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### Subject 8. Pricing Eurodollar futures, Treasury bond futures, stock index futures, and currency futures

Pricing Eurodollar Futures

A Eurodollar deposit is a dollar deposit in a U.S. or foreign bank outside the US. The settlement price when a Eurodollar futures contract matures is F = 100 - R, where R is the 90-day LIBOR.

• R is annualized relative to a 360-day money market year. Using L0(j) as the LIBOR rate on a j-day Eurodollar time deposit on day 0, if one deposits \$1, the deposit will grow to a value of 1 + L0(j) (j/360) j days later.
• The Eurodollar futures contract is structured like the T-bill contract, so its price is stated in the form of 1 - L0(j) (j/360), which is the present value of \$1 in j days.

Eurodollar futures cannot be priced easily as T-bill futures. Let's use Lh(m) as the rate (LIBOR) on a m-day Eurodollar time deposit on day h. Suppose that on day 0 we buy an (h + m)-day Eurodollar deposit that pays \$1 on day (h + m), and we sell a futures at a price of f0(h). On day h, the futures expiration day:

• The value of the underlying Eurodollar time deposit is based on 1 divided by a rate. In our example, the Eurodollar deposit has m days to go and is worth 1 / [1 + Lh(m) (m/360)].

• The expiration price of a Eurodollar futures is based on a value computed as 1 minus a rate. In our example, the futures price at expiration is fh(h) = 1 - Lh(m) m/360. The profit from the futures is f0(h) - [1 - Lh(m) (m/360)].

• Our total position on day h is {1 / [1 + Lh(m) (m/360)]} + f0(h) - [1 - Lh(m) (m/360)]. Although f0(h) is known when the transaction is initiated, Lh(m) is not determined until the futures expiration. There is no way for the Lh(m) to offset. The difference is small but not zero. Hence, Eurodollar futures do not lend themselves to an exact pricing formula based on the notion of a cost of carry of the underlying.

• This problem does not occur in the T-bill market because the spot price is a discount instrument and the futures contract is designed as a discount instrument.

Pricing Treasury Note and Bond Futures On day 0 we buy a bond at the price B0c(T+Y) and sell the futures at the price f0(T). We hold this position until the futures expiration. During this time we collect coupons and reinvest them. It's traditionally assumed that the interest rate at which these coupons are reinvested is known as r0(T). We also assume that r0(T) applies to the period from 0 to T for money borrowed and lent. At expiration, the underlying bond is delivered and the futures price is received.

The current value of the transaction, B0c(T+Y), should be the discounted value of its value at T of f0(T) + FV (CI, 0, T), where FV (CI, 0, T) is the compound value from 0 to T of any coupons paid during the time the futures contract is alive.

B0c(T+Y) = f0(T) + FV (CI, 0, T) / (1 + r0(T))T

Therefore, the futures price is:

f0(T) = B0c(T+Y) (1 + r0(T))T - FV (CI, 0, T)

This equation is a variation of our basic cost-of-carry formula. The spot price is compounded at the risk-free interest rate. We then subtract the compound future value of the reinvested coupons over the life of the contract. The coupon interest is like a negative cost of carry: it is a positive cash flow associated with holding the underlying bond.

However, now we must complicate the matter a little. Recall that a bond futures contract permits the short to choose which bond to deliver, as the Chicago Board of Trade, the Federal Reserve and the U.S. Treasury do not want a potential run on a single issue that might distort prices caused by holders of short positions scrambling to buy a single deliverable bond at expiration.

• When a given bond is delivered, the long pays the short the futures price times an adjustment term called the conversion factor. It is the price of a \$1 bond with coupon equal to that of the deliverable bond and yield equal to 6%, with calculations based on semi-annual compounding.

• Each deliverable bond has its own conversion factor. Bonds with a coupon greater (less) than 6% will have a conversion factor greater (less) than 1.

• When making the delivery decision, the short compares the cost of buying a given bond on the open market with the amount she would receive upon delivery of that bond. The bond that is the most attractive is referred to as the cheapest-to-deliver bond. The process of determining this bond is not required for Level II candidates.

Suppose the cheapest-to-deliver bond has been identified, the futures price is:

f0(T) = [B0c(T+Y) (1 + r0(T))T - FV (CI, 0, T)] / CF(T)

where CF(T) is the conversion factor.

Example 1

A \$1 face value bond pays a 10% semi-annual coupon. The annual yield is 12%. The bond has 20 years remaining until maturity, and its price is \$1.1592. Consider a futures contract calling for delivery of this bond only. The contract expires in 18 months. The risk-free rate is 6%. What is the appropriate futures price?

In this example:

• T = 1.5
• r0(T) = 0.06.
• FV(CI, 0, T) = 0.05 x 1.061 + 0.05 x 1.060.5 + 0.05 = 0.1545. Note that the first (second) coupon is paid in exactly 6 (12) months and reinvested for the one year (6 months) remaining until expiration, and the third coupon is paid in exactly one and a half years and not reinvested.
• As the underlying bond is the only deliverable bond in this example, the conversion factor is 1.0.
• f0(1.5) = 1.1592 x 1.061.5 - 0.1545 = 1.1106.

Now suppose the bond is one of many deliverable bonds. The contract specification calls for the use of a conversion factor to determine the price paid for a given deliverable bond. Suppose the bond described here has a conversion factor of 1.0416, the appropriate futures price is then 1.1106/1.0416 = 1.0662.

Pricing Stock Index Futures

Pricing stock index futures is virtually identical to that for bond futures. The cost-of-carry formula for stock index futures is:

f0(T) = S0 x (1 + r)T - FV(D, 0, T)

where:

• S0 is the currency value of the stock index.
• r is the risk-free interest rate.
• FV(D, 0, T) is the compound value over the period of 0 to T of all dividends collected and reinvested.

Some variations of this formula are:

• f0(T) = [S0 - PV(D, 0, T)] x (1 + r)T

Here the stock price is reduced by the present value of the dividends. This adjusted stock price is then compounded at the risk-free rate over the life of the futures.

• f0(T) = [S0/(1 + δ)T] x (1 + r)T

The yield δ (delta) is defined in the following manner: 1 / (1 + δ)T = 1 - FV(D, 0, T)/[S0(1 + r)T].

The stock price is discounted at the dividend yield, and this adjusted stock price is then compounded at the risk-free rate over the life of the futures.

• f0(T) = S0 x (1 - δ*) x (1 + r)T

δ* is the yield expressed as δ* = PV(D, 0, T)/S0.

The stock price is reduced by the yield and this adjusted stock price is compounded at the risk-free rate.

• f0(T) = (S0 x e-δ(c) T) x er(c) T

This formula assumes continuous compounding. The continuously compounded dividend yield is δ(c) = (1/T) x ln[(1 + δ)T]. The continuously compounded risk-free rate is defined as r(c) = ln(1 + r).

The expression in parentheses is the stock price discounted at the dividend yield rate. The result is an adjusted stock price with the present value of the dividends removed. This adjusted stock price is then compounded at the risk-free rate.

Each of these formulas is consistent with the general formula for pricing futures.

• They are each based on the notion that a futures price is the spot price compounded at the risk-free rate, plus the compound future value of any other costs minus any cash flows and benefits.
• Alternatively, one can convert the compound future value of the costs net of benefits or cash flows of holding the asset to their current value and subtract this amount from the spot price before compounding the spot price at the interest rate. The adjusted price stock price is then compounded at the risk-free interest rate to give the futures price.
• These costs, benefits, and cash flows thus represent the linkage between spot and futures prices.

Example 2

A stock index is at 1427.25. A futures contract on the index expires in 73 days. The risk-free interest rate is 5.10%. The value of the dividends reinvested over the life of the futures contract is 11.65. Find the appropriate futures price.

T = 73/365 = 0.2.

Approach 1:
The futures price using our primary formula is: f0(T) = f0(0.2) = S0(1 + r)T - FV(D, 0, T) = 1427.25 x (1.051)0.20 - 11.65 = 1429.87.

Approach 2:
We can find the present value of the dividends: PV(D, 0, T) = PV(D, 0, 0,2) = FV(D, 0, T) / (1 + r)T = 11.65/(1.051)0.20 = 11.53.
Then the futures price would be f0(T) = f0(0.2) = [S0 - PV(D, 0, T)] x (1 + r)T = (1427.25 - 11.53) x (1.051)0.20 = 1429.87.

