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Chapter 5Determination of Forward and Futures PricesIn this chapter we examine how forward and futures contracts are priced.

*Short sellingShort selling: the selling an asset that is not owned (.i.e., borrowed from a broker).The short seller is required to maintain a margin account with the broker. Similar to the margin account for futures contracts.In addition, the short seller must pay to the broker any income that would normally be received on the securities. The broker will transfer this income to the account of the client from whom the securities have been borrowed.

*Short sellingSuppose an investor instructs a broker to short 500 IBM shares: The broker will borrow the shares from another client and selling them in the market. The investor can maintain the short position for as long as desired, provided there are always shares for the broker to borrow. At some stage, however, the investor will close out the short position by purchasing 500 IBM shares. The investor takes a profit if the stock price has declined and a loss if it has risen.

*Example 1Consider the position of an investor who shorts 500 shares in April when the price per share is $120 and 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. What is the net gain to the short seller?

*Example 2Suppose you desire to short-sell 400 shares of JKI stock, which has a bid price of $25.12 and an ask price of $25.31. You cover the short position 180 days later when the bid price is$22.87 and the ask price is $23.06. Suppose that there is a 0.3% commission to engage in the short-sale (this is the commission to sell the stock) and a 0.3% commission to close the short sale (this is the commission to buy the stock back). What profit did you earn?

*Consumption vs Investment AssetsWhen considering forward and futures contracts, it is important to distinguish between investment assets and consumption assets. Investment assets are assets held by significant numbers of people purely for investment purposes.Consumption assets are assets held primarily for consumption. The easiest forward contract to price is one written on an investment asset that does not pay any income.

*Investment assets with no Income When the underlying asset pays no income, the price of a forward contract is given as: F0 = S0erT F0 = Forward price todayS0 = Spot price of the underlying asset T = the time until maturity of the contract in years r = continuously compounded risk-free rate for an investment maturing at the delivery date (i.e., in T years)

*Why is F0 = S0erT true?If F0 > S0erT then an arbitrageur can: Borrowing S0 at interest rate r for T years Buying 1 forward contract size worth of the asset for S0 Shorting a forward contract for F0 Delivering the asset at contract maturity and collect F0 Repaying the loan (with interest) of S0erT Investors profit = F0 - S0erTIf F0 < S0erT then an arbitrageur can: Short sell 1 forward contract size of the asset for S0 dollars Invest the cash from the sale S0 at the interest rate r for T years long a forward contract for F0 At contract maturity, collect S0erT from cash investment Then, buy back the asset for F0 and return it to the broker Investor makes S0erT - F0 on the transaction

*Example 1Consider a 4-month forward contract to buy a zero-coupon bond that will mature 1 year from today (the bond still has 8 months at the end of the contract). The current price of the bond is $930. Assume the 4-month risk-free rate of interest, continuously compounded is 6% per annum. What should the forward price be?

*Example 2Consider a long forward contract to purchase a non-dividend-paying stock in 3-months. Assume the current stock price is $40 and the 3-month risk-free interest rate is 5% per annum. What should the forward price be? Suppose the forward price is $43, what arbitrage opportunity exists (if any)?

*Forwards on investments with Known IncomeWe now consider pricing forward contracts with a known fixed income as part of the contract.When an investment asset provides a known dollar income, the forward price is given as, F0 = (S0 I )erT where I is the present value of the income during the life of the forward contract

AgainIf F0 > (S0 I)erT there is an arbitrage opportunity If F0 < (S0 I)erT there is an arbitrage opportunity

*Example 3Consider a long forward contract to purchase a coupon-bearing bond whose current price is $900. Suppose the forward contract matures in 9 months. Assume the coupon payment of $40 is expected after 4 months. Assume that the 4-month and 9-month risk-free continuously compounded interest rate are 3% and 4% per annum, respectively. What should the forward price be? Suppose the forward price is $870, what arbitrage opportunity exists (if any)?

*Example 4Consider a 10 month forward contract on a stock with a price of $50 and expected dividends of $0.75 per share in 3 months, 6 months, and 9 months. Assume the risk-free rate of interest, continuously compounded, is 8% per annum for all maturities. What is the price of the forward contract?

