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Alternative derivation of formulaAlternative derivation of formula

FIN 413 – RISK MANAGEMENT

Forward and Futures Prices

Topics to be covered

- Compounding frequency
- Assumptions and notation
- Forward prices
- Futures prices
- Cost of carry
- Delivery options

Suggested questions from Hull

6th edition: #4.4, 4.10, 5.2, 5.5, 5.6, 5.14

5th edition: #4.4, 4.9, 5.2, 5.5, 5.6, 5.14

Compounding frequency

- Interest can be compounded with varying frequencies.
- We will often assume that interest is compounded continuously.
- Two rates of interest are said to be equivalent if for any amount of money invested for any length of time, the two rates lead to identical future values.

A

0

n

Annual compounding- The interest earned on an investment in any one year is reinvested to earn additional interest in succeeding years.
- R ≡ EAR, effective annual rate

FV = A(1+R)n

PV = A(1+R)-n

A

A(1+R)-n

0

n

Compounding m times per year

- The year is divided into m compounding periods. Interest earned in any compounding period is reinvested to earn additional interest in succeeding periods.
- Rm ≡ the annual (or nominal) rate of interest compounded m times per year
- Rm/m ≡ the effective rate of interest for each mth of a year

A

0

n

A

Ae-R∞ n

0

n

Continuous compoundingR∞ ≡the annual rate of interest compounded continuously

FV = lim A(1+Rm/m)mn m→∞ = AeR∞n

PV = lim A(1+Rm/m)-mn m→∞ = Ae-R∞n

Natural log function

Properties:

-∞<ln(x)<∞, for 0<x<∞

ln(x)<0, for 0<x<1

ln(1) = 0

ln(x)>0, for x>1

ln(ax) = ln(a) + ln(x)

ln(a/x) = ln(a) - ln(x)

ln(ax) = xln(a)

ln(ex) = xln(e) = x

Exponential function

Properties:

ex>0, for -∞<x<∞

0<ex<1, for x<0

e0 = 1

ex>1, for x>0

e-x = 1/ex

exey= ex+y

(ex)y = exy

eln(x) = x

Short selling in the spot market

Involves selling securities that you do not own and buying them back later.

When you initiate a short sale, your broker borrows the securities from another client and sells them on your behalf in the spot market. You receive the proceeds of the sale.

Through your broker, you must pay the client any income received on the securities.

At some later stage, you must buy the securities, close your short position, and return the securities to the client from whom you borrowed.

Ignoring the income foregone, short selling yields a profit if the price of the security falls.

Buy

Sell

Example

Suppose you short sell IBM stock for 90 days. The cash flow are:

Note: Short selling is the opposite of buying.

Analysis: forward prices

- Forward contracts are easier to analyze than futures contracts.
- We begin our analysis with them.
- We will consider forward contracts on the following underlying assets:
- Assets that provide no income.
- Assets that provide a known cash income.
- Assets that provide a known yield.
- Commodities
- Later we will consider futures contracts.

Assumptions

There are some market participants (such as large financial institutions) that:

- pay no transactions costs (brokerage fees, bid-ask spreads) when they trade.

- are subject to the same tax rate on all profits.

- can borrow or lend at the risk-free rate of interest.

- exploit arbitrage opportunities as they arise.

Note: The quality of any theory is a direct result of the quality of the underlying assumptions. The assumptions determine the degree to which the theory matches reality.

Notation

T : the time (in years) until the delivery date of a forward contract

S (or S0): the current spot price of the asset underlying a forward contract

K : the delivery price specified in a forward contract

F (or F0): the current forward price

f : the current value of a forward contract to the long

-f : the current value of a forward contract to the short

r : the risk-free interest rate (expressed as an annual, continuously compounded rate) for an investment maturing in T years

Note: In practice, r is set equal to the LIBOR with a maturity of T years.

LIBOR

- LIBOR: London Interbank Offer rate
- The rate at which large international banks are willing to lend to other large international banks for a specified period.
- The rate at which large international banks fund most of their activities.
- A variable interest rate.
- A commercial lending rate, higher than corresponding Treasury rates.

Analysis

- Objective: to derive formulas for F and f.
- We will use arbitrage pricing methods.
- Note: The basis of any arbitrage is to sell what is relatively overvalued and to buy what is relatively undervalued.

