Determination of Forward and Futures Prices Chapter 5

1. Determination of Forward and Futures PricesChapter 5

2. The participants HEDGERS: OPEN FUTURES POSITIONS IN ORDER TO ELIMINATE SPOT PRICE RISK. SPECULATORS: OPEN RISKY FUTURES POSITIONS FOR EXPECTED PROFITS. ARBITRAGERS: OPEN SIMULTANEOUS FUTURES AND SPOT POSITIONS IN ORDER TO MAKE ARBITRAGE PROFITS.

3. Supply and demand for forwards and futures will determine their market prices. BUT The forwards and futures markets are NOT independent of the spot market. If spot and futures prices do not maintain A SPECIFIC relationship, dictated by economic rationale, then, arbitragers will enter these markets. Their activities will tend to force the prices to realign.

4. A static model of price formation in the futures markets. In the following slides we analyze the Demand and Supply of futures by: Long and Short Hedgers Long and Short Speculators And then, the Arbitrageurs activities in the spot and futures markets.

5. Ft (k) 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. a 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. Expt [St+k] b c 0 Od Quantity of long positions Demand for LONG futures positions by long HEDGERS

6. Ft (k) 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. d 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. e Expt [St + k] f 0 QS Quantity of short positions Supply of SHORT futures positions by short HEDGERS.

7. Ft (k) S Supply schedule D Ft (k)e Premium Expt [St + k] Demand schedule S D 0 QS Qd Quantity of positions Equilibrium in a futures market with a preponderance of long hedgers.

8. Ft (k) S D Supply schedule Expt [St + k] Discount Ft (k)e Demand schedule S D 0 Qd QS Quantity of positions Equilibrium in a futures market with a preponderance of short hedgers.

9. Ft (k) Speculators will not demand any long positions if the settlement price exceeds the expected future spot price. a Speculators demand more long positions the greater the discount of the settlement price below the expected future spot price. Expt [St + k] b c 0 Quantity of long positions Demand for long positions in futures contracts by speculators.

10. Ft (k) Speculators supply more short positions the greater the premium of the settlement price over the expected future spot price d Expt [St + k] e Speculators will not supply any short positions if the settlement price is below the the expected future spot price f 0 Quantity of short positions Supply of short positions in futures contracts by speculators.

11. Ft (k) S D Increased supply from speculators Expt [St + k] Discount Ft (k)e Increased demand from speculators S D 0 Qd QE Qs Quantity of positions Equilibrium in a futures market with speculators and a preponderance of short hedgers.

12. Ft (k) S Increased supply from speculators D Ft (k)e Increased demand from speculators Premium Expt [St + k] S D 0 QE Quantity of positions Equilibrium in a futures market with speculators and a preponderance of long hedgers.

13. Ft (k); St Excess supply of the asset when the spot market price is St Spot supply } Ft (k)e Premium Expt [St + k] Spot demand 0 QE Quantity of the asset Equilibrium in the spot market

14. Ft (k) Schedule of excess demand by hedgers and speculators Expt [St + k] } Premium Ft (k)e Excess demand for long positions by hedgers and speculators when the settlement price is Ft (k)e 0 Q Net quantity of long positions held by hedgers and speculators Equilibrium in the futures market

15. 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.

16. ARBITRAGE WITH FUTURES: Arbitragers trade in both, the futures and the spot markets simultaneously. Then, they wait until delivery time and close their positions in both markets. Note: their profit is guaranteed when open their positions.