*Forwards on investments with Known YieldWe now consider pricing forward contracts with fixed known yield instead of a known income. In particular the income is expressed as a percent of the assets value at the time the income is paid.Let q be the average yield per annum on an asset during the life of a forward contract with continuous compounding.When an asset provides a known yield rather than a know dollar income, the forward price is, F0 = S0 e(r q)T

*Example 5Consider a 6 month forward contract on an asset that is expected to provide an income equal to 2% of the asset price once during a 6-month period. The asset price is $25.The risk-free rate of interest with continuous compounding is 10% per annum. What is the price of the forward contract?

*Valuing Forward ContractsIt is important to note the difference between the forward price today (denoted F0) and the value of the forward contract today (denoted f ).Forward contracts initially have a value of zero (f = 0), because the delivery price in the contract (denoted K) is usually set to equal the forward price (denoted F0).However, once the deal is negotiated both the value of the forward price and the value of the forward contract will change over time.Banks are required to value all the contracts in their trading books each day.

*Valuing Forward ContractsSupposeK = the delivery price in a forward contractF0 = the forward price that would apply to the contract todayThe value of a long forward contract is f = (F0 K)erTThe value of a short forward contract is f = (K F0)erT With a little algebra, it can be shown that for a long position: Value of forward contracts with no income: f = S0 KerTValue of forward contracts with known income: f = S0 I KerTValue of forward contracts with known yield: f = S0eqT KerT

*Example 6A long forward contract on a non-dividend-paying stock was entered in the past. It currently has 6 months to maturity. The risk-free rate of interest with continuous compounding is 10% per annum, the stock price is $25, and the delivery price is $24. What is the price of the forward contract?

*Pricing futures contractsThe pricing technique of a futures contract is usually the same as that for a forward contract, with the exception of Eurodollar futures contracts.However, when interest rates are uncertain futures and forward are priced slightly different.A positive correlation between interest rates and the asset price implies that the futures price is slightly higher than the forward priceA negative correlation implies the reverse!

*Futures prices of Stock IndicesA stock index can be regarded as the price of an investment asset that pays dividends.The investment asset is the stock portfolio underlying the index.The dividend paid by this investment asset reflects the dividends that would be received by the holder of the portfolio of stocks underlying the index.It is usually assumed that the stock index pays a known dividend yielddenoted qTherefore, a stock index futures price is F0 = S0 e(r q)T

*Example 7Consider a 3 month futures contracts on the S&P 500. Suppose that the stocks underlying the index provide a continuously compounded dividend yield of 1% per annum. Assume also that the current value of the index is 1300, and that the continuously compounded risk-free interest rate is 5% per annum. What is the price of a futures contract?

*Example 8The risk free rate is 7% per annum with continuous compounding, and the dividend yield on the stock index is 3.2% per annum. The current value of the index is 150. What is the 6 month futures price?

*Index ArbitrageIn practice the dividend yield on the stock portfolio underlying an index varies week by week throughout the year.So there may be stock index arbitrage opportunities:When F0 > Se(r q)T an arbitrageur buys the stocks underlying the index and sells futures on the indexWhen F0 < Se(r q)T an arbitrageur buys futures on the index and shorts (or sells) the stocks underlying the indexThese strategies are known as Index Arbitrage. The process involves simultaneous trades in futures and many different stocks.

*Example 9Describe an arbitrage opportunity using the quotes given below. Assume that the dividend yield on the S&P 500 index is 2.5% per year (continuously compounded).Quotes: On March 12, 1998, the S&P 500 Index June Futures settlement price was 1080.10. One futures contract is for 250 times the index value. The risk-free rate was 5.07% (continuously compounded), and the number of days to maturity was 98. The spot index value was 1068.47.

*Pricing Currency Forwards and FuturesUnlike other underlying assets, when we buy and store a currency, it earns interest at an appropriate rate, so that one unit of the currency grows to more than one unit over time.In fact, a foreign currency is analogous to a security paying a dividend yield, where the dividend yield is the risk-free interest rate in the foreign country.In this case, the price of a contract is F0 = S0e(r rf )TNote, this is the well-known interest rate parity relationship from international finance.

*Example 10Suppose that the 2-year interest rates in Australia and the United States are 5% and 7%, respectively, (continuously compounded) and the spot exchange between the Australian dollar (AUD) and the US dollar (USD) is 0.6200 USD per AUD. What is the 2-year forward exchange rate? Suppose the forward exchange rate is 0.63, what arbitrage opportunity exists (if any)?