Forward contract: UA provides no income

Examples: forward contracts on non-dividend-paying stocks and zero-coupon bonds.

Proposition: F = SerT, in the absence of arbitrage opportunities

Note: F = SerT > S

Forward contract: UA provides no income

Proposition: F = SerT, in the absence of arbitrage opportunities

Proof: Suppose F > SerT.

Arbitrage strategy (to be implemented today):

- Buy one unit of the UA in the spot market by borrowing S dollars for T years at rate r.
- Short a forward contract on one unit if the UA.

At time T:

- Sell the UA for F dollars under the terms of the forward contract.
- Repay the bank SerT dollars.

Arbitrage profit per unit of UA = [F – SerT ] > 0.

S is bid up and F is bid down.

Forward contract: UA provides no income

Suppose F <SerT.

Arbitrage strategy (to be implemented today):

- Go long a forward contract on one unit if the UA.
- Sell or short sell one unit of the UA. This leads to a cash inflow of S dollars. Invest this for T years at rate r.

At time T:

- The proceeds from the sale/short sale have grown to SerT dollars.
- Buy the UA for F dollars under the terms of the forward contract.
- Return the UA to your portfolio or to the client from whom it was borrowed.

Arbitrage profit per unit of UA = [SerT – F ] > 0.

F is bid up and S is bid down.

Thus: F =SerT

Alternative derivation of formula

- Spot transaction
- Price agreed to.
- Price paid/received.
- Item exchanged.
- Prepaid forward contract
- Price agreed to.
- Price paid/received.
- Item exchanged in T years.
- Forward contract
- Price agreed to
- Price paid/received in T years.
- Item exchanged in T years.

Alternative derivation of formula

Underlying asset provides no income:

FP = S

Explanation: With a prepaid forward contract, as compared to a spot transaction, physical exchange of the asset is delayed T years. But since the asset, by assumption, pays no income to the holder, the holder neither receives nor foregoes income due to the delay.

F = FP erT = SerT

Explanation: The forward contract allows the long to delay payment for T years and requires the short to delay receipt. The long can earn interest on the cash that would otherwise have been paid. The short foregoes this interest. The forward price (which is arrived at by multiplying the prepaid forward price, equal to S, by erT) compensates the short for the delay.

Forward contract: UA provides no income

Proposition: f = S – Ke-rT

Proof:

In general: f = (F – K )e-rT

We derived: F = SerT

Thus: f = (SerT – K )e-rT = S – Ke-rT

Also: -f = -(F – K )e-rT= (K – F )e-rT = Ke-rT - S

The value today of the UA in the spot market.

The value today of the price that the long has agreed to pay for the asset in T years.

Forward contract: UA provides no incomeWe derived: f = S – Ke-rT

Thus: f > 0 iff S > Ke-rT

K

0

T

The value today of the price that the short has agreed to receive in T years for the UA.

The value today of the UA in the spot market.

Forward contract: UA provides no incomeWe derived: -f = Ke-rT – S

Thus: -f > 0 iff Ke-rT > S

K

0

T

Example: #5.9, page 121

T = 1 year

S = $40

r = 10%

(a) F = SerT = $40e(0.10×1) = $44.21

f = S – Ke-rT

= $40 – $44.21e-(0.10×1)

= 0

0

1

Example (continued)

(b) T = ½ year

S = $45

r = 10%

F = S erT = $45e(0.10×0.5) = $47.31

f = S – Ke-rT

= $45 – $44.21e-(0.10×0.5)

= $2.95

0

0.5

1

Creating a forward contract synthetically

A security is “created synthetically” by assembling a portfolio of traded assets that replicates the payoff to the security.

A long position in aforward contract can be created synthetically by:

- Buying the UA with borrowed funds.
- Buying a call option and writing a put option.

Creating a forward contract synthetically

Method 1:

Consider a forward contract on a stock with a delivery date in T years. The stock will pay no dividends during the next T years.

The forward contract can be created synthetically by buying the stock with borrowed funds.

r≡ the annual, continuously compounded rate at which funds can be borrowed.

S0 ≡ the current price of the stock.