Approach 3:
One specification based on the yield δ is 1/(1 + δ)T = 1 - FV(D, 0, T)/[S0(1 + r)T] = 1 - 11.65/[1427.25 x 1.0510.20] = 0.9919. So (1 + δ)T is 1/0.9919 = 1.0082. Then the futures price is f0(T) = f0(0.20) = [S0/(1 + δ)T] x (1 + r)T = (1427.25/1.0082) x 1.0510.20 = 1429.80.
The difference comes from rounding.

Approach 4:
δ* = PV(D, 0, T)/S0 = 11.53 / 1427.25 = 0.0081.
f0(T) = f0(0.20) = S0 x (1 - δ*) x (1 + r)T = 1427.25 x (1 - 0.0081) x (1 + 0.051)0.2 = 1429.84.

Approach 5:
If we assume continuous compounding of interest and dividends, r(c) = ln(1 + r) = ln(1.051) = 0.049742, and δ(c) = (1/T) x ln[(1 + δ)T] = (1/0.2) x ln(1.0510.2) = 0.0408. The futures price is f0(T) = (S0 x e-δ(c) T) x er(c) T = (1427.25 x e-0.0408 x 0.2) x e0.049742 x 0.2 = 1429.81.

Pricing Currency Futures

A currency futures contract can be viewed like a stock index futures, whereby the dividend yield is analogous to the foreign interest rate. Currency futures prices are determined by buying the underlying currency and selling a futures contract on the currency. The position is held, and the underlying currency pays interest at the foreign risk-free rate. At expiration, the currency is delivered and the futures price is received. The correct futures price is the one that prevents this transaction from earning an arbitrage profit.

Futures price = Spot exchange rate discounted by Foreign interest rate ) compounded at Domestic interest rate:

• Discrete interest: f0(T) = [S0/(1 + rf)T] x (1 + r)T
• Continuous interest: f0(T) = (S0 x e-r(fc) T) x er(c) T

Where:

• rf: foreign risk-free rate.
• r: domestic risk-free rate.
• r(fc): the continuously compounding foreign risk-free interest rate. r(fc) = ln(1 + rf).
• r(c): the continuously compounding domestic risk-free interest rate. r(c) = ln(1 + r).

See the last basic question for an example of using these two formulas.

#### Practice Question 1

Consider a \$1000 face value Treasury bond that pays interest at 7% semi-annually. Each coupon is thus \$35. The bond has exactly ten years remaining so during that time it will pay 20 coupons, each six months apart. The yield on the bond is 8%, and the risk-free rate is 5.5%. Now consider a futures contract that expires in one year and nine months. What should the futures price be?

First find the price of the bond: it is found by calculating the present value of both the 20 coupons and the face value: the price is \$942.11.
T = 1.75.
The accumulated value of the coupons and the interest on them is 35 x 1.055 1.25 + 35 x 1.0550.75 + 35 x 1.055 0.25 = \$109.33.
The futures price is then f0(T) = 942.11 x 1.0551.75 - 109.33 = \$925.68.

#### Practice Question 2

The risk-free interest rate is 4.5%. A stock index is at 892.35, and a futures contract on the index expires in six months. The future value of the dividends reinvested over the life of the futures is 32.32. What should the futures price be?

T = 6/12 = 0.5.
f0(T) = f0(0.5) = S0(1 + r)T - FV(D, 0, T) = 892.35 x (1.045)0.50 - 32.32 = 879.89.

#### Practice Question 3

The spot exchange rate for the Euro dollar is 0.9576. The U.S. interest rate is 6% and the foreign interest rate is 5.25%. A futures contract expires in 92 days. What is the appropriate futures price?

T = 92/365 = 0.2521.
Discrete interest: f0(T) = f0(0.2521) = 0.9576/1.0525 0.2521 x 1.060.2521 = 0.9593.
Continuous interest: r(fc) = ln(1.0525) = 0.05117, and r(c) = ln(1.06) = 0.0583.
The futures price is then f0(0.2521) = (0.9576 x e-0.05117 x 0.2521) x e0.0583 x 0.2521 = 0.9593.

#### Practice Question 4

Currently, the S & P 500 Index is at 856.50 and the annualized risk-free rate is 6.2%. If the annualized dividend on the index is \$13.54, what should be the futures price on the current 6-month stock index future? (Ignore the effects of the timing of dividends.)

A. 864.51.
B. 875.88.
C. 859.54.

Step 1. Calculate the 6-month equivalent dividend rate: 6/12 (13.54) = 6.77.

Step 2. Compute futures price. F0 = S0 [ 1 + RF](t/360) - Dividend = 856.50 [ 1 + (0.062)](180/360) - 6.77 = 875.88

#### Practice Question 5

A \$1000 face value bond pays an 8% semi-annual coupon. The annual yield is 6%. The risk-free rate is 4%. The bond has exactly 10 years remaining until maturity. Suppose the bond is one of many deliverable bonds, and its conversion factor is 1.0354. Consider a futures contract calling for delivery of this bond only. The contract expires in 15 months. What is the appropriate futures price?

A. \$1132.4
B. \$1213.992
C. \$1086.45

First find the price of the bond: it is found by calculating the present value of both the 20 coupons and the face value: the price is \$1148.78
T = 1.25.
The accumulated value of the coupons and the interest on them is 40 x 1.04 0.75 + 40 x 1.040.25 = \$81.59
The futures price is then f0(T) = [1148.78 x 1.041.25 - 81.59]/1.0354 = \$1086.45.

#### Practice Question 6

The risk-free interest rate is 4.5%. A stock index is at 892.35, and a futures contract on the index expires in six months. The future value of the dividends reinvested over the life of the futures is 32.32. What should the futures price be under the assumption of continuous compounding of interest and dividends?

A. 879.90.
B. 899.12.
C. 912.18.

T = 6/12 = 0.5.
1/(1 + δ)T = 1 - FV(D, 0, T)/[S0(1 + r)T] = 1 - 32.32/[892.35 x 1.0450.50] = 0.9646. So (1 + δ)T is 1/0.9646 = 1.0367.
r(c) = ln(1 + r) = ln(1.045) = 0.044, and δ(c) = (1/T) x ln[(1 + δ)T] = (1/0.5) x ln(1.0367) = 0.0721.
The futures price is f0(T) = (S0 x e-δ(c) T) x er(c) T = (892.35 x e-0.0721 x 0.5) x e0.044 x 0.5 = 879.90.

It should be noted that any of five approaches to calculate stock index futures price will lead to the same answer.

#### Practice Question 7

The risk-free interest rate is 6.25%. A stock index is at 1421.35, and a futures contract on the index expires in 84 days. At expiration, the value of the dividends on the index is 17.47. The appropriate futures price should be ______ in terms of a specification of the dividend yield.

A. 1423.94.
B. 1431.78.
C. 1429.32.

T = 84/365 = 0.23.
1/(1 + δ)T = 1 - FV(D, 0, T)/[S0(1 + r)T] = 1 - 17.47/[1421.35 x 1.06250.23] = 0.9879. So (1 + δ)T is 1/0.9879 = 1.0122. Then the futures price is f0(T) = f0(0.23) = [S0/(1 + δ)T] x (1 + r)T = (1421.35/1.0122) x 1.06250.23 = 1423.94.

It should be noted that any of five approaches to calculate stock index futures price will lead to the same answer.

#### Practice Question 8

Consider a futures contract expiring in 78 days on the Canadian dollar. The spot exchange rate is \$0.7236. The Canadian interest rate is 4.25% and the U.S. interest rate is 2.6%. What is the appropriate futures price under the assumption of continuous compounding?

A. 0.7341.
B. 0.7211.
C. 0.7198.

T = 78/365 = 0.2137.
r(fc) = ln(1.0425) = 0.04162, and r(c) = ln(1.026) = 0.02567.
The futures price is then f0(0.2137) = (0.7236 x e-0.04162 x 0.2137) x e0.02567 x 0.2137= 0.7211.

#### Practice Question 9

If the actual 6-month forward rate is 125¥/\$ When it really should be 120¥/\$, which of the following transactions would not be part of a trader's strategy in order to earn an arbitrage profit?

A. Buy dollars in the spot market.
B. Invest at the Japanese risk-free rate for the 6-month period.
C. Sell 6-month dollar futures today.