SHORT SELLING STOCKS In terms of cash flows per share: S t is the cash flow/share from selling the stock short thereby, opening a SHORT POSITION on date t. -S T is the cash flow from purchasing the stock back on date T (and delivering it to the lender thereby, closing the SHORT POSITION.)

79. 79 Options Risk-Return Tradeoffs at expiration PROFIT PROFILE OF A STRATEGY: A graph of the profit/loss as a function of all possible market prices of the underlying asset We will begin with profit profiles at the option’s expiration; I.e., an instant before the option expires.

80. 80 Options Risk-Return Tradeoffs at Expiration 1.Only at expiration  T-t = 0 2.No time value! Only intrinsic value! The CALL at Expiration: is exercised if the CALL is in-the money:S T > K and the Cash flow/share = S T – K. expires worthless if the CALL is out-of-the money:S T  K and the Cash flow/share = 0. Algebraically: Cash Flow/share = Max{0, S T – K}

81. 81 Options Risk-Return Tradeoffs at Expiration 1.Only at expiration  T-t = 0 2.No time value! Only intrinsic value! The PUT at Expiration: is exercised if the PUT is in-the money:S T < K and the Cash flow/share = K - S T. expires worthless if the PUT is out-of-the money:S T > K and the Cash flow/share = 0. Algebraically: Cash Flow/share = Max{0, S T – K}

82. 82 3.All parts of the strategy remain open till the option’s expiration. 4.All parts of the strategy are closed out at option’s expiration. 5.A Table Format The analysis of every strategy is done with a table of cash flows. Every row is one part (leg) of the strategy. Every row is analyzed separately. The cash flow of the entire strategy is the vertical sum of the rows.

83. 83 The algebraic expressions of cash flows per share: ICF(t) CF at Expiration(T) Long stock:-S t + S T Short stock: S t - S T Long call:-c t +Max{0, S T -K} Short call: c t - Max{0, S T -K} Long put:-p t +Max{0, K- S T } Short put: p t - Max{0, K - S T } The profit/loss per share is the cash flow at expiration plus the initial cash flow of the strategy, disregarding the time value of money.

84. 84 The algebraic expressions of P/L per share at expiration: P/L per share at Expiration Long stock:-S t + S T Short stock: S t - S T Long call:-c t + Max{0, S T -K} Short call: c t - Max{0, S T -K} Long put:-p t + Max{0, K- S T } Short put: p t - Max{0, K - S T }

85. 85 6.A Graph of the profit/loss profile at expiration The P/L per share from the strategy as a function of all possible prices of the underlying asset at expiration.