ST

-1 × what is owing to the bank = -1 × S0 erT

Creating a forward contract syntheticallyValue at time T of a long position in a forward contract = fT= FT - K = ST – K= ST – S0erT

Value at time T of replicating portfolio:

fT

ST

Forward contract: UA provides a known cash income

Examples: forward contracts on dividend-paying stocks and coupon bonds.

I≡ the present value of the income to be received over the remaining life of the forward contract

Proposition: F = (S – I )erT, in the absence of arbitrage opportunities

Forward contract: UA provides a known cash income

Note: F = (S – I )erT < SerT

This price is lower than if the asset didn’t pay income.

Forward contract: UA provides a known cash income

Proposition: F = (S – I )erT, in the absence of arbitrage opportunities

Proof: Suppose F > (S – I )erT.

Arbitrage strategy (to be implemented today):

- Buy one unit of the UA in the spot market by borrowing S dollars for T years at rate r.
- Short a forward contract on one unit if the UA.

Use the income from the asset to repay the loan.

At time T:

- Sell the UA for F dollars under the terms of the forward contract.
- Repay the bank (S – I )erT dollars.

Arbitrage profit per unit of UA = [F – (S – I )erT] > 0.

S is bid up and F is bid down.

Forward contract: UA provides a known cash income

Suppose F < (S – I )erT.

Arbitrage strategy (to be implemented today):

- Go long a forward contract on one unit if the UA.
- Sell or short sell one unit of the UA. This leads to a cash inflow of S dollars. Invest this for T years at rate r.

At time T:

- The proceeds from the sale/short sale have grown to (S – I )erT dollars.
- Buy the UA for F dollars under the terms of the forward contract.
- Return the UA to your portfolio or to the client from whom it was borrowed.

Arbitrage profit per unit of UA = [(S – I )erT – F] > 0.

F is bid up and S is bid down.

Thus: F = (S – I )erT

Alternative derivation of formula

- Spot transaction
- Price agreed to.
- Price paid/received.
- Item exchanged.
- Prepaid forward contract
- Price agreed to.
- Price paid/received.
- Item exchanged in T years.
- Forward contract
- Price agreed to
- Price paid/received in T years.
- Item exchanged in T years.

Alternative derivation of formula

Underlying asset provides a known cash income:

FP = S - I

Explanation: With a prepaid forward contract, as compared to a spot transaction, physical exchange of the asset is delayed T years. As a result of the delay, the long foregoes income with present value I and the short receives this income. Thus, the price paid by the long and received by the short is reduced by amount I.

F = FP erT = (S – I )erT

Explanation: The forward contract allows the long to delay payment for T years and requires the short to delay receipt. The long can earn interest on the cash that would otherwise have been paid. The short foregoes this interest. The forward price (which is arrived at by multiplying the prepaid forward price, equal to S - I, by erT) compensates the short for the delay.

Forward contract: UA provides a known cash income

Proposition: f = S – I – Ke-rT

Proof:

In general: f = (F – K )e-rT

We derived: F = (S – I )erT

Thus: f = [(S – I )erT – K]e-rT = (S – I )– Ke-rT

Also: -f = Ke-rT – (S – I )

The value today of the UA in the spot market.

The value today of the price that the long has agreed to pay for the asset in T years.

The value today of the income the long foregoes as a result of delaying purchase of the asset for T years.

Forward contract: UA provides a known cash incomeWe derived: f = S – I – Ke-rT

Thus: f > 0 iff S > Ke-rT + I

K

0

T

The value today of the price at which the short has agreed to sell the asset in T years.

The value today of the income the short receives as a result of delaying sale of the asset for T years.

The value today of the UA in the spot market.

