Determination of Forward and Futures Prices. Chapter 5

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1 Determination of Forward and Futures Prices Chapter 5 1

2 Consumption vs Investment Assets Investment assets are assets hed by a significant number of peope purey for investment purposes (Exampes: stocks, bonds, god, siver athough a few peope might aso hod them for industria purposes for exampe). Consumption assets are assets hed primariy for consumption (Exampes: copper, oi, pork beies) and not usuay for investment purposes. Arbitrage arguments wi not work the same way for consumption assets as they do for investment assets. 2

3 Short Seing (Page ) Short seing or shorting invoves seing securities you do not own. Your broker borrows the securities from another cient and ses them in the market in the usua way. 3

4 Short Seing (continued) At some stage you must buy the securities so they can be repaced in the account of the cient. You must pay dividends and other benefits the owner of the securities receives. There may be a sma fee for borrowing the securities. 4

5 Exampe You short 100 shares when the price is $100 and cose out the short position three months ater when the price is $90. During the three months a dividend of $3 per share is paid. What is your profit? What woud be your oss if you had bought 100 shares? 5

6 Exampe The profit from shorting is 100 x [ ] = $700. Note that the dividend payment owered the price of the stock, and so the origina owner of the stock, for whom the dividend was intended, must be compensated for it. If you had bought 100 shares instead, the oss woud have been: 100 x [ ] = $700. The profits/osses are mirror images of one another. 6

7 Notation for Vauing Futures and Forward Contracts S 0 : Spot price today F 0 : Futures or forward price today T: Time unti deivery date (expressed in years) r: Risk-free interest rate for maturity T (aso expressed in years) 7

8 An Arbitrage Opportunity? Suppose that: The spot price of a non-dividend-paying stock is $40. The 3-month forward/futures price is $43. The 3-month US$ interest rate is 5% per annum. Is there an arbitrage opportunity? 8

9 An Arbitrage Opportunity? Yes. Suppose that you borrow $40 for 3 months at the risk-free rate to buy the stock, and you use the money to buy the stock. At the same time, you enter into a short futures contract. In 3 months, you must deiver the stock (since you entered a contract to se) and you coect $43 (the futures price). You repay the oan of $40x(1+0.05) (3/12) = $40.49 You get to keep: $43 - $40.49 = $2.51 with no risk/investment. 9

10 Another Arbitrage Opportunity? Suppose that: The spot price of a non-dividend-paying stock is $40. The 3-month forward price is $39. The 3-month US$ interest rate is 5% per annum. Is there an arbitrage opportunity? 10

11 Another Arbitrage Opportunity? Yes. Suppose that you short the stock, receive $40, and invest the $40 for 3 months at the risk-free rate. At the same time, you enter into a ong futures contract. In 3 months, you must buy the stock (since you entered a contract to buy) and you pay $39 (the futures price). You receive from your investment $40x(1+0.05) (3/12) = $40.49 You get to keep: $ $39 = $1.49 with no risk/investment. 11

12 The Forward/Futures Price What we concude from the ast two exampes is that one can make an arbitrage profit as the difference between the futures prices F 0 and $40x(1+0.05) (3/12). Therefore one can make an arbitrage profit by taking advantage of the difference between F 0 and S 0 (1+r) T. To eiminate the presence of arbitrage, we must therefore have: F 0 = S 0 (1+r) T 12

13 The Forward/Futures Price If the spot price of an investment asset is S 0 and the futures price for a contract deiverabe in T years is F 0, then: F 0 = S 0 (1+r) T where r is the T-year risk-free rate of interest. In our exampes, S 0 =40, T=0.25, and r=0.05 so that the no-arbitrage forward/futures price shoud be: F 0 = 40(1.05) 0.25 =

14 When using continuous compounding F 0 = S 0 e rt This equation reates the forward price and the spot price for any investment asset that provides no income and has no storage costs. If short saes are not possibe when the forward price is ower than the noarbitrage price, the no-arbitrage formua (continuous or reguar compounding) sti works for an investment asset because investors who hod the asset (for investment purposes but not consumption) wi se it, invest the proceeds at the risk-free rate, and buy forward contracts. If short saes are not possibe when the forward price is higher than the no-arbitrage price, however, it s a non-issue since the strategy woud ca for buying the stock (instead of short-seing it). 14

15 When an Investment Asset Provides a Known Doar Income When an investment asset provides income with a present vaue of I during the ife of the forward contract, we can generaize the previous formua as: F 0 = (S 0 I )e rt where I is the present vaue of the income during the ife of the forward contract. 15

16 When an Investment Asset Provides a Known Doar Income: Exampe Consider a 10-month forward contract on a $50 stock, with a continuous riskess rate of 8% per annum, and $0.75 dividends expected after 3 months, 6 months, and 9 months. The present vaue of the dividends, I, is given by: I = 0.75e -0.08x3/ e -0.08x6/ e -0.08x9/12 = The no-arbitrage forward price therefore must be: F 0 = ( ) e 0.08x10/12 = $

