note5b_determiningforwardpricesi_pt2

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Winter 2014 Determination of Forward and Futures Prices II Prof. J. Page

FIN 411: Financial Derivatives

Determination of Forward and Futures Prices, Part 2

In the previous reading, we introduced the basic principles that determine the relation-

ship between the forward price and the spot price of the underlying asset. In this reading,

we discuss how the the value of an existing forward contract can be determined, the po-

tential differences between forward and futures prices, and discussed the determination

of futures prices for stock index futures, currency futures, and commodity futures.

1. Valuing Forward Contracts

When a forward contract is created, the forward price is set so that the value of the contract

is zero. However, as prices fluctuate after the contract has been initiated, the value of

the forward contract may become positive or negative. Banks are required to value allcontracts on their trading books each day (this is referred to as marking to market). It is

also important for other types of companies to be able to monitor the value of the contracts

which they are party to. In this section we consider how the value of an existing forward

contract can be determined.

Consider a forward contract that was initiated some time ago, with a delivery date that

isTyears from today. To avoid confusion, we will useKto denote the forward price in the

existing contract, and will use F0 to represent what the current forward price would be if

we were to initiate a new contract today with the same delivery date. Letrbe the current

interest rate for maturity at timeT.

Fact 1. The current value of a long forward contract for delivery at timeT is

V = (F0 K)erT (1)

whereF0 is the current forward price for delivery at timeT, K is the delivery price of the

contract, andris the current risk-free rate of interest for maturity at timeT. The value of

a short forward contract is

V = (K F0)erT

(2)

To see why, compare a long forward contract created today with an existing contract

with delivery priceK. The only difference between these contracts is the price that will be

paid for the asset at time T. The value of the existing contract is the difference between

the current contract price, K, and the forward price that could be secured in a contract

initiated today,F0. Because this difference in cash outflows, F0K, would not be realized

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until timeT, the value of the existing contract today is the present value of the difference

in prices,(F0 K)erT.

A long forward contract on a non-dividend paying stock was entered into sometime ago. It currently has 6 months remaining to expiration. The current 6-month

risk-free rate is 10%, the stock price is $25, and the delivery price is $24. Then

the current 6-month forward price is given by

F0 = 25e0.10.5 =$26.28

and the value of the existing forward contract is

V

= (26.28 24)e0.10.5 =$2.17

Example 1

Note that since the current forward price for an asset with no intermediate income is

F0= S0erT, we can write the value of the forward contract as

V =S0 KerT (3)

In other words, the value of the forward contract to the long party is the difference between

the spot value of the asset and the present value of the forward price.

Similarly, we can write the value of a long forward contract on an asset that provides

intermediate income as

V =S0 KerT (4)

Where is the present value of the income paid out by the asset during the life of the

forward contract. If the income generated by the asset is best expressed as a yield, the

value of the forward contract can be written as

V=S0eqT KerT (5)

whereq is the yield or rate of income generated by the asset.

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2. Futures Prices

2.1. Are Futures Prices and Forward Prices Equal?

Because futures contracts are settled daily, the timing of cash flows is different than for

an otherwise equivalent forward contract, even though the cumulative gains or losses on

both contracts is the same. Because of this difference in timing, the futures price can be

different from the forward price if interest rates fluctuate randomly.

Suppose, for example, that interest rates change randomly but are positively corre-

lated, so that on average interest rates go up when the futures price increases. In this

scenario, the margin balance would grow when interest rates are higher and shrink when

interest rates are lower, so that on average, a long futures position would outperform a

long forward contract. Conversely, if interest rate were random and negatively correlated

with futures prices, a long futures contract would perform worse than the correspondingforward contract on average.

This suggests that when changes in interest rates are positively correlated with changes

in futures prices, the futures price would exceed the forward price for an otherwise equiv-

alent contract. This is because an investor would be willing to agree to a higher futures

price in order to take advantage of marking-to-market. Similarly, when changes in interest

rates are negatively correlated with futures price changes, so that marking-to-market be-

comes a disadvantage, the futures price would be lower than the corresponding forward

price to compensate.

In practice, there tends to be very little difference between futures prices and the

theoretical forward price. For many types of futures contracts, there is little meaningful

correlation between interest rate changes and futures price changes, and for shorter-

lived contracts the effect would be negligible anyway. The difference can be significant for

longer-lived interest rate futures, where there is sure to be correlation between interest

rate and futures price changes. For our purposes, we will ignore the potential differences

and assume that futures prices should equal the theoretical forward price.

2.2. Stock Index futures Prices

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 that comprise the index, and the

dividends are the aggregate dividends that paid by the stocks in the index. Because stock

a index is generally composed of many stocks that pay dividends at different times, the

dividend income is usually characterized as a known dividend yield, rather than discrete

dividend payments as with an individual stock. Ifq is the dividend yield on the index, then

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the futures price on the index is

F0= S0erq)T (6)

whereS0 is the current level of the index.

Consider a three-month futures contract on the S&P 500. Suppose that the

stocks underlying the index generate a dividend yield of 1% per annum, the cur-

rent level of the index is 1,300, and the three-month continuously compounded

interest rate is 5%. The futures price of the index should thus be

F0= 1,300e0.050.01)0.25 =$1,313.07

Example 2

In practice, the dividend yield on the portfolio underlying an index varies throughout

the year. For example, a large proportion of the dividends on NYSE stocks are paid in the

first weeks of February, May, August, and November. The dividend yield on an index like

the S&P 500 is therefore higher at these times and lower during other parts of the year

when fewer stocks are paying dividends. The value of q should represent the average

dividend yield during the life of the contract.

2.2.1. Index Arbitrage

If F0 > S0e(rq)T, an arbitrage profit can be made by buying the stocks underlying the

index and shorting futures contracts. IfF0 < S0e(rq)T, a trader can profit by doing the

reverse: shorting the stocks in the index and going long in the futures contract. This is

known as index arbitrage. To effectively do index arbitrage, a trader must be able to

trade both the index futures and the portfolio of stocks underlying the index very quickly

at quoted market prices. For indices involving many stocks, index arbitrage is sometimes

accomplished by trading a relatively small, representative sample of the stocks in the

index (which, although it can work well in practice, is no longer a true arbitrage due to the

risk that the return on the representative sample of stocks deviates significantly from that

of the index). Index arbitrage is usually implemented through program trading, where acomputer system is used to automatically generate trades. Most of the time, the activities

of arbitrageurs ensures that Equation 6 holds, but occasionally market conditions make

effective arbitrage impossible, allowing the futures price to get out of line with the spot

price.

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2.3. Currency Futures and Forwards

From perspective of a US investor, the underlying asset of currency forwards and futures

is one unit of foreign currency. The holder of a foreign currency can earn interest at the

risk-free rate prevailing in the foreign country (e.g., the holder can invest the currency in

a foreign-denominated bond)

Suppose you have 1000 units of a foreign currency and you want to convert it to dollars

at timeT. There are two ways you could do so:

Buy dollars forward at timeT

Buy dollars at the spot rate today, and invest at the US risk free rate until timeT

The outcomes of these two strategies must be equal to preclude arbitrage, hence we can

solve for the forward price:

Fact 2. The forward price of foreign currency, to be exchanged at timeT, is

F0= S0e(rr)T (7)

whereS0 is the spot price in US dollars of one unit of foreign currency, F0 is the forward

or futures price in US dollars of one unit of foreign currency,ris the US risk-free rate, and

ris the foreign risk free rate.

Note that this is identical to our earlier equation for an investment asset with a knownyield, substituting r for. We can think of the foreign currency as an investment asset

that pays a known yield of r.

Suppose the 2-year interest rates in Australia and the US are 5% and 7%, re-

spectively, and that the spot exchange rate between the Australian dollar (AUD)

and the US dollar (USD) is 0.6200 USD per AUD. The 2-year forward exchange

rate should thus be

F0= 0.62e(0.070.05)2 =0.6453

Example 3

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Suppose the 2-year forward rate were too low, say 0.6300. Then an arbitrageur

can:

1. Borrow 1,000 AUD at 5% per annum for 2 years, convert to 620 USD, andinvest the USD at 7%; and

2. Take a long position in a forward contract to buy 1,000e0.052 =

1,105.17 AUD for 1,105.17 0.63 = 696.26 USD.

The 620 USD invested at 7% will grow to 620e0.072 = 713.17. Of this,

696.26 USD will be used to purchase 1,105.17 under the terms of the futures

contracts, which in turn is used to repay the borrowed AUD. These leaves

713.17 696.26 USD of profit for the arbitrageur.

On the other hand, suppose that the 2-year forward rate is 0.6600. In this case,

and arbitrageur can:

1. Borrow 1,000 USD at 7% per annum, convert to 1,000/0.23 =

1,612.90, AUD and invest the AUD at 5%;

2. Enter a short forward contract to sell 1,612.90e0.052 =1,782.53 AUD

for1,782.53 0.66 = 1,176.47 USD.

The proceeds of the sale under the forward contract will be used to repay the

1000e0.072 =1,150.27 owed on the borrowed USD, leaving the arbitrageur

with a profit of1,176.47 1,150.27 = 26.20 USD.

Example 3, continued

2.4. Futures on commodities

For futures contracts on commodities, we must account for the fact that commodities

are physical assets that are costly to store and transport. Furthermore, the distinction

between investment assets and consumption assets becomes important. We will first

examine contracts on commodities that can be considered investment assets, such as

gold and silver, then extend our results to contracts on consumption assets.

2.4.1. Income and Storage Costs

Owners of gold such as central banks earn income from their gold holdings by lending

it to investment banks who need to short gold in order to offset the risks generated by

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the hedging products they create for gold-producers. Gold owners charge interest on the

borrowed gold called the gold lease rate. The same is true of silver. At the same time

however, physical assets like gold and silver have storage costs.

In the absence of storage costs and income, the forward price of a commodity that is

an investment asset is given byF0= S0e

rT

Storage costs can be treated as negative income, so that if U is the present value of

storage costs during the life of the contract, net of any income generated by the asset,

then the forward price should be equal to

F0 = (S0+ U)erT (8)

Consider a 1-year futures contract on an investment asset that provides no in-

come. It costs $2 per unit per year to store the asset, and the cost accrues at the

end of the year. Suppose the spot price is $450 per unit and the risk-free rate is

7% per annum for all maturities. Then

U= 2e0.07 =1.865

and the futures price F0, is given by

F0= (450 +1.865)e0.071

Example 4

If the storage costs (net of income) are proportional to the price of the commodity, then

they can be treated as a negative yield. In this case, we can write the futures price as

F0= S0e(r+)T (9)

2.4.2. Consumption Commodities

Commodities that are consumption assets rather than investment assets usually provide

no income, but can entail significant storage costs. Now consider the arbitrage strategies

that would be used to determine futures prices from spot prices. Suppose F0 > (S0+

U)erT. How would you take advantage of this situation? As usual, an arbitrageur would:

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1. BorrowS0+ U at the risk-free rate and use it to pay for the commodity plus storage

costs; and

2. Short a futures contract on the commodity

The arbitrageur thus locks in a profit of F0 (S0+ U)erT

at timeT. There is no problemwith doing this.

However, suppose thatF0< (S0+ U)erT. Normally an arbitrageur would

1. Sell the commodity, save the storage costs, and invest at the risk-free rate; and

2. Take a long position in the futures contract

This argument cannot be used for a consumption commodity because the holder of the

commodity usually plans to use it in some way. They are reluctant to sell the commodity in

the spot market because they cant use forward contracts in the manufacturing process,

etc. Hence the most we can say for a consumption asset is

F0 (S0+ U)erT

2.4.3. Convenience Yields

For consumption commodities, the futures price is often less than(S0+ U)erT because

ownership of the physical commodity provides benefits that are not obtained by holders

of futures contracts. For example, an oil refiner can use crude oil held in inventory as

an input to the refining process, but can not do the same with a futures contract. Thebenefits from holding the physical commodity are referred to as the convenience yield,

and is defined as the yieldysuch that

F0eyT = (S0+ U)e

rT (10)

If net storage costs are a constant proportion of the spot price, we can write

F0eyT =S0e

(r+)T

or

F0= S0e(r+y)T (11)

3. Cost of Carry

We can summarize the relationship between futures prices and spot prices in terms of

the more general concept of cost of carry. This is equal to any storage cost plus the

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interest paid to finance the asset, less any income earned on the asset during the life of

the contract:

For a non-dividend paying stock or zero coupon bond, the cost of carry isr, because

there are no storage costs and no income provided by the asset;

For an income producing financial asset such as a stock index with dividend yieldq,

the cost of carry is r q;

For currencies, it isr r;

For a commodity with that provides income at rate q and requires storage costs at

rate, it is r q +; and so on.

If we denote the cost of carry as c, we can thus write the futures price in general terms as

F0= S0ecT (12)

or for a consumption commodity with convenience yield y,

F0= S0e(cy)T (13)

4. TL;DR

When forward contracts are initiated, the forward price is set so that the value ofthe contract is zero to both parties. However, the value of the contract can become

positive or negative as prices evolve after the contract is initiated.

The value of an existing forward contract is equal to the difference between the

contract price (K)and the forward price that one could secure today for a contract

with the same delivery date (F0), discounted to the present. The formulas for forward

or futures price, and for the value of existing contracts are summarized below:

Asset Forward/Futures Price

Value of long forward

contract with delivery

priceK

Provides no income S0erT S0 Ke

rT

Provides known income

with present value (S0 )erT S0 Ke

rT

Provides known yieldq S0e(rq)T S0e

qT KerT

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For consumption assets, it is not possible to obtain an exact futures price as a func-

tion of the spot price and other observable variables. This is because of the inher-

ent value that users derive from physically owning the asset. This benefit of physical

ownership prevents us from obtaining an upper bound to the theoretical futures price

using the usual arbitrage arguments. We can, however, back out the implied value ofthis benefit from observed futures prices. This quantity, known as the convenience

yield, is defined as the valueysuch that.

F0eyT = (S0+ U)e

rT

or

F0= S0e(r+y)T

The concept of cost of carry can be a useful way of summarizing the factors thatinfluence the relationship between the forward or futures price and the spot price.

The cost of carry is the cost of financing the underlying asset plus any storage costs,

minus any income provided by the asset. The futures price is greater than the spot

price by an amount reflecting the cost of carry (net of the convenience yield in the

case of consumption assets).

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note5b_determiningforwardpricesi_pt2

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