Forward contract: UA provides a known cash incomeWe derived: -f = Ke-rT – (S – I)

Thus: -f > 0 iff Ke-rT + I > S

K

0

T

Example: #5.23, page 123

S = $50

r = 8%

T = 6/12

(a) I = $1e-(0.08×2/12) + $1e-(0.08×5/12) = $1.9540

F = (S – I )erT = (50 – 1.9540)e(0.08×6/12) = $50.0068

-f = -(S – I – Ke-rT)

= -(50 – 1.9540 – 50.0068e-(0.08×6/12)) = 0

$1

$1

6/12

0

2/12

5/12

Example (continued)

(b) S = $48

r = 8%

T = 3/12

I = $1e-(0.08×2/12) = $0.9868

F = (S – I)erT = (48 – 0.9868)e(0.08×3/12) = $47.9629

-f = -(S – I – Ke-rT)

= -(48 – 0.9868 – 50.0068e-(0.08×3/12)) = $2.00

$1

$1

6/12

0

2/12

3/12

5/12

Example (continued)

S = $50

T = 6/12

(a) I = $1e-(0.078×2/12) + $1e-(0.082×5/12) = $1.9535

$1

$1

6/12

0

2/12

5/12

Term structure of interest rates:

Forward contract: UA provides a known yield

Examples: forward contracts on stock portfolios and currencies.

q≡ the average yield per annum expressed as a continuously compounded rate

Proposition: F = Se(r-q)T, in the absence of arbitrage opportunities

Forward contract: UA provides a known yield

Note: F = Se(r-q)T < SerT

This price is lower than if the asset didn’t pay income.

Forward contract: UA provides a known yield

Proposition: F = Se(r-q)T, in the absence of arbitrage opportunities

Proof: Suppose F > Se(r-q)T.

Arbitrage strategy (to be implemented today):

- Buy one unit of the UA in the spot market by borrowing S dollars for T years at rate r.
- Short a forward contract on one unit if the UA.

Use the income from the asset to repay the loan.

At time T:

- Sell the UA for F dollars under the terms of the forward contract.
- Repay the bank Se(r-q)T dollars.

Arbitrage profit per unit of UA = [F – Se(r-q)T ] > 0.

S is bid up and F is bid down.

Forward contract: UA provides a known yield

Suppose F <Se(r-q)T.

Arbitrage strategy (to be implemented today):

- Go long a forward contract on one unit if the UA.
- Sell or short sell one unit of the UA. This leads to a cash inflow of S dollars. Invest this for T years at rate r.

At time T:

- The proceeds from the sale/short sale have grown to Se(r-q)T dollars.
- Buy the UA for F dollars under the terms of the forward contract.
- Return the UA to your portfolio or to the client from whom it was borrowed.

Arbitrage profit per unit of UA = [Se(r-q)T – F ] > 0.

F is bid up and S is bid down.

Thus: F =Se(r-q)T

Alternative derivation of formula

- Spot transaction
- Price agreed to.
- Price paid/received.
- Item exchanged.
- Prepaid forward contract
- Price agreed to.
- Price paid/received.
- Item exchanged in T years.
- Forward contract
- Price agreed to
- Price paid/received in T years.
- Item exchanged in T years.

Alternative derivation of formula

Underlying asset provides a known yield:

FP = Se-qT

Explanation: FP equals the investment required in the asset today that will yield one unit of the asset in T years when physical delivery occurs. e-qT units of the asset will grow to e-qT× eqT= 1 unit of the asset in T years, assuming that the income provided by the asset is reinvested in the asset. e-qT units of the asset cost Se-qT today. Therefore, FP = Se-qT .

F = FP erT = Se-qTerT = Se(r-q)T

Explanation: The forward contract allows the long to delay payment for T years and requires the short to delay receipt. The long can earn interest on the cash that would otherwise have been paid. The short foregoes this interest. The forward price (which is arrived at by multiplying the prepaid forward price, equal to Se-qT, by erT) compensates the short for the delay.

Forward contract: UA provides a known yield

Proposition: f = Se-qT – Ke-rT

Proof:

In general: f = (F – K )e-rT

We derived: F = Se (r-q)T

Thus: f = [Se(r-q)T – K ]e-rT = Se(r-q)T/erT– Ke-rT

= Se-qT – Ke-rT

Also: -f = Ke-rT – Se-qT

Example: #5.11, page 122

r = 9%

S = 300

T = 5/12

q = (5% + 2% + 2% + 5% + 2%)/5 = 3.2%

F = Se (r-q)T = 300e((0.09-0.032)×5/12) = 307.34

Forward prices & futures prices

- Like forward contracts, futures contracts are contracts for deferred delivery.
- But, unlike forward contracts, futures contracts are marked to market daily.
- Consider “corresponding” forward and futures contracts:
- Same underlying asset.
- Delivery date in two days.
- The contracts are identical except:
- Forward contract is settled at maturity.
- Futures contract is settled daily.
- Ignore taxes, transaction costs, and the treatment of margins.
- F≡ the forward price
- G ≡ the futures price

