WORKBOOK RAMON RABINOVITCH 1 DERIVATIVES ARE CONTRACTS Two

WORKBOOK RAMON RABINOVITCH 1

DERIVATIVES ARE CONTRACTS Two parties Agreement Underlying security 2

DERIVATIVES FORWARDS FUTURES OPTIONS SWAPS 3

FORWARD MARKET THE MARKET FOR DEFERRED DELIVERY SELLER = SHORT BUYER = LONG THE TWO PARTIES MAKE A CONTRACT THAT DETERMINES THE TRANSACTION TO BE MADE ON A FUTURE DATE DELIVERY AND PAYMENT WILL TAKE PLACE IN THE FUTURE AS SPECIFIED BY THE CONTRACT BETWEEN THE SHORT AND LONG. 4

A FORWARD CONTRACT THE SHORT WILL SELL TO THE LONG 25, 764 AUNCES OF GOLD OF A CERTAIN QUALITY. THE PRICE WILL BE $300/AUNCE. DELIVERY AND PAYMENT WILL TAKE PLACE IN A SPECIFIED PLACE EXACTLY 95 DAYS FROM TODAY. RISKS 5

THE FUTURES MARKET FUTURES CONTRACTS ARE STANDARDIZED FORWARDS TRADED ON ORGANIZED EXCHANGES STANDARDIZED ARE THE COMMODITIES, THE DELIVERY PROCEEDURES, THE DELIVERY TIMES AND PAYMENTS. CLEARINGHOUSE 6

THE MARKET FOR SWAPS A SWAPS IS AN AGREEMENT BETWEEN TWO PARTIES ARRANGING CASH FLOWS DISTRIBUTION BETWEEN THE TWO PARTIES. THE CASH FLOWS ARE BASED ON THE VALUE OF AN UNDERLYING COMMODITY 7

THE MARKET FOR SWAPS FOR THE NEXT 5 YEARS, PARTY B WILL PAY PARTY C THE FIXED RATE OF 7% ON $100, 000, WHILE PARTY C WILL PAY PARTY B THE 6 -MONTHS LIBOR. PAYMENTS WILL TAKE PLACE TWICE A YEAR. ONLY THE NET CASH FLOW WILL EXCHANGE HANDS 8

THE MARKET FOR SWAPS $3, 500, 000 => B C <= 6 -month LIBOR THE UNDERLYING SECURITY IS $100, 000 9

OPTIONS • A contingent claim: The option’s value is contingent upon the value of the underlying asset Two Types of Options • Calls • Puts 10

Option Buyer, holder or long. In exchange for making a payment of money the premium, the owner of an option-the long-has a call-the right, but not the obligation, to buy or a put-the right to sell- a specified quantity of the underlying commodity at the execise price before the option’s expiration date. 11

Option Seller, writer or short. In exchange for receiving the premium, the option’s writer has the obligation to sell the underlying asset (in case of a call) or purchase the underlying asset ( in case of a put) at the predetermined exercise price upon being served with an exercise notice during the life of the option, I. e. , before the option expires. 12

WHY TRADE DERIVATIVES? PRICE RISK UNCERTAINTY VOLATILITY IS THE FUNDAMENTAL REASON FOR TRADING DERIVATIVES 13

PRICE RISK At time zero, the asset’s price at time t is not known. Probability distributio St S 0 0 t 14 time

OPTIONS NOTATIONS: S– the underlying asset’s market price X- the exercise price t – the current date T – the expiration date T-t - time till expiration c, p - European call, put premiums C, P –American call, put premiums 15

Buyer of a call option has the right to buy the underlying at the strike price, X, before the call expires at T. Thus => expects the price of the underlying commodity to increase during the period of the option contract. 16

Seller of a call option must sell the underlying asset for X, if exercised. Thus => expects the price of the underlying asset to remain below or at the exercise price during the option’s life. This way the writer keeps the premium 17

