Solutions to Homework 4

FM 5021 Mathematical Theory Applied to Finance

4.15 Use the rates in Problem 4.14 (given in the table below) to value anFRA where you will pay 5% for the third year on $1 million.

Maturity (years) Rate (% per annum)

1 2.02 3.03 3.74 4.25 4.5

The forward interest rate for the third year with continuous compounding is

0.037(3) 0.03(2)3 2 = 0.051, i.e. 5.1%,

which measured with annual compounding is equal to

e0.051 1 = 0.05232 or 5.232%.Since the 3-year interest rate is 3.7% with continuous compounding, the value of the FRAis

1, 000, 000 (0.05232 0.05) 1 e0.0373 = $2, 078.85.4.16 A 10-year 8% coupon bond currently sells for $90. A 10-year 4% coupon

bond currently sells for $80. What is the 10-year zero rate? (Hint: Considertaking a long position in two of the 4% coupon bonds and a short position inone of the 8% coupon bonds.)A long position in two 4% coupon bonds and a short position in one 8% coupon bond leadto a cash flow of $90 2 $80 = $70 in year 0 and a cash flow of $200 $100 = $100in year 10. Since the coupons cancel out, $100 in 10 years is equivalent to $70 today.Therefore, the 10-year rate with continuous compounding satisfies

100 = 70e10R.

Hence, R = 110ln 100

70= 0.0357, i.e. 3.57% per annum.

5.2 What is the difference between the forward price and the value of aforward contract?The forward price of an asset is the price at which you would agree to buy or sell the assetat a future time. The value of a forward contract is zero at the time when it is first entered

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into. As time passes, the underlying asset price changes, and the value of the contract maybecome positive or negative.Consider, for example, a forward contract on an investment asset with price S0 that providesno income. The forward price F0 is given by

F0 = S0erT ,

where r is the risk-free rate, and T is the time to maturity. If K is the delivery price for acontract that was negotiated some time ago, the value of the forward contract f today is

f = (F0 K)erT = S0 KerT .5.3 Suppose that you enter into a 6-month forward contract on a non-

dividend-paying stock when the stock price is $30 and the risk-free interestrate (with continuous compounding) is 12% per annum. What is the forwardprice?The forward price is

$30e0.120.5 = $31.86.

5.5 Explain carefully why the futures price of gold can be calculated from thespot price and other observable variables whereas the futures price of coppercannot.Gold is an investment asset. Its futures price as a function of the spot price of gold S0, therisk-free rate r, and time to maturity T is

F0 = S0erT .

If the futures price is relatively high (F0 > S0erT ), investors will buy gold and short fu-

tures contracts. If the futures price is too low (F0 < S0erT ), they will sell or short gold

and long futures contracts.Copper is a consumption asset. If the futures price is too high, buying copper and shortingfutures contracts is profitable. However, because investors do not in general hold the asset,the strategy of selling copper and buying futures is not available to them. Therefore, thereis an upper bound, but not a lower bound, to the futures price.

5.6 Explain carefully the meaning of the terms convenience yield and cost ofcarry. What is the relationship between futures price, spot price, convenienceyield, and cost of carry?Ownership of the physical commodity may provide benefits that are not obtained by holdersof futures contracts. These benefits from owing the physical asset are referred to as theconvenience yield provided by the commodity.The cost of carry is the interest that is paid to finance the asset plus the storage cost lessthe income earned on the asset.The relationship between the futures price (F0), the spot price (S0), the convenience yield(y), and the cost of carry (c) is

F0 = S0e(cy)T ,

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where T is the time to maturity of the futures contract.

5.14 The 2-month interest rates in Switzerland and the United States are,respectively, 3% and 8% per annum with continuous compounding. The spotprice of the Swiss franc is $0.6500. The futures price for a contract deliverablein 2 months is $0.6600. What arbitrage opportunities does this create?The theoretical futures price for a contract deliverable in two months is

0.6500e(0.080.03)212 = $0.6554.

The actual futures price is, therefore, too high. An arbitrageur would buy Swiss francs andshort Swiss francs futures.

5.16 Suppose that F1 and F2 are two futures contracts on the same commod-ity with times to maturity, t1 and t2, where t2 > t1. Prove that

F2 F1er(t2t1),

where r is the interest rate (assumed constant) and there are no storage costs.For the purposes of this problem, assume that a futures contract is the sameas a forward contract.Assume that

F2 > F1er(t2t1).

It is easily seen that this situation gives rise to arbitrage opportunitties. An investor canmake a riskless profit by1. Taking a long position in a futures contract which matures at time t1.2. Taking a short position in a futures contract which matures at time t2.When the first futures contract matures, the asset is purchased for F1 using funds borrowedat rate r. It is then held until time t2 at which point it is exchanged for F2 under the secondcontract. The costs of the funds borrowed and accumulated interest at time t2 is F1e

r(t2t1).A positive profit of

F2 F1er(t2t1)is then realized the time t2. This arbitrage opportunity would not exist for a long time.Therefore,

F2 F1er(t2t1).5.20 Show that equation (5.3) is true by considering an investment in the assetcombined with a short position in a futures contract. Assume that all incomefrom the asset is reinvested in the asset. Use an argument similar to that infootnotes 2 and 4 and explain in detail what an arbitrageur would do if equation(5.3) did not hold.Suppose we buy N units of the asset and the income from the asset is reinvested in theasset. The income from the asset causes the holding of the asset to grow at a continuouslycompounded rate q. By time T the holding has grown to NeqT units of the asset. We,

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therefore, buy N units of the asset at time zero at a cost of S0 per unit, and enter into aforward contract to sell NeqT units for F0 per unit at time T . The cash flows from thisstrategy are

Time 0 : NS0Time T : NF0e

qT

Because there is no uncertainty about these cash flows, the present value of the time Tinflow must equal the time zero outflow when we discount at the risk-free rate.

NS0 = (NF0eqT )erT ,

i.e.F0 = S0e

(rq)T ,

which is exactly equation (5.3).If F0 > S0e

(rq)T , an arbitrageur would borrow money at rate r and buy N units of theasset. At the same time the arbitrageur would enter into a forward contract to sell NeqT

units of the asset at time T . As income is received, it is reinvested in the asset. At time Tthe loan is repaid and the arbitrageur makes a profit of N(F0e

qT S0erT ).If F0 < S0e

(rq)T , an arbitrageur would short N units of the asset investing the proceedsat rate r. At the same time the arbitrageur would enter into a forward contract to buyNeqT units of the asset at time T . When income is paid on the asset, the arbitrageur owesmoney on the short position. The investor meets this obligation from the cash proceeds ofshorting further units. The result is that the number of units shorted grows at rate q toNeqT . The cummulative short position is closed out at time T and the arbitrageur makesa profit of N(S0e

rT F0eqT ).

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