17 When an Investment Asset Provides a Known Yied Sometimes the underying asset provides a known yied rather than a known cash income. The yied can be viewed as the income as a percentage of the asset price at the time it is paid. The no-arbitrage formua then becomes: F 0 = S 0 e (r q )T where q is the average yied during the ife of the contract (expressed with continuous compounding) 17

18 When an Investment Asset Provides a Known Yied: Exampe Consider a 6-month forward contract on a $25 asset expected to produce a yied per annum of 3.96%, and a riskfree rate of 10%. The no-arbitrage price thus is: 25e ( )x0.5 = $25.77 Note that if you were not given the continuous yied directy but tod that the income provided woud be equa to 2% of the asset price during a 6- month period, you woud need to convert from discrete to continuous: 2% in 6 months is equivaent to 3.96% per annum for 6 months with continuous compounding, because: = e q(1/2) Backing out q as 2n(1+0.02) we get q =

19 Vauing a Forward Contract A forward contract is worth zero (except for bid-offer spread effects) when it is first negotiated. Banks are required to vaue a the contracts in their trading books each day, as ater the contract may have a positive or negative vaue. Suppose that K is the deivery price and F 0 is the forward price for a contract that woud be negotiated today. The deivery is in T years from today and the continuousy compounded risk-free rate is r. 19

20 or f = (F 0 K)e rt or f = S 0 e qt Ke rt 20 Vauing a Forward Contract By considering the difference between a contract with deivery price K and a contract with deivery price F 0 we can deduce that: The vaue, f, of a ong forward contract is f = (F 0 K)e rt or f = S 0 Ke rt The vaue of a short forward contract is f = (K F 0 )e rt or f = Ke rt S 0 If the asset provides a known income with present vaue I or a known yied q, for a ong forward contract position, we have: f = (F 0 K)e rt or f = S 0 I Ke rt

21 Vauing a Forward Contract: Intuition Suppose that, a whie back, you entered into a ong futures contract to buy an asset for K. Today, the same futures contracts has a price F 0 that happens to be higher than K. The oder futures contract that you entered aows you to ony pay K instead of the higher F 0 at expiration, a vaue or benefit at expiration equa to F 0 -K. Thus the present vaue today of that benefit is: f = (F 0 K)e rt 21

22 Vauing a Forward Contract: Intuition Suppose that, a whie back, you entered into a short futures contract to se an asset for K. Today, the same futures contracts has a price F 0 that happens to be ower than K. The oder futures contract that you entered aows you to receive K instead of the ower F 0 at expiration, a vaue or benefit at expiration equa to K-F 0. Thus the present vaue today of that benefit is: f = (K F 0 )e rt 22

23 Forward vs. Futures Prices When the maturity and asset price are the same, forward and futures prices are usuay assumed to be equa. (Eurodoar futures are an exception, we wi see this ater). When interest rates are uncertain, futures and forward prices are, in theory, sighty different: A strong positive correation between interest rates and the asset price impies the futures price is sighty higher than the forward price. A strong negative correation impies the reverse. For most practica purposes, however, we wi assume that forward and futures prices are the same, equa to F 0. 23

24 Stock Index (Page 116) Can be viewed as an investment asset paying a dividend yied. The futures price and spot price reationship is therefore F 0 = S 0 e (r q )T where q is the dividend yied on the portfoio represented by the index during the ife of the contract. 24

25 Stock Index (continued) For the formua to be true, it is important that the index represent an investabe asset. In other words, changes in the index must correspond to changes in the vaue of a tradabe portfoio. The Nikkei index viewed as a doar number does not represent an investment asset, since Nikkei futures have a doar vaue of 5 times the Nikkei. 25

26 Stock Index: Exampe Consider a 3-month futures contract on the S&P 500 index where the index yieds 1% per annum. The current index vaue is 1,300 and the continuousy compounded riskess rate is 5%. The futures price F 0 must be: 1,300e ( )x0.25 Thus F 0 = $1,

27 Index Arbitrage The arbitrage strategies are the same as when deaing with an individua security. When F 0 > S 0 e (r-q)t an arbitrageur buys the stocks underying the index and ses futures. When F 0 < S 0 e (r-q)t an arbitrageur buys futures and shorts or ses the stocks underying the index. 27

28 Index Arbitrage (continued) Index arbitrage invoves simutaneous trades in the index futures contract and in many different stocks. Very often a computer is used to generate the trades: this is an exampe of program trading, the trading of many stocks simutaneousy with a computer. Occasionay simutaneous trades are not possibe and the theoretica no-arbitrage reationship between F 0 and S 0 does not hod, as on October 19, 1987 ( Back Monday ). 28

29 Futures and Forwards on Currencies (Page ) A foreign currency is anaogous to a security providing a yied because the hoder of the currency can earn interest at the risk-free interest rate prevaiing in the foreign country. The yied is the foreign risk-free interest rate. It foows that if r f is the foreign risk-free interest rate, ( F S e r r f ) T =

30 Expanation of the Reationship Between Spot and Forward 1000 units of foreign currency (time zero) e r f T 1000 units of foreign currency at time T 1000S 0 doars at time zero r f T 1000F 0 e doars at time T 1000S 0 e rt doars at time T 30

31 Arbitrage in Foreign Exchange Markets Suppose that the 2-year interest rates in Austraia and the United States are 5% and 7%. The spot exchange rate is 0.62 USD per AUD. The 2-year forward exchange rate or price shoud be: 0.62e ( )x2 = What is the arbitrage if the forward rate is 0.63? What is the arbitrage if the forward rate is 0.66? 31

32 Arbitrage in Foreign Exchange Markets If the forward rate is 0.63, the forward is underpriced (<0.6453). Take a ong position in the forward contract: an agreement to buy AUD in 2 years at the rate of 0.63 Borrow 1,000 AUD 5% per year, convert the amount immediatey to 1,000x0.62 = 620 USD and 7%. Two years ater, 620 USD grow to 620e 0.07x2 = USD. Must repay the AUD oan: owe 1,000e 0.05x2 = AUD. Use the forward to buy AUD, costing x0.63 or USD and repay the oan baance of AUD. Keep the difference = USD. 32

33 Arbitrage in Foreign Exchange Markets If the forward rate is 0.66, the forward is overpriced (>0.6453). Take a short position in the forward contract: an agreement to se AUD in 2 years at the rate of 0.66 Invest in AUD today by first borrowing 1,000 7% per year and converting the amount immediatey to 1,000/0.62 = 1, AUD. Therefore invest 1, %. Two years ater, 1, AUD grow to 1,612.90e 0.05x2 = 1, AUD. Must repay the USD oan: owe 1,000e 0.07x2 = 1, USD. Use the forward to se 1, AUD, giving you 1,782.53x0.66 or 1, USD and repay the oan baance of 1, USD. Keep the difference 1, , = USD. 33

34 Futures on Commodities that are Investment Assets (God, Siver, ) With commodity futures, one has to take into account possibe storage costs of the commodity (because impementing an arbitrage strategy might impy the buying and thus the storing of the commodity). Storage costs can be viewed as negative income or yied. Letting u be the storage cost per unit time as a percent of the asset vaue, we have: F 0 = S 0 e (r+u )T Aternativey, etting U be the present vaue of the storage costs, we have: F 0 = (S 0 +U )e rt 34

35 Futures on Commodities that are Consumption Assets (Copper, Oi, ) With commodities that are consumption assets, if the futures price is above the no-arbitrage eve, one ses the futures and buys the underying commodity, driving prices back in equiibrium. However, if the futures price is beow the no-arbitrage eve, the arbitrage strategy woud entai buying the futures and seing the underying commodity. But if hed for consumption purposes, it won t be sod. Thus the futures price wi stay beow its no-arbitrage eve. Letting u be the storage cost per unit time as a percent of the asset vaue, we therefore have: F 0 S 0 e (r+u )T Aternativey, etting U be the present vaue of the storage costs, we therefore have: F 0 (S 0 +U )e rt 35

36 Convenience Yied With consumption asset commodities we thus have: F 0 S 0 e (r+u )T Therefore the difference between F 0 and S 0 e (r+u )T can be viewed as refecting the convenience of hoding the physica consumption asset (rather than the futures contract on it). So if we et y be the parameter that wi equate the two: we then must have: F 0 e yt = S 0 e (r+u )T y is referred to as the convenience yied, and we have: F 0 = S 0 e (r+u -y)t 36

37 The Cost of Carry (Page 124) The cost of carry, c, is the interest cost - income earned + storage costs, so c = r q + u For an investment asset F 0 = S 0 e ct For a consumption asset F 0 S 0 e ct The convenience yied on the consumption asset, y, is defined so that F 0 = S 0 e (c y )T 37

38 Futures Prices & Expected Future Spot Prices Suppose k is the expected rate of return required by investors on an asset. We can invest F 0 e r T at the risk-free rate and enter into a ong futures contract to create a cash infow of S T at maturity. This shows that: Fe = E( S ) e or F = E( S ) e rt kt ( r k ) T 0 T 0 T 38

39 Futures Prices & Future Spot Prices No Systematic Risk (β=0) k = r F 0 = E(S T ) Positive Systematic Risk (β>0) k > r F 0 < E(S T ) Negative Systematic Risk (β<0) k < r F 0 > E(S T ) Positive systematic risk: stock indices (Norma Backwardation) Negative systematic risk: god (at east for some periods, Contango) However, the terms backwardation and contango are aso often used to describe whether the futures price is beow or above the spot price S 0. 39

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