Forward prices & futures prices

Day 0F00

Day 1F10

Day 2F2 = S2F2 – K = S2 – F0

Forward price:Payoff to buyer:

Day 0G00

Day 1G1G1 – G0

Day 2G2 = S2G2 – G1 = S2 – G1

Futures price:Payoff to buyer:

Forward prices & futures prices

Example: Suppose we have:

Day 0: G0 = $2

Day 1: G1 = $1 with a 50% probability

= $3 with a 50% probability

Day 2: G2 = S2 since the futures contract terminates.

Example (continued)

Suppose that the interest rate is a constant 10% (effective per day).

On day 1, if G1 = $1: the futures buyer has a loss = (G0 – G1) = $1. S/he would borrow this amount at r = 10% and have to repay $1.10 on day 2.

On day 1, if G1 = $3: the futures buyer has a gain = (G1 – G0) = $1. S/he would invest this amount at r = 10% and have $1.10 on day 2.

Since there is a 50% chance of paying interest of $0.10 and a 50% chance of earning interest of $0.10, there is no expected benefit from marking to market on day 1.

Since the futures contract offers no benefit as compared to the forward contract, G0 = F0.

Example (continued)

Now suppose that the interest rate is not constant. Suppose that r = 12% on day 1 if G1 = $3 and r = 8% on day 1 if G1 = $1.

On day 1, if G1 = $1: the futures buyer has a loss = (G0 – G1) = $1. S/he would borrow this amount at r = 8% and have to repay $1.08 on day 2.

On day 1, if G1 = $3: the futures buyer has a gain = (G1 – G0) = $1. S/he would invest this amount at r = 12% and have $1.12 on day 2.

Now there is an expected benefit from marking to market = (50% × $0.12 – 50% × $0.08) = $0.02.

Since the futures contract offers a benefit as compared to the forward contract, G0 must exceed F0.

Example (continued)

Now suppose that the interest rate is not constant. Suppose that r = 8% on day 1 if G1 = $3 and r = 12% at day 1 if G1 = $1.

On day 1, if G1 = $1: the futures buyer has a loss = (G0 – G1) = $1. S/he would borrow this amount at r = 12% and have to repay $1.12 on day 2.

On day 1, if G1 = $3: the futures buyer has a gain = (G1 – G0) = $1. S/he would invest this amount at r = 8% and have $1.08 on day 2.

Now the expected gain from marking of market = (50% × $0.08 – 50% × $0.12) = -$0.02.

Since the forward contract offers a benefit as compared to the futures contract, F0 must exceed G0.

Forward prices & futures prices

- With this reasoning:

- G0 = F0 when interest rates are uncorrelated with the futures price.

- G0 > F0 when interest rates are positively correlated with the futures price.

- F0 > G0 when interest rates are negatively correlated with the futures price.

- Empirical evidence:

- Differences between the forward and futures prices are usually trivial once factors such as taxes, transaction costs, and the treatment of margin are controlled for.

- Exceptions:

. Contracts on fixed income instruments, like T-bills. The prices of T-bills are highly negatively correlated with interest rates. F0 > G0

. Long-lived contracts.

- Formulas for F : use to calculate both forward prices and futures prices.

Stock index futures contracts

- Heavily traded. See National Post website.
- Stock index: a weighted average of the prices of a selected number of stocks.
- Underlying: the portfolio of stocks comprising the index.
- Examples of stock indices (futures exchanges):
- S&P/TSX Canada 60 Index (ME)
- S&P 500 Composite Index (CME)
- NYSE Composite Index (NYFE)

Stock index futures contracts

- A futures contract on an asset that provides income.
- Formulas:
- F = Se(r-q)T
- F = (S – I )erT
- S denotes the current value of the index.
- Index arbitrage: what kind of trader might engage in this arbitrage?
- F > Se(r-q)T
- F < Se(r-q)T

Stock index futures contracts

- Cash-settled contracts.
- More likely to lead to delivery.
- On the last trading day, the settlement price is set equal to the closing value of the index.
- Multiplier(m):
- S&P 500 composite index futures, m = 250
- S&P/TSX Canada 60 index futures, m = 200
- The long gains if F2 > F1. The short gains if F2 < F1:
- F1: the futures price at the time the position is initiated.
- F2: the futures price at the time the position is terminated.

Example

On May 20, 2005, you go long two March 2006 futures contracts on the S&P 500 Composite Index. The contract is trading at 1206.60. Suppose you hold the contract to expiration and the index is at 1193.50 at that time. What is your gain/loss?

Solution:

F1 = 1206.60

F2 = 1193.50

Your loss = ((F1 – F2)×$250×2) = ((1206.60 – 1193.50)×$250×2) = $6,550

Note:

- If you had shorted the contracts, you would have gained $6,550.
- If m = 1, your loss would have equaled $26.20.

Stock index futures contracts

- S&P 500 composite index futures: m = 250
- Mini S&P 500 futures: m = 50
- Both of these contracts trade on CME.
- See www.cme.com
- Question: Who trades the mini? Designed for individual investors, rather than professional portfolio managers.

Forward and futures contracts on currencies

See National Post website.

Foreign currency: a security that provides a known yield at rate q = rf

Our earlier formula, F = Se(r-q)T, becomes F = Se(r-rf )T

Notation:

r ≡ the domestic risk-free interest rate

rf≡ the foreign risk-free interest rate

S≡ the spot price of the foreign currency (or spot exchange rate) expressed in units of the domestic currency, e.g., 1 CAD = 0.9270 USD

F ≡ the forward or futures price of the foreign currency expressed in units of the domestic currency, e.g., 1 CAD = 0.9342 USD (1-year forward)

Forward and futures contracts on currencies

Proposition: F = Se(r-rf )T, in the absence of arbitrage opportunities

Proof: Suppose F > Se(r-rf )T.

Arbitrage strategy (to be implemented today):

- Buy one ₤ in the spot market by borrowing S dollars for T years at rate r.
- Short a forward contract on one ₤.

Use the income from the invested ₤ to repay the loan.

At time T:

- Sell the ₤ for F dollars under the terms of the forward contract.
- Repay the bank Se(r-rf)T dollars.

Arbitrage profit per ₤ = [F – Se(r-rf)T ] > 0.

S is bid up and F is bid down.

Forward and futures contracts on currencies

Suppose F <Se(r-rf)T.

Arbitrage strategy (to be implemented today):

- Go long a forward contract on one ₤.
- Sell one ₤. This leads to a cash inflow of S dollars. Invest this for T years at rate r.

At time T:

- The proceeds from the sale have grown to Se(r-rf )T dollars.
- Buy one ₤ for F dollars under the terms of the forward contract.
- Return the ₤ to your portfolio.

Arbitrage profit per ₤ = [Se(r-rf)T – F ] > 0.

F is bid up and S is bid down.

Thus: F =Se(r-rf )T

Forward contract: UA is a foreign currency

For an asset that provides a known yield, we had:

f = Se-qT – Ke-rT

-f = Ke-rT – Se-qT

Foreign currency: a security that provides a known yield at rate q = rf

Thus, for a forward contract on a foreign currency, we have:

f = Se-rfT – Ke-rT

-f = Ke-rT – Se-rfT

Futures on commodities

Commodity: bulky, entails storage costs if held

Types:

- Investment commodity: held primarily for investment purposes, e.g., gold, silver
- Consumption commodity: held primarily to be used, e.g., oil, copper, canola

Investment commodities

Examples: gold, silver

Ignoring storage costs, these are assets that pay no income. Thus: F = SerT.

But storage costs can be treated as negative income.

Letting U≡ the present value of the storage costs incurred during the life of a forward/futures contract:

F = (S – I )erT = (S – (–U ))erT = (S + U )erT

Investment commodities

Proposition: F = (S + U )erT, in the absence of arbitrage opportunities

Proof: Suppose F > (S + U )erT.