Buyer of a put option has the right to sell the underlying for X before the put expires at T. Thus => expects the market price of the underlying commodity, S, to decrease during the put’s life. 18

Seller of a put must buy the underlying for X if exercised. Thus => expects the S to remain at or above X during the put’s life. This way the put writer keeps the premium. 19

$ X X t T S 20

Types of Options • American Options exercisable any time before expiration • European Options exercisable only on expiration date Asian Options European style on the underlying average price during its life 21

At-the-money S=X In this case the intrinsic value for both calls and puts is zero: S-X = X-S = 0 and the premium consists of the extrinsic value only. 22

In-the-money Calls Puts S>X S<X S-X >0 X-S>0 The Intrinsic value is positive 23

Out-of-the-money Calls S<X S-X<0 Puts S>X X-S>0 The intrinsic value is zero and the premium consists of the extrinsic value only 24

Option Price = Premium Two Components: Intrinsic Value The amount by which an option is in-the-money. Extrinsic (time) Value The amount by which the price of an option exceeds its intrinsic value. 25

Option Market Price Premium = Intrinsic value + extrinsic value Intrinsic value Calls Max{0, S-X) Puts Max{0, X-S) Intrinsic value cannot be negative 26

Option Parameters Underlying Price = S Strike Price = X Time to Expiration =T-t Annual Volatility = s Annual Interest Rate = r Annual Dividend Rate= q 27

Summary CALLS PUTS S X 50 35 50 40 50 50 50 60 Feb Mar May 18 19 21 12 13. 5 16 6. 5 8. 25 12 3 4 9 Feb Mar Apr. 05. 15. 27. 25. 34. 50. 75 1 1. 15 11 12 15 All prices are in $s Expirations: At the rd market close on the 3 Friday of the expiration month 28

INTEL Thursday, September 21, 2000. S = $61. 48 CALLS - LAST PUTS - LAST X oct nov jan apr 40 22 --- 23 --- --- 0. 56 --- 50 12 --- --- 0. 63 --- --- 55 8. 13 --- 11. 5 --- 1. 25 --- 3. 63 --- 60 4. 75 --- 8. 75 --- 2. 88 4 5. 75 --- 65 2. 50 3. 88 5. 75 8. 63 6. 00 6. 63 8. 38 10 70 0. 94 --- 3. 88 --- 9. 25 --- 11. 25 --- 75 0. 31 --- 13. 38 --- 16. 79 80 --- 1. 63 --- 90 --- 0. 81 --- --- 95 --- 0. 44 --- --- 5. 13 --- ----- 29

WHY TRADE OPTIONS ON ORGANIZED EXCHANGES? IN THE OVER-THE-COUNTER MARKET (OTC) INVESTORS ARE EXPOSED TO Credit risk Operational risk Liquidity risk 30

1. CREDIT RISK 2. Does the other party have enough resources to meet its obligation? 31

2. Operational risk: Will the other party deliver the underlying if I exercise my call? Will the other party take delivery of the underlying if I exercise my put ? 32

3. Market liquidity. 4. In case the long wishes to get out of the market, what are the obstacles? In case the short wishes to quit its obligation, what to do? In other words: how can the two parties come out of their respective obligations? 33

THE GUARANTEE The exchanges understood that there will exist no efficient options markets until the above problems are resolved. So they have created the: OPTIONS CLEARING CORPORATION 34

THE OPTION CLEARING CORPORATION PLACE IN THE MARKET EXCHANGE CORPORATION OPTIONS CLEARING CORPORATION CLEARING NONCLEARING MEMBERS MEMEBRS OCC MEMBER CLIENTES BROKERS 35

The Options Clearing Corporation (OCC) gives all the LONGS the absolute guarantee of the completion of its side of the contract: You will always be able to exercise your option!!! 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 36 be no operational default.

