1. 0 Determination of Forward and Futures Prices Chapter 3

2. 1 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.

3. 2 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.

4. 3 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.

5. 4 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 lender. 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 derivatives. In terms of cash flows: S t is the cash flow from selling the stock short on date t. -S T is the cash flow from purchasing the back on date T.

6. 5 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 LandingRisk-Free Borrowing And Landing: By purchasing the risk-free asset, investors lend their capital and earn the risk-free rate. By selling the risk-free asset, investors borrow capital at the risk-free rate.

7. 6 The One-Price Law: There exists only one risk-free rate in an efficient economy. Continuous Compounding and Discounting: Calculating the future value of a series of cash flows or, the present value of the cash flows, respectively, in a continuous time framework.

8. 7 Compounded Interest Any principal amount, P, invested at an annual interest rate, r, compounded annually, for T years would grow toA T = P(1 + r) T. If compounded Quarterly: A T = P(1 +r/4) 4T. In general, with m compounding periods every year, the periodic rate becomes r/m and mT is the number of compounding periods. Thus, P grows to: A T = P(1 +r/m) mT.

9. 8 Monthly compounding becomes: A T = P(1 +r/12) 12T and daily compounding yields: A T = P(1 +r/365) 365T Eample: T =10 years; r =12%; P = $100. 1. Simple annual 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) 3,650 = $331.94.

10. 9 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. Observe that continuous time means that the number of compounding periods every year, m, increases without limit. This implies that the length of every compounding time period goes to zero and thus, the periodic interest rate, r/m, becomes smaller and smaller.

11. 10 This reasoning implies that we need to solve:

12. 11 EXAMPLE, continued: First, recall that: example:x e 12 102.59374246 1,0002.71692393 10,0002.71814592 In the limit e = 2.718281828…

13. 12 EXAMPLE, continued: Recall that in our example: T= 10 years and r = 12% and P=$100. Thus, P=$100 invested at an annual rate of 12%. will grow to by the factor: CompoundingFactor Simple3.105848208 Quarterly3.262037792 Monthly3.300386895 Daily3.319462164 Continuously 3.320116923

14. 13

15. 14 Continuous Compounding (Page 43) In the limit as we compound more and more frequently we obtain continuously compounded interest rates. $100 grows to $100e RT when invested at a continuously compounded rate R for time T. $100 received at time T discounts to $100e -RT at time zero when the continuously compounded discount rate is R.

16. 15 Conversion Formulas (Page 44) Define R c : continuously compounded rate R m : same rate with compounding m times per year

17. 16 FUTURES and SPOT PRICES: AN ECONOMICS MODEL of DEMAND and SUPPLY SPECULATORS:WILL OPEN RISKY FUTURES POSITIONS FOR EXPECTED PROFITS. HEDGERS: WILL OPEN FUTURES POSITIONS IN ORDER TO ELIMINATE ALL PRICE RISK. ARBITRAGERS: WILL OPEN SIMULTANEOUS FUTURES AND CASH POSITIONS IN ORDER TO MAKE ARBITRAGE PROFITS.

18. 17 HEDGERS: HEDGERS TAKE FUTURES POSITIONS IN ORDER TO ELIMINATE PRICE RISK. THERE ARE TWO TYPES OF HEDEGES A LONG HEDGE TAKE A LONG FUTURES POSITION IN ORDER TO LOCK IN THE PRICE OF AN ANTICIPATED PURCHASE AT A FUTURE TIME A SHORT HEDGE TAKE A SHORT FUTURES POSITION IN ORDER TO LOCK IN THE SELLING PRICE OF AN ANTICIPATED SALE AT A FUTURE TIME.

19. 18 ARBITRAGE WITH FUTURES: SPOT MARKETFUTURES MARKET Contract to buy the product LONG futures Contract to sell the product SHORT futures

20. 19 Demand for LONG futures positions by long HEDGERS Long hedgers want to hedge all of their risk exposure if the settlement price is less than or equal to the expected future spot price. c b a OdOd 0Quantity of long positions Long hedgers want to hedge a decreasing amount of their risk exposure as the premium of the settlement price over the expected future spot price increases. F t (k) Exp t [S t+k ]

21. 20 Supply of SHORT futures positions by short HEDGERS. Short hedgers want to hedge a decreasing amount of their risk exposure as the discount of the settlement price below the expected future spot price increases. f e d QSQS 0 Quantity of short positions Short hedgers want to hedge all of their risk exposure if the settlement price is greater than or equal to the expected future spot price. F t (k) Exp t [S t + k ]

22. 21 Equilibrium in a futures market with a preponderance of long hedgers. D S D QdQd 0 Quantity of positions F t (k) Exp t [S t + k ] S F t (k) e Supply schedule Demand schedule Premium QSQS

23. 22 Equilibrium in a futures market with a preponderance of short hedgers. S D QdQd 0 Quantity of positions F t (k) Exp t [S t + k ] S F t (k) e Supply schedule Demand schedule Discount D QSQS

24. 23 Demand for long positions in futures contracts by speculators. 0Quantity of long positions F t (k) Exp t [S t + k ] Speculators will not demand any long positions if the settlement price exceeds the expected future spot price. Speculators demand more long positions the greater the discount of the settlement price below the expected future spot price. c b a

25. 24 Supply of short positions in futures contracts by speculators. 0 Quantity of short positions F t (k) Exp t [S t + k ] Speculators supply more short positions the greater the premium of the settlement price over the expected future spot price Speculators will not supply any short positions if the settlement price is below the the expected future spot price f e d

26. 25 Equilibrium in a futures market with speculators and a preponderance of short hedgers. S D Q d Q E Q s 0Quantity of positions F t (k) Exp t [S t + k ] S F t (k) e Increased supply from speculators Discount D Increased demand from speculators

27. 26 Equilibrium in a futures market with speculators and a preponderance of long hedgers. S D 0Quantity of positions F t (k) Exp t [S t + k ] S F t (k) e Increased supply from speculators Premium D QEQE Increased demand from speculators

28. 27 Equilibrium in the spot market 0 Quantity of the asset F t (k); S t F t (k) e Premium QEQE Spot demand Excess supply of the asset when the spot market price is S t } Spot supply Exp t [S t + k ]

29. 28 Equilibrium in the futures market 0 Net quantity of long positions held by hedgers and speculators F t (k) Exp t [S t + k ] F t (k) e Premium Q } Excess demand for long positions by hedgers and speculators when the settlement price is F t (k) e Schedule of excess demand by hedgers and speculators

30. 29 ARBITRAGEIN PERFECT MARKETS CASH -AND-CARRY DATESPOT MARKETFUTURES MARKET NOW1. BORROW CAPITAL.3. SHORT FUTURES. 2. BUY THE ASSET IN THE SPOT MARKET AND CARRY IT TO DELIVERY. DELIVERY 1. REPAY THE LOAN 3. DELIVER THE STORED COMMODITY TO CLOSE THE SHORT FUTURES POSITION

31. 30 ARBITRAGEIN PERFECT MARKETS REVERSE CASH -AND-CARRY DATESPOT MARKETFUTURES MARKET NOW1. SHORT SELL ASSET3. LONG FUTURES 2. INVEST THE PROCEEDS IN GOV. BOND DELIVERY:2. REDEEM THE BOND 3. TAKE DELIVERY ASSET TO CLOSE THE LONG FUTURES POSITION 1. CLOSE THE SPOT SHORT POSITION

32. 31 Notation S0:S0:Spot price today. (Or S t ) F 0,T :Futures or forward price today for delivery at T. ( or F t,T ) T:Time until delivery date r:Risk-free interest rate for delivery date.

33. 32 Gold Example (From Chapter 1) For gold F 0 = S 0 (1 + r ) T (assuming no storage costs) If r is compounded continuously instead of annually F 0 = S 0 e rT PROOF:

34. 33 ARBITRAGEIN PERFECT MARKETS CASH -AND-CARRY DATESPOT MARKETFUTURES MARKET NOW1. BORROW CAPITAL: S 0 3. SHORT FUTURES 2. BUY THE ASSET INF 0,T THE SPOT MARKET AND CARRY IT TO DELIVERY DELIVERY 1. REPAY THE LOAN 3. DELIVER THE STORED COMMODITY TO CLOSE THE SHORT FUTURES POSITION S 0 e rT  F 0,T