Arbitrage strategy (to be implemented today):

- Buy one ounce of gold in the spot market, and arrange to store it, by borrowing (S+U ) dollars for T years at rate r.
- Short a forward contract on one ounce of gold.

At time T:

- Sell the ouncefor F dollars under the terms of the forward contract.
- Repay the bank (S+U )erT dollars.

Arbitrage profit per ounce = [F – (S+U )erT ] > 0.

S is bid up and F is bid down.

Investment commodities

Suppose F < (S+U )erT.

Arbitrage strategy (to be implemented today):

- Go long a forward contract on one ounce of gold.
- Sell one ounce of gold and forego storage costs. This leads to a cash inflow of (S+U )dollars. Invest this for T years at rate r.

At time T:

- The proceeds from the sale have grown to (S+U )erT dollars.
- Buy one ounce for F dollars under the terms of the forward contract.
- Return the ounceto your portfolio.

Arbitrage profit per ounce = [(S+U )erT – F ] > 0.

F is bid up and S is bid down.

Thus: F = (S+U )erT

- Spot transaction
- Price agreed to.
- Price paid/received.
- Item exchanged.
- Prepaid forward contract
- Price agreed to.
- Price paid/received.
- Item exchanged in T years.
- Forward contract
- Price agreed to
- Price paid/received in T years.
- Item exchanged in T years.

Alternative derivation of formula

Underlying requires the payment of storage costs (expressed in present value dollar terms):

FP = S + U

Explanation: With a prepaid forward contract, as compared to a spot transaction, physical exchange of the asset is delayed T years. As a result, the long forgoes storage costs with present value U and the short has to pay these costs. Thus, the price paid by the long and received by the short is increased by amount U.

F = FP erT = (S + U )erT

Explanation: The forward contract allows the long to delay payment for T years and requires the short to delay receipt. The long can earn interest on the cash that would otherwise have been paid. The short foregoes this interest. The forward price (which is arrived at by multiplying the prepaid forward price, equal to S + U, by erT) compensates the short for the delay.

Investment commodities

As an alternative, storage costs can be expressed as a proportion or percentage of the current spot price of the commodity.

Storage costs can then be treated as a negative yield.

Letting u≡ storage costs per annum as a proportion or percentage of the spot price:

F = Se(r-q)T = Se(r-(-u))T = Se(r+u)T

- Spot transaction
- Price agreed to.
- Price paid/received.
- Item exchanged.
- Prepaid forward contract
- Price agreed to.
- Price paid/received.
- Item exchanged in T years.
- Forward contract
- Price agreed to
- Price paid/received in T years.
- Item exchanged in T years.

Alternative derivation of formula

Underlying requires the payment of storage costs (expressed as a percentage of the spot price):

FP = SeuT

Explanation: FP equals the investment required in the asset today that will yield one unit of the asset in T years when physical delivery occurs. euT units of the asset will grow to euT× e-uT = 1 unit of the asset in T years, taking into consideration the storage costs that must be paid. euT units of the asset cost SeuT. Therefore, FP = SeuT.

F = FP erT = SeuTerT = Se(r+u)T

Explanation: The forward contract allows the long to delay payment for T years and requires the short to delay receipt. The long can earn interest on the cash that would otherwise have been paid. The short foregoes this interest. The forward price (which is arrived at by multiplying the prepaid forward price, equal to SeuT, by erT) compensates the short for the delay.

Examples: copper, oil, canola

Proposition: F≤ (S + U )erT

F ≤ Se(r+u)T

Consumption commoditiesF> (S + U )erT (F > Se(r+u)T)

F< (S + U )erT (F < Se(r+u)T)

Sell

Buy

Buy

Sell

Traders will respond. S will be bid up and F will be bid down.

Traders may not respond. If they don’t, S will not be bid down and F will not be bid up.

Consumption commodities

Note: We can convert the inequalities to equalities by using the concept of convenience yield: a measure of the benefits of holding the physical commodity.

Letting y ≡ the convenience yield, expressed as an annual, continuously compounded rate:

F = (S + U )e(r-y )T

F = Se(r+u-y )T

Estimating convenience yield

Provide an estimate of the convenience yield of oil:

It is May 2007.

Current spot price (WTI) = $64.35

The August 2007 contract (NYMEX) is trading at $66.52.

