Determination of Forward and Futures Prices Chapter 3

1. Determination of Forward and Futures PricesChapter 3

2. 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. 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. 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. 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: St is the cash flow from selling the stock short on date t. -ST is the cash flow from purchasing the back on date T.

6. 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 earn the risk-free rate. By selling the risk-free asset, investors borrow capital at the risk-free rate.

7. 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. Compounded Interest Any principal amount, P, invested at an annual interest rate, r, compounded annually, for T years would grow to AT = P(1 + r)T. If compounded Quarterly: AT = 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: AT = P(1 +r/m)mT.

9. Monthly compounding becomes: AT = P(1 +r/12)12T and daily compounding yields: AT = P(1 +r/365)365T Eample: T =10 years; r =12%; P = $100. 1. Simple annual compounding yields: A10 = $100(1+ .12)10 = $310.58 2. Monthly compounding yields: A10 = $100(1 + .12/12)120  = $330.03 3. Daily compounding yields: A10 = $100(1 + .12/365)3,650 = $331.94.

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

12. EXAMPLE, continued: First, recall that: example: xe 1 2 10 2.59374246 1,000 2.71692393 10,000 2.71814592 In the limit e = 2.718281828…

13. 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 Simple 3.105848208 Quarterly 3.262037792 Monthly 3.300386895 Daily 3.319462164 Continuously 3.320116923

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

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

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

20. 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

21. 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.

22. 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.

23. 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.

24. 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.

25. 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.

26. 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.

27. 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.

28. 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

29. 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

30. 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

31. 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

32. Notation

33. Gold Example (From Chapter 1) • For gold F0 = S0(1 + r )T (assuming no storage costs) • If ris compounded continuously instead of annually F0 = S0erT PROOF:

34. ARBITRAGE IN PERFECT MARKETS CASH -AND-CARRY DATESPOT MARKETFUTURES MARKET NOW 1. BORROW CAPITAL: S0 3. SHORT FUTURES 2. BUY THE ASSET IN F0,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 S0erT F0,T

35. ARBITRAGE IN PERFECT MARKETS REVERSE CASH -AND-CARRY DATESPOT MARKETFUTURES MARKET NOW 1. SHORT SELL ASSET: S0 3. LONG FUTURES 2. INVEST THE PROCEEDS F0,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 S0erT  F0,T

36. Extension of the Gold Example(Page 46, equation 3.5) • For any investment asset that provides no income and has no storage costs F0 = S0erT

37. When an Investment Asset Provides a Known Dollar Income (page 48, equation 3.6) F0 = (S0 – I )erT where Iis the present value of the income

38. When an Investment Asset Provides a Known Yield (Page 49, equation 3.7) F0 = S0e(r–q )T where q is the average yield during the life of the contract (expressed with continuous compounding)

39. Valuing a Forward ContractPage 50 • Suppose that K is delivery price in a forward contract, F0,T is forward price today for delivery at T • The value of a long forward contract, ƒ, is ƒ = (F0,T – K )e–rT • Similarly, the value of a short forward contract is (K – F0,T )e–rT

40. 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. Stock Index (Page 52) • Can be viewed as an investment asset paying a dividend yield • The futures price and spot price relationship is therefore F0 = S0e(r–q )T where q is the dividend yield on the portfolio represented by the index

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

44. 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 F0,T and S0 does not hold

45. Futures and Forwards on Currencies (Page 55-58) • 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

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

49. 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. 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 RUS = 7.85% ; RGB = 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 20 RECEIVE 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