35. 34 ARBITRAGEIN PERFECT MARKETS REVERSE CASH -AND-CARRY DATESPOT MARKETFUTURES MARKET NOW1. SHORT SELL ASSET: S 0 3. LONG FUTURES 2. INVEST THE PROCEEDS F 0,T IN GOV. BOND DELIVERY: 2. REDEEM THE BOND 3. TAKE DELIVERY ASSET TO CLOSE THE LONG FUTURES POSITION 1. CLOSE THE SPOT SHORT POSITION S 0 e rT  F 0,T

36. 35 Extension of the Gold Example (Page 46, equation 3.5) For any investment asset that provides no income and has no storage costs F 0 = S 0 e rT

37. 36 When an Investment Asset Provides a Known Dollar Income (page 48, equation 3.6) F 0 = (S 0 – I )e rT where I is the present value of the income

38. 37 When an Investment Asset Provides a Known Yield (Page 49, equation 3.7) F 0 = S 0 e (r–q )T where q is the average yield during the life of the contract (expressed with continuous compounding)

39. 38 Valuing a Forward Contract Page 50 Suppose that K is delivery price in a forward contract, F 0,T is forward price today for delivery at T The value of a long forward contract, ƒ, is ƒ = (F 0,T – K )e –rT Similarly, the value of a short forward contract is (K – F 0,T )e –rT

40. 39 Forward vs Futures Prices Forward and futures prices are usually assumed to be the same. When interest rates are uncertain they are, in theory, slightly different: A strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward price A strong negative correlation implies the reverse

41. 40 Stock Index (Page 52) Can be viewed as an investment asset paying a dividend yield The futures price and spot price relationship is therefore F 0 = S 0 e (r–q )T where q is the dividend yield on the portfolio represented by the index

42. 41 Stock Index (continued) For the formula to be true it is important that the index represent an investment asset In other words, changes in the index must correspond to changes in the value of a tradable portfolio The Nikkei index viewed as a dollar number does not represent an investment asset

43. 42 Index Arbitrage When F 0 >S 0 e (r-q)T, an arbitrageur buys the stocks underlying the index and sells futures. When F 0 <S 0 e (r-q)T, an arbitrageur buys futures and shorts or sells the stocks underlying the index.

44. 43 Index Arbitrage (continued) Index arbitrage involves simultaneous trades in futures and many different stocks Very often a computer is used to generate the trades Occasionally (e.g., on Black Monday) simultaneous trades are not possible and the theoretical no-arbitrage relationship between F 0,T and S 0 does not hold

45. 44 A foreign currency is analogous to a security providing a dividend yield The continuous dividend yield is the foreign risk-free interest rate It follows that if r f is the foreign risk- free interest rate Futures and Forwards on Currencies (Page 55-58)

46. 45 THE INTEREST RATES PARITY Wherever financial flows are unrestricted, exchange rates, the forward rates and the interest rates in any two countries must maintain a NO- ARBITRAGE relationship: Interest Rates Parity.

47. 46 NO ARBITRAGE: CASH-AND-CARRY TIMECASHFUTURES t(1) BORROW $A. r DOM (4) SHORT FOREIGN CURRENCY (2) BUY FOREIGN CURRENCY FORWARD F t,T ($/FC) A/S($/FC) [=AS(FC/$)] AMOUNT: (3) INVEST IN BONDS DENOMINATED IN THE FOREIGN CURRENCY r FOR T(3) REDEEM THE BONDS (4) DELIVER THE CURRENCY TO EARN CLOSE THE SHORT POSITION (1) PAY BACK THE LOAN RECEIVE: IN THE ABSENCE OF ARBITRAGE:

48. 47 NO ARBITRAGE: REVERSE CASH – AND - CARRY TIMECASHFUTURES t(1) BORROW FC A. r FOR (4) LONG FOREIGN CURRENCY (2) BUY DOLLARS FORWARD F t,T ($/FC) AS($/FC) AMOUNT IN DOLLARS: (3) INVEST IN T-BILLS FOR R DOM TREDEEM THE T-BILLS TAKE DELIVERY TO CLOSE EARN THE LONG POSITION PAY BACK THE LOAN RECEIVE IN THE ABSENCE OF ARBITRAGE:

49. 48 FROM THE CASH-AND-CARRY STRATEGY: FROM THE REVERSE CASH-AND-CARRY STRATEGY: THE ONLY WAY THE TWO INEQUALITIES HOLD SIMULTANEOUSLY IS BY BEING AN EQUALITY:

50. 49 ON MAY 25 AN ARBITRAGER OBSERVES THE FOLLOWING MARKET PRICES: S(USD/GBP) = 1.5640 S(GBP/USD) =.6393 F(USD/GBP) = 1.5328 F(GBP/USD) =.6524 R US = 7.85% ; R GB = 12% CASH AND CARRY TIMECASHFUTURES MAY 25(1) BORROW USD100M AT 7. 85%SHORT GBP 68,477,215 FORWARD FOR 209 DAYS FOR DEC. 20, FOR USD1.5328/GBP (2) BUY GBP63,930,000 (3) INVEST THE GBP63,930,000 IN BRITISH BONDS DEC 20RECEIVE GBP68,477,215 DELIVER GBP68,477,215 FOR USD104,961,875.2 REPAY YOUR LOAN: ARBITRAGE PROFIT: USD104,961,875.2 - USD104,597,484.3 = USD364,390.90

51. 50 Futures on Consumption Assets ( Page 59) F 0  S 0 e (r+u )T where u is the storage cost per unit time as a percent of the asset value. Alternatively, F 0  (S 0 +U )e rT where U is the present value of the storage costs.

52. 51 The Cost of Carry (Page 60) The cost of carry, c, is the storage cost plus the interest costs less the income earned. For an investment asset F 0 = S 0 e cT For a consumption asset F 0 = S 0 e cT The convenience yield on the consumption asset, y, is defined so that F 0 = S 0 e (c–y )T

53. 52 ARBITRAGE IN THE REAL WORLD TRANSACTION COSTS DIFFERENT BORROWING AND LENDING RATES MARGINS REQUIREMENTS RESTRICTED SHORT SALES AN USE OF PROCEEDS STORAGE LIMITATIONS *BID - ASK SPREADS **MARKING - TO - MARKET *BID - THE HIGHEST PRICE ANY ONE IS WILLING TO BUY AT NOW ASK - THE LOWEST PRICE ANY ONE IS WILLING TO SELL AT NOW. **MARKING - TO - MARKET: YOU MAY BE FORCED TO CLOSE YOUR POSITION BEFORE ITS MATURITY.

54. 53 FOR THE CASH - AND - CARRY: BORROW AT THE BORROWING RATE: r B BUY SPOT FOR:S ASK SELL FUTURES AT THE BID PRICE:F(BID). PAY TRANSACTION COSTS ON: BORROWING BUYING SPOT SELLING FUTURES PAY CARRYING COST PAY MARGINS

55. 54 THE REVERSE CASH - AND - CARRY SELL SHORT IN THE SPOT FOR:S BID. INVEST THE FACTION OF THE PROCEEDS ALLOWED BY LAW:f;0 ≦ f ≦ 1. LEND MONEY (INVEST) AT THE LENDING RATE:r L LONG FUTURES AT THE ASK PRICE:F(ASK). PAY TRANSACTION COST ON: SHORT SELLING SPOT LENDING BUYING FUTURES PAY MARGIN

56. 55 With these market realities, a new no-arbitrage condition emerges: B L < F < B U As long as the futures price fluctuates between the bounds there is no possibility to make arbitrage profits BUBU BLBL BUBU BLBL time F

57. 56 Example S 0,BID (1 - c)[1 + f(r BID )] < F 0, t < S 0,ASK (1 + c)(1 + r ASK ) c is the % of the price which is a transaction cost. Here, we assume that the futures trades for one price. In order to understand the LHS of the inequality, remember that the rule in the USA is that you may invest only a fraction, f, of the proceeds from a short sale. So, in the reverse cash and carry, the arbitrager sells the asset short at the bid price. Then (1-f)S 0,BID cannot be invested while fS 0,BID (1+r BID ) is invested. Thus, the inequality becomes: F 0,T  (1-f)S 0 + fS 0 (1+r BID ) F 0,T  S 0 (1 + fr BID )

58. 57 EXAMPLE 1. S 0,BID (1 - T)[1 + f(r L )] < F 0, t < S 0,ASK (1 + T)(1 + r B ) S 0, ASK = $20.50 / bbl S 0, BID = $20.25 / bbl r ASK = 12 % r BID = 8 % c = 3 % $20.25(.97)[1+f(.08)]<F 0,t < $20.50(1.03)(1.12) $19.6425 + f($1.57) < F 0,t < $23.6488 DEPENDING ON f, ANY FUTURES PRICE BETWEEN THE TWO LIMITS WILL LEAVE NO ARBITRAGE OPPORTUNITIES. THE CASH-AND-CARRY WILL COST $23.6488/bbl. THE REVERSE CASH-AND-CARRY WILL COST 19.6425 + f(1.62). IF f=0.5 THE LOWER BOUND IS $20.45. IN THE REAL MARKET, f = 1, FOR SOME LARGE ARBITAGE FIRMS AND THEIR LOWER BOUND IS $21.26. THUS, IT IS CLEAR THAT THERE ARE DIFFERENT ARBITRAGE BOUNDS APPLICABLE TO DIFFERENT INVESTORS. THE TIGHTER THE BOUNDS, THE GREATER ARE THE ARBITRAGE OPPORTUNITIES.

59. 58 Example 2.: THE INTEREST RATES PARITY In the real markets, buyers pay the ask price while sellers receive the bid price. Moreover, borrowers pay the ask interest rate while lenders only receive the bid interest rate. Therefore, in the real markets, it is possible for the forward exchange rate to fluctuate within a band of rates without presenting arbitrage opportunities.Only when the market forward exchange rate diverges from this band of rates arbitrage exists.

60. 59 NO ARBITRAGE: CASH - AND - CARRY TIMECASHFUTURES t(1) BORROW $A. r D,ASK (4) SHORT FOREIGN CURRENCY FORWARD (2) BUY FOREIGN CURRENCY A/S ASK ($/FC) F BID ($/FC) (3) INVEST IN BONDS DENOMINATED IN THE FOREIGN CURRENCY r F,BID TREDEEM THE BONDS DELIVER THE CURRENCY TO CLOSE THE SHORT POSITION EARN: PAY BACK THE LOAN RECEIVE: IN THE ABSENCE OF ARBITRAGE:

61. 60 NO ARBITRAGE: REVERSE CASH - AND - CARRY TIMECASHFUTURES t(1) BORROW FCA. r F,ASK (4) LONG FOREIGN CURRENCY FORWARD FOR FASK(USD/FC) (2) EXCHANGE FOR AS BID (USD/FC) (3) INVEST IN T-BILLS FOR r D,BID TREDEEM THE T-BILLS TAKE DELIVERY TO CLOSE THE LONG POSITION EARN RECEIVE in foreign currency, the amount: PAY BACK THE LOAN IN THE ABSENCE OF ARBITRAGE:

62. 61 From Cash and Carry: From reverse cash and Carry Notice that The RHS(1) > RHS(2) Define:RHS(1)  B U RHS(2)  B L (3)And F ASK ($/D) > F BID ($/D) Always!

63. 62 BU BLBU BL F ASK F ASK ($/D) > F BID ($/D). CONCLUSION: Arbitrage exists only if both ask and bid futures prices are above B U, or both are below B L. F BID F($/D) BUBU BLBL

64. 63 A numerical example: Given the following exchange rates: SpotForwardInterest rates S(USD/NZ) F(USD/NZ) r(NZ) r(US) ASK0.4438 0.4480 6.000% 10.8125% BID0.4428 0.4450 5.875% 10.6875% Clearly, F(ask) > F(bid). ( USD0.4480NZ > USD0.4450/NZ) We will now check whether or not there exists an opportunity for arbitrage profits. This will require comparing these forward exchange rates to: B U and B L

65. 64 Inequality (1): 0.4450 < (0.4438)e (0.108125 – 0.05875)/12 = 0.4456 = B U 0.4480 > (0.4428)e (0.106875 – 0.06000)/12 = 0.4445 = B L  No arbitrage.  Lets see the graph Inequality (2):

66. 65 BUBLBUBL Clearly: F ASK ($/FC) > F BID ($/FC). An example of arbitrage:F BID = 0.4465 F ASK = 0.4480 F BID = 0.4450 F BUBU F ASK = 0.4480 BLBL 0.4445 0.4456