Determination of Forward and Futures Prices

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1 Determination of Forward and Futures Prices Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012

2 Short selling A popular trading (arbitrage) strategy is the shortselling or shorting. This trade involves selling an asset that is not owned by the seller, and that the seller has borrowed. Short selling is motivated by the belief that an asset s price will decline, enabling it to be bought back at a lower price to make a profit. Short selling is largely prompted by speculation. Short sellers take advantage of any likely arbitrage opportunities. 2

3 Short selling Investor A instructs a broker to short 500 shares who does not own. The broker will borrow the shares from another client, say Investor B, and sell them in the market. Investor A can maintain the short position for as long as desired, provided there are always shares to borrow. At some stage, Investor A will close out the position by purchasing the 500 shares. The 500 shares are replaced in the account of Investor B. 3

4 Short selling Investor A takes a profit if the stock price has declined and a loss if it has risen. If at any time while the contract is open the broker is not able to borrow shares, investor A is forced to close out the position, even if not ready to do so. Investor A must pay dividends and other benefits to the owner of the shares, i.e. to Investor B via the broker. A broker fee is also charged to Investor A for borrowing the shares. 4

5 Short selling Investor A is required to maintain a margin account with the broker. The margin account consists of cash or marketable securities deposited by the Investor with the broker to guarantee that the investor will not walk away from the short position if the share price increases. An initial margin is required and if there are adverse movements (i.e., increases) in the price of the asset that is being shorted, additional margin may be required. If the additional margin is not provided, the short position is closed out. 5

6 Short selling: Example Consider the position of Investor A who shorts 500 shares in April when the price per share is $120. Investor A closes out the position by buying them back in July when the price per share is $100. Suppose that a dividend of $1 per share is paid in May. Investor A receives 500 x $120 = $60,000 in April when the short position is initiated. The dividend leads to a payment by Investor A of 500 x $1 = $500 in May. Investor A also pays 500 x $100 = $50,000 for shares when the position is closed out in July. The net gain is: $60,000 - $500 - $50,000 = $9,500 assuming there is no fee for borrowing the shares. 6

7 Short selling Massive short selling is a practice that is often observed after the burst of a price bubble. Examples are the Dutch Tulip-mania in the seventeenth century, the U.S. stock price crash in 1929, the NASDAQ price bubble of , etc. Since the practice of short selling is alleged to magnify the decline of asset prices, it has been banned and restricted many times during history. As such, short selling bans have been commonly used as a regulatory measure to stabilise prices during downturns in the economy. 7

8 Short selling The most recent example was in September of 2008 with the prohibition of short selling by the U.S. Securities and Exchange Commission (SEC) for 799 financial companies in an effort to stabilise those companies. At the same time the U.K. Financial Services Authority (FSA) prohibited short selling for 32 financial companies. On September 22, Australia enacted even more extensive measures with a total ban of short selling. 8

9 Pricing Futures and Forward Contracts S 0 : Spot price today F 0 : Futures or forward price today T: Time until delivery date r: Risk-free interest rate for maturity T r is the rate at which money is borrowed or lent when there is no credit risk, so that the money is certain to be repaid. Participants in derivatives markets have traditionally assumed that LIBOR rates are the relevant risk-free rates. 9

10 Pricing Futures and Forward Contracts LIBOR is a reference interest rate; it is designed to reflect the rate of interest at which banks are prepared to make large wholesale deposits with other banks. LIBOR is quoted in all major currencies for maturities up to 12 months: E.g., 1-month LIBOR is the rate at which 1-month deposits are offered. A deposit with a bank can be regarded as a loan to that bank. A bank must satisfy certain creditworthiness criteria in order to be able to receive deposits from another bank at LIBOR. Typically, it must have a AA credit rating. LIBID is the rate which a AA bank is prepared to pay on deposits from another bank 10

11 Pricing Futures and Forward Contracts Arbitrage arguments are used to determine the forward and futures prices of an asset from its spot price and other observable market variables. In other words, it is the trading activities of market participants to take advantage of arbitrage opportunities as they occur that determine the relationship between forward and spot prices. 11

12 Forward price of an asset Suppose an asset that provides the holder with no income: non-dividend-paying stocks, or zero-coupon bonds. Consider a long forward contract to purchase a nondividend-paying stock in 3 months. Assume the current stock price is $40 and the 3-month risk-free interest rate is 5% per annum. Suppose that the forward price is relatively high at $43. An arbitrageur can borrow $40 at the risk-free interest rate of 5% per annum, buy one share, and short a forward contract to sell one share in 3 months. 12