17. ARBITRAGE IN PERFECT MARKETS CASH -AND-CARRY DATESPOT MARKETFUTURES MARKET NOW 1. 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

18. ARBITRAGE IN PERFECT MARKETS REVERSE CASH -AND-CARRY DATESPOT MARKETFUTURES MARKET NOW 1. SHORT SELL ASSET 3. 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

19. Notation

20. ARBITRAGE IN PERFECT MARKETS(P103) CASH -AND-CARRY DATESPOT MARKETFUTURES MARKET NOW 1. BORROW CAPITAL: S0 3. SHORT FUTURES t=0 2. BUY THE ASSET IN F0,T THE SPOT MARKET AND CARRY IT TO DELIVERY DELIVERY 1. REPAY THE LOAN 3. DELIVER THE STORED T COMMODITY TO CLOSE THE SHORT FUTURES POSITION S0erT F0,T

21. ARBITRAGE IN PERFECT MARKETS REVERSE CASH -AND-CARRY DATESPOT MARKETFUTURES MARKET NOW 1. SHORT SELL ASSET: S0 3. LONG FUTURES t=0 2. INVEST THE PROCEEDS F0,T IN GOV. BOND DELIVERY: 2. REDEEM THE BOND 3. TAKE DELIVERY T ASSET TO CLOSE THE LONG FUTURES POSITION 1. CLOSE THE SPOT SHORT POSITION S0erT  F0,T

22. Conclusion (p.103):When an Investment Asset Provides NO INCOME and the only carrying cost is the interest F0,T = S0erT

23. When an Investment Asset Provides a Known Dollar Income (p.105) F0,T = (S0 – I )erT where Iis the present value of the income

24. When an Investment Asset Provides a Known annual Yield, q. (P.107) F0,T = S0e(r–q )T where q is the average yield during the life of the contract (expressed with continuous compounding)

25. Valuing a Forward Contract(Page 107) For the sake of comparison: K = Ft,T is the forward price today ,t , for delivery at T. At a later date, j, F0 = Fj,T. ƒj , is the forward value at any time j; t ≤ j ≤ T. Ft,T Fj,T Date: t j T

26. Valuing a Forward Contract(p.108) Again: • Suppose that, Ft,T is forward price today ,t , for delivery at T and Fj,T is the forward price at date j, for delivery at T. • At j, t ≤ j ≤ T, thevalueof a long forward contract, ƒj[L], is fj[L] = (Fj,T – Ft,T )e–r(T-j)

27. Valuing a Forward Contract(p.108) • At j, t ≤ j ≤ T, the value of a short forward contract fj[SH] is fj[SH] = (Ft,T – Fj,T )e–r(T-j)

28. 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

29. Stock Index (P. 111) • Can be viewed as an investment asset paying a dividend yield • The futures price and spot price relationship is therefore F0,T = S0e(r–q )T where q is the dividend yield on the portfolio represented by the index

30. 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

31. Stock Index Arbitrage When F0,T > S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futures. When F0,T < S0e(r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index.

32. Index Arbitrage • 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 F0,T and S0 does not hold

33. Futures and Forwards on Currencies (P113) • 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 rf is the foreign risk-free interest rate

34. The same parameters used in my slides are noted as follows:

35. THE INTEREST RATES PARITY If financial flows are unrestricted, the SPOT and FORWARD exchange rates and the INTEREST rates in any two countries must satisfy the Interest Rates Parity:

36. In the following derivations of the Theoretical Interest Rate Parity and the practical Interest Rate Parity in the real world we denote: DC = The Domestic currency. FC = The Foreign currency. DOM = domestic. FOR = foreign. Q = Amount borrowed domestically. P = Amount borrowed abroad.

37. NO ARBITRAGE: CASH-AND-CARRY TIMECASHFUTURES t (1) BORROW Q. rDOM (4) SHORT FOREIGN CURRENCY (2) BUY FOREIGN CURRENCY FORWARD Ft,T(DC/FC) [Q]/S(DC/FC) = [Q]S(FC/DC)] AMOUNT: (3) INVEST IN BONDS DENOMINATED IN THE FOREIGN CURRENCY rFOR T (3) REDEEM THE BONDS EARN (4) DELIVER THE CURRENCY TO CLOSE THE SHORT POSITION (1) PAY BACK THE LOAN RECEIVE: IN THE ABSENCE OF ARBITRAGE:

38. NO ARBITRAGE: REVERSE CASH – AND - CARRY TIMECASHFUTURES t (1) BORROW [P] . rFOR (4) LONG FOREIGN CURRENCY (2) BUY DOLLARS FORWARD Ft,T(DC/FC) [P]S(DC/FC) AMOUNT IN DOLLARS: (3) INVEST IN T-BILLS FOR RDOM T REDEEM THE T-BILLS EARN TAKE DELIVERY TO CLOSE THE LONG POSITION PAY BACK THE LOAN RECEIVE IN THE ABSENCE OF ARBITRAGE:

39. 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:

40. Example: The six-months rates in the USA and the EC are 4% and 7%, respectively. The current spot exchange rate is: S(USD/EUR) = USD1.49/EUR. The no arbitrage six-months forward rate is: F(USD/EUR) = 1.49e –[.04 - .07](.5) F(USD/EUR) = USD1.5125185/EUR If the Forward market rate is other than the above, arbitrage is possible.

41. 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.

42. FOR THE CASH - AND - CARRY: BORROW AT THE BORROWING RATE: rB BUY SPOT FOR: SASK SELL FUTURES AT THE BID PRICE: F(BID). PAY TRANSACTION COSTS ON: BORROWING BUYING SPOT SELLING FUTURES PAY CARRYING COST PAY MARGINS

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

44. With these market realities, a new no-arbitrage condition emerges: BL< FBID < FASK< BU As long as the futures price fluctuates between the bounds there is no possibility to make arbitrage profits BU BU F BL BL time

45. Example 1: S0,BID (1 - c)[1 + f(rBID )] < F0, T< S0,ASK (1 + c)(1 + rASK) 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 in the USA the rule 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)S0,BID cannot be invested. Only fS0,BID is invested. Thus, the inequality becomes: F0,T  (1-f)(1-c)S0,BID + fS0,BID(1-c)(1+rBID) F0,T  S0,BID(1-c)(1 + frBID)

46. S0,BID(1-c)[1+f(rBID )]< F0,T< S0,ASK(1+c)(1+rASK) S0,ASK = $90.50 / bbl S0,BID = $90.25 / bbl rASK = 12 % rBID = 8 % c = 3 % $90.25(.97)[1+f(.08)]<F0,T< $90.50(1.03)(1.12) $87.5425 + f($7.0034) < F0,T< $104.4008

47. EXAMPLE 1. $87.5425 + f($7.0034) < F0,T< $104.4008 THE CASH-AND-CARRY costs: $104.4008/bbl. THE REVERSE CASH-AND-CARRY costs: 87.5425+ f($7.0034). IF f=0.5 the lower bound of the futures becomes: $91.0042. In the real market, f = 1, for some large arbitrage firms and thus, for these firms the lower bound is: $94.5459.

48. Example 2: THE INTEREST RATES PARITY In the real markets the forward exchange rate fluctuates within a band of rates without presenting arbitrage opportunities.Only when the market forward exchange rate diverges from this band of rates arbitrage exists. Given are: Bid and Ask domestic and foreign spot rates; forward rates and interest rates.

49. NO ARBITRAGE: CASH - AND - CARRY TIMECASHFUTURES t (1) BORROW [Q]. rD,ASK (4) SHORT FOREIGN CURRENCY FORWARD (2) BUY FOREIGN CURRENCY [Q]/SASK(DC/FC) FBID (DC/FC) (3) INVEST IN BONDS DENOMINATED IN THE FOREIGN CURRENCY rF,BID T REDEEM THE BONDS DELIVER THE CURRENCY TO CLOSE THE SHORT POSITION EARN: PAY BACK THE LOAN RECEIVE: IN THE ABSENCE OF ARBITRAGE:

50. NO ARBITRAGE: REVERSE CASH - AND - CARRY TIMECASHFUTURES t (1) BORROW [P] . rF,ASK (4) LONG FOREIGN CURRENCY FORWARD FOR FASK(DC/FC) (2) EXCHANGE FOR [P]SBID (DC/FC) (3) INVEST IN T-BILLS FOR rD,BID T REDEEM THE T-BILLS EARN TAKE DELIVERY TO CLOSE THE LONG POSITION RECEIVE in foreign currency, the amount: PAY BACK THE LOAN IN THE ABSENCE OF ARBITRAGE: