Table of Contents

Forward and futures prices

–  brief review,

–  assets paying discrete and continuous income,

–  forward foreign exchange contracts,

–  assets incurring discrete and continuous cost,

–  cost of carry.

Forward and Futures Prices

Brief Review

We assume market is frictionless, and short sales are permitted without

any restrictions, fees, and other costs.

The market is free of arbitrage opportunities: if two portfolios A and B have the same cash flows and have the same value at some future time T , then they have the same value for all times t < T .

Income paid on short sold assets is paid to the owner of the assets, and

the owner of the assets pays costs associated with short sold assets.

Forward and futures prices agree when interest rates are deterministic.

Investment assets are held purely for investment purposes and do not have any consumption value – stocks, bonds, etc.

Consumption assets may require additional encouragement to entice the owner to sell – timber, copper, grains, etc.

Forward price on investment assets: Ft = St er (T t) .

We now consider the price of forward and futures on an assets that pay

income or incurs costs.

Investment  Assets  Providing  Known Discrete Income

Investment Assets Providing Known Discrete Income

Let St be the price of an investment asset at time t and suppose that it pays a known amount D at time tD , where t tD T .

–  for example, D may be a dividend on a stock or a coupon from a bond.

Let r be the risk free rate and for simplicity assume that it is constant and the same for all maturities.

Income from the asset received at time tD can be invested at the risk free rate r from time tD to T , and accumulates to Der (T tD ) .

Der (T tD)

D

ST

StD

0 t tD T

Recall that in deriving the forward price for assets that do not provide

any income or incur any costs, we considered the following portfolios:

–  A consisting of a long forward contract with delivery price Ft and a cash amount Ft e r(T t)  invested at the risk free rate r over the period from t to T ,

–  B consisting of the underlying asset.

If we try to use the same portfolios in this case:

–  income D at time tD from portfolio B can be invested at the rate r for the period from tD to T .

–  so the value at time T of portfolio B will be VT (B) = ST + Der (T tD) .

To ensure that the value of portfolio A coincides with VT (B), we need to adjust the cash amount so that we have enough to meet the forward obligation and the additional amount Der (T tD )  at time T .

Hence, the required cash deposit at time t must be

Ft e r(T t) + Der (T tD ) e r(T t)     = Ft e r(T t) + De r(tD t) :

Let It = De r(tD t)  be the present value of the income from the asset. Then the appropriate portfolios A and B at time t are:

–  A consisting of one long position in the T-maturity forward contract with delivery price Ft and cash amount Ft e r(T t)  + It invested at the risk free rate until time T , so that Vt (A) = Ft e r(T t)  + It .

–  B consisting of one long position in the underlying asset, so Vt (B) = St .

At time T , both A and B consist of the underlying asset together with cash amount It er (T t)  = Der (T tD ) , and so

VT (A) = ST + Der (T tD )  = VT (B):

Since there are no intermediate net cash flows for both portfolios, the

assumption of absence of arbitrage implies Vt (A) = Vt (B). That is, Ft e r(T t) + It = St , or equivalently

Ft = (St It ) er (T t) :                                  (1)

More generally, if the underlying asset provides multiple payments prior to forward maturity, then applying the same argument shows that (1)  holds if It is set equal to the sum of all the discounted payments up to forward maturity.

~

price decreases approximately by the dividend amount.

~  Since the forward contract matures at time T > tD , the price must be

~

part of its value prior to the forward maturity.

~  The forward price formula (1) deducts from the future value of the

~  The forward price formula (1), in the case of one cash payment prior to

form Ft = (St er (tD t) D) er (T tD ) .

~  This has the following intuitive interpretation:

~  The case when there are two payments D1  and D2  at times t1  and t2

Example 1

Suppose BHP shares are currently trading at $34:50, and the BHP divi- dends and the corresponding risk free rates are:

What is the price of a forward contract on BHP with maturity 6 months and with 2 years?

–  Firstly, consider the 6 month forward contract. Computing the present value of the dividends paid prior to 6 months gives

I0 (0:5) = 0:60e 0:040:25 0:5940:

–  The adjusted asset price is S0 I0 (0:5) 34:50 0:5940 = 33:9060, and applying the forward price formula (1) gives

F0 (0:5) = (S I0 (0:5))e0:0410:5 $34:61:

–  Next, consider the 2 year forward contract. Computing the present value of the dividends paid prior to 2 years gives

I0 (2) = 0:60e 0:040:25  + 0:65e 0:0420:75

+ 0:60e 0:0425 1:25  + 0:65e 0:043 1:75

2:3957:

–  The adjusted asset price is S0 I0 (2) 34:50 2:3957 = $32:1043, and so F0 (2) = (S0 I0 (2))e0:04352 (34:50    2:3957)e0:04352 $35:02:

Example 2

If in Example 1, the 6 month forward price is F = $34:80, then show that there exists an arbitrage opportunity by constructing an appropriate portfolio. Account carefully for all cash flows.

– From the previous example, the no-arbitrage forward price is F0  = $34:61.

–  Since F > F0 , the 6 month forward is overvalued relative to the underlying asset in the market.

–  So we should take a short position in the 6 month forward, which does not involve any cash flow.

– Rearranging (1) gives Ft (St It )er (T t) , and since Ft and St appear with opposite signs, we need to take the opposite position in the underlying asset.

– So we buy the underlying asset with associated cash flow    $34:50.

–  Net initial cash flow is    $34:50, and we can fund it by borrowing $34:50 at the risk free rate for 6 months.

–  However, this ignores the dividend that will be received in 3 months time.

–  The neatest way to deal with the dividend is to split the required loan of $34:50 (to buy the underlying share) into two components:

-  since we will receive $0:60 dividend in 3 months times, we can borrow     0:60e 0:040:25 0:5940 of $34:50 for 3 months, and repay this loan in 3 months time with the dividend we receive.

-  borrow the remaining 34:50 0:5940 = $33:9060 for 6 months.

– This way, we can ensure that the net intermediate cash flows are zero.

~  Maintaining zero net intermediate cash flows is not necessary (we only need

cash flow at maturity is non-negative.

– More precisely, the required portfolio is:

-  short position in the forward with delivery price F = $34:80,

-  long position in the BHP share,

-  borrow 0:60e 0:040:25 $0:5940 at 4:0% for 3 months,

-  borrow the remaining $33:9060 at 4:1% for 6 months.

–  Note that the net initial cash flow for this portfolio is zero, so that it does not cost anything to set up.

–  We now need to consider each time a cash flow occurs.

–  In 3 months time, we receive a dividend of $0:60 from the long position in the BHP share, and use this to repay the 3 month loan.

– The net cash flow at this time is hence zero, and in particular non-negative.

–  Finally, consider the situation at maturity.

-  meet the obligation on the forward contract by delivering the BHP share we bought initially and receive $34:80.

-  this clears our position in the BHP share.

-  repay the 6 month loan which has accumulated to 33:9060e0:0410:5 $34:61:

– The net cash flow at maturity is hence

34:80 34:61 = $0:19

Example 3

If in  Example 1 the 6  month forward  price  is F =  34:40,  then show that there exists an arbitrage opportunity by constructing an appropriate portfolio.

–  In this case, the forward contract is undervalued relative to the underlying BHP shares in the market.

–  So we take a long position in the forward contract, a short position in the BHP share, and deposit $34:50 received on the sale of the BHP share.

–  Again, we need to to split the deposit into two components, and account carefully for the dividend in 3 months time.

–  The required portfolio in this case is:

-  long position in the forward with delivery price $34:40.

-  short position in the underlying BHP share.

-  deposit 0:60e 0:040:25 $0:5940 at 4:0% for 3 months.

-  deposit the remaining $33:9060 at 4:1% for 6 months.

–  Again, the net initial cash flow from this portfolio is zero.

–  In three months time, the first deposit matures and we receive $0:60, which we use pay to the owner of the BHP share we short sold initially.

–  This ensures that the net cash flow at this time is zero.

–  Finally, at maturity:

-  second deposit matures and we receive $34:61.

-  we use $34:40 of this to meet our forward obligation to buy the underlying BHP share.

-  we use this share to close out our short position in the BHP share.

– The net cash flow at maturity is hence

34:61 34:40 = $0:21

which is a profit.

Example 4

Consider a 3-year 6% semiannual coupon bond with face value $100; 000. If the risk free rate is 4:25% for all maturities, then what is the 1-year

forward  price on the  bond?   Assume that the forward contract  matures

just after the second coupon.

– In this case, the bond provides discrete income in the form of coupons.

– Since only the first two coupons are paid prior to maturity,

I0  = 3; 000 e 0:04250:5  + e 0:04251 5; 812:09:

– Next, the bond price can be computed by discounting the cash flows:

6

S0  = 3; 000 X e 0:04250:5i + 100; 000 e 0:04253 104; 750:17:

=1

–  Hence, the forward price of the bond is

F0 (1) = (104; 750:17    5; 812:09) e0:0425 1 $103; 233:58:

~  Note that this could have been obtained by discounting the cash flows with

Let St be the price of an investment asset that pays continuous income at the yield of q – recall that all rates are annualised and continuously compounded unless stated otherwise.

This means that over each time increment t , the income paid by the asset is qSt t .

~

Let n(t) be the number of assets held at time t , and suppose all the income received from the asset is reinvested to buy additional assets.

The change in n(t) over a small time interval t is just the additional number of assets bought, so that

income over t n(t) qSt t

St St

This leads to the differential equation dn(t) = qn(t) dt, with solution n(t) = n(t0 )eq(t t0 ) :

~  Differentiate the above expression for n(t) to verify that it is a solution

So if we begin with one long asset at time t , so that n(t) = 1, then at time T we will have n(T) = eq(T t)  assets.

In constructing the portfolios to determine the forward price, we must

ensure that the forward contract in portfolio A is for the purchase of eq(T t)  assets at time T , and that we invest sufficient funds to buy eq(T t)  assets at the delivery price of Ft per asset.

Hence, the appropriate portfolios A and B at time t are:

–  A consisting of long eq(T t)  forwards with maturity T and price Ft , and cash amount Ft e(q r)(T t)  = Ft eq(T t) e r(T t)  invested until time T .

–  B consisting of one long position in the underlying asset, with continuous income reinvested by buying additional assets at all times.

Then A and B both consist of eq(T t)  assets at time T , so that

VT (A) = ST eq(T t)  = VT (B):

Since there are no net intermediate cash flows associated with either portfolio prior to forward maturity, it follows from the assumption of absence of arbitrage that Vt (A) = Vt (B).

That is, Ft e(q r)(T t)  = St , or equivalently

Ft = St e(r q)(T t) :

At first, it may seem unrealistic to assume that an asset pays income

continuously, but for assets such as stock indices and currencies this is a reasonable approximation.

Example 5

Consider a 6-month forward contract on an investment asset that pays a continuous dividend yield of 5% per annum.  If the current asset price is $20 and the risk free rate is 4%, then what is the forward price?

– In this example S0  = 20, r = 0:04, q = 0:05 and T = 0:5.

– So applying the expression for the forward price gives

F0  = 20 e(0:04   0:05)0:5 $19:90:

Two most common examples of “assets” that are modelled as paying

continuous income are:

–  stock index: stocks in the index pay dividends at different times, and it is common to approximate this by continuous dividend yield q .

–  foreign exchange: underlying asset is one unit of foreign currency, which can be deposited to earn interest at the foreign risk free rate, and so in this case the asset earns continuous income at rate q = rf .

Example 6

Consider a 3-month forward FX contract to buy US$1,000,000. If the spot

AUD/USD exchange rate is 0:74, the 3-month risk free rate in Australia is 1:4% and the corresponding rate in the US is 1:2%, then what value of AUD is required in 3-months?

–  In this case, the underlying asset is one unit of USD, and the spot price of this asset is S0  = 1=0:74 = AU$1:35135.

–  Underlying asset can be invested at the US risk free rate to earn continuous income at the yield q = rf = 1:2%, and so the forward exchange rate is

F0  = S0 e(r rf)(T t) 1:35135 e(0:014   0:012)0:25 1:3520272:

– So AU$1,352,027.20 is required to buy US$1,000,000 in 3 months time.