The OCC also clears all options trading and maintains the list of all long and short positions. Every long position must be MATCHED with a short position. Hence, the total sum of all options traders positions must be ZERO at all times. The OCC’s absolute guarantee together with the trading list makes the market very liquid. ONE – traders are not afraid to enter the market TWO – traders can quit the market at any point in time by OFFSETTING their original position. 37

OFFSETTING POSITIONS A trader with a LONG position who wishes to get out of the market must open a SHORT position with equal number of the same options on the same underlying asset for the same month of expiration and for the same exercise price. EX: LONG 5, SEP, $125, IBM puts offsets this position by SHORT 5, SEP, $125, IBM puts. The above trades offset each other and zero out the trader’s position with the OCC. This trader is out of the market. The premiums paid and received upon entering the above positions determine 38 the trader’s profit or loss.

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 on the same underlying asset for the same month of expiration and for the same exercise price. EX: SHORT 25, JAN, $75, BA calls offsets this position by LONG 25, JAN, $75, BA calls. The above trades offset each other and zero out the trader’s position with the OCC. This trader is out of the market. The premiums paid and received upon entering the above positions determine 39 the trader’s profit or loss.

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. 40

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. 41

A proof by contradiction: is a method of proving that an assumption, or a set of assumptions, is incorrect by showing that the implication of the assumptions contradicts these very same assumptions. 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. 42

The One-Price Law: There exists only one risk-free rate in an efficient economy. 43

Compounded Interest Any principal amount, P, invested at an annual interest rate, r, compounded annually, for n years would grow to: An = P(1 + r)n. If compounded Quarterly: An = P(1 +r/4)4 n. In general, with m compounding periods every year, the periodic rate becomes r/m and nm is the total compounding periods. Thus, P grows to: An = P(1 +r/m)nm. 44

Monthly compounding becomes: An = P(1 +r/12)n 12 and daily compounding yields: An = P(1 +r/12)n 12. EXAMPLES: n =10 years; r =12%; P = $100 1. Simple compounding yields: A 10 = $100(1+. 12)10 = $310. 58 2. Monthly compounding yields: A 10 = $100(1 +. 12/12)120 = $330. 03 3. Daily compounding yields: A 10 = $100(1 +. 12/365)3650 = $331. 94. 45

In the early 1970 s, 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 in the formula: Observe that continuous time means that the number of compounding periods, m, increases without limit, while the periodic interest rate, r/m, becomes smaller and smaller. 46

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: 47

EXAMPLE, continued: First, we remind you that the number e is defined as: For example: x e 1 2 10 2. 59374246 100 2. 70481382 1, 000 2. 71692393 10, 000 2. 71814592 1, 000 2. 71828046 In the limit e = 2. 71828…. . 48

Recall that in our example: N = 10 years and r = 12% and P=$100. Thus, P=$100 invested at a 12% annual rate, continuously compounded for ten years will grow to: Continuous compounding yields the highest return to the investor: Compounding Factor Simple 3. 105848208 Quarterly 3. 262037792 Monthly 3. 300386895 Daily 3. 319462164 continuously 3. 320116923 49

Continuous Discounting This expression may be rewritten as: 50

EXAMPLE, continued: First, we remind you that the number e is defined as: 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: Therefore, we can write the continuously discounted value of $320. 01 is: 51

PURE ARBITRAGE PROFIT: A PROFIT MADE 1. WITHOUT EQUITY and 2. WITHOUT ANY RISK. 52

Risk-free lending and borrowing Arbitrage: A market situation in which 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. 53

Risk-free lending and borrowing PURE ARBITRAGE PROFIT: A PROFIT MADE 1. WITHOUT EQUITY INVESTMENT and 2. WITHOUT ANY RISK We will assume that the options market is efficient. This assumption implies that one cannot make arbitrage profits in the options markets 54

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: Pt = NPV{the T-bill Face-Value} We will only use continuous discounting 55

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 riskfree rate. I. e. , investors may write IOU notes, promising the risk-free rate to their buyers, thereby, raising capital at the risk-free rate. 56

Risk-free lending and borrowing The One-Price Law: There exists only one risk-free rate in an efficient economy. Proof: If two risk-free rates exist in the market concurrently, all investors will try to borrow at the lower rate and simultaneously try to invest at the higher rate for an immediate arbitrage profit. These activities will increase the lower rate and decrease the higher rate until they coincide to one unique risk-free rate. 57

Risk-free lending and borrowing By purchasing the risk-free asset, investors lend capital. By selling the risk-free asset, investors borrow capital. Both activities are at the risk-free rate. 58

Continuous Discounting: Recall that continuous compounding and discounting use the number e, which in itself is used as the result of “continualizing” the simple compounding formula as follows: 59

EXAMPLE: First, we remind the reader that the number e is defined as: On your own calculator you may try: x e 1 2 10 2. 59374246 100 2. 70481382 1, 000 2. 71692393 10, 000 2. 71814592 1, 000 2. 71828046. In the limit e = 2. 71828…. . 60

This expression may be rewritten as: But first, QUESTION: Given P and r, how long it takes to double our money? - “the 72 rule” Ans. : 2 P = Pert ; t = [ln 2]/r t = 69. 31/r. r = 10% ==> t = 6. 931 yrs. 61

Continuous Discounting This expression may be rewritten as: 62

Again, 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: The continuously discounted value of $332. 01 is: 63

We are now ready to calculate the current value of a T-Bill. Pt = NPV{the T-bill Face-Value}. Thus: the current time, t, T-bill price, Pt , which pays FV upon its maturity on date T, is: Pt = [FV]e-r(T-t) Clearly, r is the risk-free rate in the economy. 64

EXAMPLE: Consider a T-bill that promises its holder FV = $1, 000 when it matures in 276 days, with a yield-to-maturity of 5%: Inputs for the formula: FV = $1, 000 r =. 05 T-t = 276/365 yrs Pt = [FV]e-r(T-t) Pt = [$1, 000]e-(. 05)276/365 Pt = $962. 90. 65

EXAMPLE: The yield-to -maturity of a bond which sells for $945 and matures in 100 days, promising the FV = $1, 000 is: r = ? Pt = $945; FV = $1, 000; T-t= 100 days. Inputs for the formula: FV = $1, 000; Pt= $945; T-t = 100/365. Solving Pt = [FV]e-r(T-t) for r: r = [365/100]ln[$1, 000/$945] r = 10. 324%. 66

SHORT SELLING STOCKS 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. 67

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. 68

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. In terms of cash flows: St is the cash flow from selling the stock short on date t, and -ST is the cash flow from purchasing the back on date T. 69

Options Risk-Return Tradeoffs PROFIT PROFILE OF A STRATEGY A graph of the profit/loss as a function of all possible market values of the underlying asset We will begin with profit profiles at the option’s expiration; I. e. , an instant before the option expires. 70

Options Risk-Return Tradeoffs At Expiration 1. Only at expiry; T-t = 0 2. No time value; T-t = 0 3. At maturity CALL is exercised If S>X expires worthless If S X Cash Flow = Max{0, S – X} PUT is exercised If S<X expires worthless If S X Cash Flow = Max{0, X – S} 71

4. All legs of the strategy remain open till expiry. 5. A Table Format Every row is one leg of the strategy. Every row is analyzed separately. The total strategy is the vertical sum of the rows. The profit is the cash flow at expiry plus the initial cash flows at the of the strategy, disregarding the time value of money 72

6. A Graph of the profit/loss profile The profit/loss from the strategy as a function of all possible prices of the underlying asset at expiration. 73

The algebraic expressions Of profit/loss at expiration: Cash Flows: Long stock: ST – S 0 Short stock: S 0 - ST Long call: -c + Max{0, ST -X} Short call: c + Min{0, X- ST } Long put: -p + Max{0, X- ST} Short put: p + Min{0, ST -X} 74