*Futures on CommoditiesWe now move to consider futures contracts on commodities. Some commodities pay income, but almost all of them have storage costs. Storage cost can be treat as negative income. Let U be the present value of the net storage costs, so that the futures price is given asF0 = (S0+U )erT

If the net storage costs is expressed as a percent of the asset value, denoted u, the futures price is F0 = S0 e(r+u )T

*Example 11Consider a 1 year futures contract on an investment asset that provides no income. It costs $2 per unit to store the asset, with the payment being made at the end of the year. Assume that the spot price $600 per unit and the risk-free rate is 5% per annum for all maturities. What is the price of the futures contract? Describe a potential arbitrage strategy.

*Convenience Yield Users of the consumption asset place a different value to the commodity than investors (who use the asset to speculate).As a result, consumption asset have a convenience yield. This measures the extent to which users of the commodity feel that ownership of the physical asset provides benefits that are not obtained by holding the future contract. If the storage cost is known and has a present value, U, the convenience yield, y, is defined so that F0eyT = (S0+U )erT If the cost of storage is proportional to the spot price, u, then the convenience yield, y, is defined so that F0eyT = S0e(r+u)T or F0 = S0e(r+u-y)T

*Convenience Yield The convenience yield reflects the markets expectation concerning the future availability of the commodity. The greater the possibility that a shortage of the commodity may occur in the future, the higher the convenience yield .

The larger the inventories of the commodity, the smaller the convenience yield.

*The Cost of CarryThe storage cost of the underlying asset plus the interest costs of financing it minus the income received from it is called the cost of carry, denoted c.The concept of the cost of carry summarizes all the ideas we have discussed in this chapter. For instance, Non-dividend paying stocks: the cost of carry is r (no storage & no income)Stock index: the cost of carry is r q (income earned at rate q)Currency futures: the cost of carry is r rf (foreign currency earn income at rate rf (foreign risk-free rate))Commodities with income q and storage costs at rate u: the cost of carry is r q + u

*The Cost of CarryThus, with cost of carry the futures price for a contract on an investment asset can be written as: F = S0ecT

For a consumption asset it isF = S0e(c y)Twhere y is the convenience yield on the consumption asset

*The Implied Repo Rate The implied repo rate is the interest rate embedded in futures or forward prices. It is the interest rate that would make observed forward or futures prices be equal to the theoretical prices predicted under the no-arbitrage pricing method. For example, suppose that there is a forward contract on an asset that involves no payouts. Then, the forward and spot prices are related by the expression F0 = S0erT With some basis algebra, implied repo rate given by F0, S0, and T is

*The Implied Repo Rate Similarly, if we consider an asset that has a continuous dividend yield of d, the forward and spot prices are linked via:F = S0e (r - q)T Therefore, the implied repo rate in this case is given by:

Similarly, if the asset has storage costs at rate u, the implied repo rate in this case is given by:

*Example 1Let the underlying asset on a forward contract be a stock on which no dividends are expected over the next three months. Suppose the current spot price of the stock is S0 = $25 and the forward price for delivery in three months is F0 = $26. What is the implied repo rate?

*The Implied Repo Rate and ArbitrageThe implied repo rate represents the rate at which an investor can borrow synthetically by simultaneously going short spot and long forward. Suppose the implied repo rate is r, and you can borrow at a rate rb < r . Then you can create an arbitrage opportunity by borrowing at the rate rb and investing synthetically at the rate r.That is, by borrowing at the rate rb, buying the asset in the spot market, and selling the forward contract.

*The Implied Repo Rate and ArbitrageSuppose the implied repo rate is r and you can lend at a rate rl > r. Then you can create an arbitrage opportunity by synthetically borrowing at the rate r and lending at rl That is, by buying the forward contract, selling the asset in the spot market, and lending at the rate rl. Note that, arbitrage is precluded as long as rl < r < rb. This means that there is an interval of forward (or futures) prices that is consistent with no-arbitrage when borrowing and lending rates differ. when rl = rb, we obtain a unique forward (or futures) price consistent with no-arbitrage.

*Example 2Suppose the current spot price of gold is $330 per oz, and the forward price for delivery in one month is $331.35. Suppose also that the one-month borrowing and lending rates you face are 5% and 4.85%, respectively. Finally, suppose that it costs nothing to store gold. Is there an arbitrage opportunity?

*Example 3Consider a futures contract on a stock index. Suppose that the current index level is 1400, the three-month index futures level is 1425, the dividend yield on the index is 2%, and you can borrow for three months at 8%. Is there an arbitrage opportunity present here?

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