13 Forward price of an asset If the spot price of an investment asset is S 0 and the futures price for a contract deliverable in T years is F 0, then: F 0 = S 0 e rt where r is the T-year risk-free rate of interest. 13

14 Interest Rates Compounding When we compound m times per year at rate R an amount A grows to A(1+R/m) m in one year Compounding frequency Value of $100 in one year at 10% Annual (m=1) Semiannual (m=2) Quarterly (m=4) Monthly (m=12) Weekly (m=52) Daily (m=365)

15 Interest Rates Continuous Compounding 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. 15

16 Forward price of an asset At the end of the 3 months, the arbitrageur delivers the share and receives $43.00 The sum of money required to pay off the loan is: Hence, the arbitrageur locks in a profit of $ $40.50 = $2.50 at the end of the 3-month period. 16

17 Forward price of an asset Suppose that the forward price is at $39.00 An arbitrageur can short one share, invest the money brought in from a transaction of the short sale at 5% per annum for 3 months, and take a long position in a 3-month forward contract. The proceeds of the short sale in 3 months grow to: At the end of the 3 months, the arbitrageur pays $39, takes delivery of the share under the terms of the forward contract, and uses it to close out the short position. A net gain of $ $39.00 = $1.50 is made at the end of the 3 months. 17

18 Forward price of an asset A net gain of $ $39.00 = $1.50 is made at the end of the 3 months. Overall: a) The first arbitrage works when the forward price is greater than $ b) The second arbitrage works when the forward price is less than $ We deduce that for there to be no arbitrage the forward price must be exactly $

19 Forward price of an asset Short sales are not possible for all assets and sometimes is banned from authorities. This does not matter: we can still derive and use the same formula (F 0 = S 0 e rt ). All that we require is that there be a significant number of people who hold the asset purely for investment (and by definition this is always true of an investment asset). If the forward price is too low, they will find it attractive to sell the asset and take a long position in a forward contract. 19

20 A Known Income Consider a forward contract on an asset that will provide a perfectly predictable cash income to the holder. Examples are stocks paying known dividends and coupon-bearing bonds. We adopt the same approach as before and obtain: F 0 = (S 0 I )e rt where I is the present value of the income during life of forward contract. 20

21 A Known Yield Consider the situation where the asset underlying a forward contract provides a known yield. This means that the income is known when expressed as a percentage of the asset s price at the time the income is paid. Suppose that an asset is expected to provide a yield of 5% per annum. This could mean that income is paid once a year and is equal to 5% of the asset price at the time it is paid. The formula is: F 0 = S 0 e (r q )T where q is the average yield during the life of the contract (expressed with continuous compounding). 21

22 Forward Prices and Futures Prices When the short-term risk-free interest rate is constant, the forward price for a contract with a certain delivery date is in theory the same as the futures price for a contract with that delivery date. When interest rates vary, forward and futures prices are in theory no longer the same. We can get a sense of the nature of the relationship by considering the situation where the price of the underlying asset, S, is strongly positively correlated with interest rates. 22

23 Forward Prices and Futures Prices When S increases, an investor who holds a long futures position makes an immediate gain because of the daily settlement procedure. The positive correlation indicates that it is likely that interest rates have also increased. Similarly, when S decreases, the investor will incur an immediate loss. 23

24 Forward Prices and Futures Prices An investor holding a forward contract rather than a futures contract is not affected in this way by interest rate movements. A long futures contract will be slightly more attractive than a similar long forward contract. Hence, when S is strongly positively correlated with interest rates, futures prices will tend to be slightly higher than forward prices. When S is strongly negatively correlated with interest rates, a similar argument shows that forward prices will tend to be slightly higher than futures prices. 24

25 Futures Prices of Stock Indices We have introduced futures on stock indices and showed how a stock index futures contract is a useful tool in managing equity portfolios. We now consider how index futures prices are determined. A stock index can usually be regarded as the price of an investment asset that pays dividends. The investment asset is the portfolio of stocks underlying the index, and the dividends paid by the investment asset are the dividends that would be received by the holder of this portfolio. 25

26 Futures Prices of Stock Indices It is usually assumed that the dividends provide a known yield rather than a known cash income. The following formula gives the futures price of stock indices: F 0 = S 0 e (r q )T where q is the average dividend yield on the portfolio represented by the index during life of contract. 26

27 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. To do index arbitrage, a trader must be able to trade both the index futures contract and the portfolio of stocks underlying the index very quickly at the prices quoted in the market. 27

28 Forward and Futures Prices: Summary 28

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