Let u = 10%.

There are 3 months to maturity of the contract.

3-month LIBOR = 5.32%

F = Se(r+u-y)T

66.52 = 64.35e(0.0532+0.10-y)(3/12)

y = 2.0537%

Example: #5.15, page 122

S = $9

Storage costs = $0.24 per year payable quarterly in advance

r = 10%

T = 9/12

U = ($0.24/4) +($0.24/4)e-(0.10×3/12) + ($0.24/4)e-(0.10×6/12) = $0.1756

F = (S + U )erT = (9 + 0.1756)e(0.10×9/12) = $9.89

$0.24/4

$0.24/4

$0.24/4

0

3/12

6/12

9/12

No-Arbitrage Bounds

The analysis has ignored transaction costs: trading fees, bid-ask spreads, different interest rates for borrowing and lending, and the possibility that buying or selling in large quantities will cause prices to change.

With transaction costs, there is not a single no-arbitrage price but rather a no-arbitrage region.

Example

A trader owns silver as part of a long-term investment portfolio. There is a bid-offer spread in the market for silver. The trader can buy silver for $12.02 per troy ounce and sell for $11.97 per troy ounce. The six-month risk-free interest rate is 5.52% per annum compounded continuously. For what range of six-month forward prices of silver does the trader have an arbitrage opportunity?

Solution: For silver: F = (S + U )erT

F = Se(r+u)T

Assume U = u = 0 since we are given no information on storage costs.

Thus, F = SerT in the absence of arbitrage opportunities.

Buy

Buy

Sell

Example (continued)There is an arbitrage opportunity if:

1) F > SerT = $12.02e(0.0552×6/12) = $12.36

2) F < SerT = $11.97e(0.0552×6/12) = $12.31

The trader has an arbitrage opportunity for F > $12.36 and F < $12.31. There is no arbitrage opportunity for $12.31 ≤ F ≤ $12.36.

Example (continued)

Now suppose that the trader must pay a $0.10 transaction fee per ounce of silver.

There is an arbitrage opportunity if:

1) F > SerT = ($12.02 + $0.10)e(0.0552×6/12) = $12.46

2) F < SerT = ($11.97 - $0.10)e(0.0552×6/12) = $12.20

The trader has an arbitrage opportunity for F > $12.46 and F < $12.20. There is no arbitrage opportunity for $12.20 ≤ F ≤ $12.46.

Forward and futures contracts on currencies

- If interest rates are expressed as annual rates compounded continuously:
- If interest rates are expressed as equivalent effective annual rates:

Cost of carry

Cost of carry (c): the cost of holding an asset, including the interest paid to finance purchase of the asset plus storage costs minus income earned on the asset.

c can be positive, zero, or negative.

The concept allows us to express our formulas for F in a more general way:

- Investment asset: F = SecT
- Consumption asset: F = Se(c-y)T

Cost of carry

Investment asset: F = SecT

Consumption asset: F = Se(c-y)T

T = 0 implies F = Se0 = S

That is, the forward/futures price of an asset equals its spot price at the time the contract expires.

Cost of carry

Investment asset: F = SecT

Consumption asset: F = Se(c-y)T

∂F/∂S = the amount by which the forward (futures) price changes in response to an infinitesimal change in the spot price, ceteris paribus

Investment asset: ∂F/∂S = ecT > 0

Consumption asset: ∂F/∂S = e(c-y)T > 0

F and S are positively correlated.

c > 0 implies ecT > 1 and F > S

Normal, contango market

c < 0 implies 0 < ecT < 1 and F < S

Inverted market, backwardation

Inverted Market

Normal Market

F

S

S

F

1st trading day

1st trading day

1st trading day

DP

DP

Cost of carryInvestment asset: F = SecT

c > y implies e (c-y)T > 1 and F > S

Normal, contango market

c < y implies 0 < e (c-y)T < 1 and F < S

Inverted market, backwardation

Inverted Market

Normal Market

F

S

S

F

1st trading day

1st trading day

1st trading day

DP

DP

Cost of carryConsumptiom asset: F = Se (c-y)T

Cost of carry

Investment asset: F = SecT

Consumption asset: F = Se(c-y)T

∂F/∂T = the amount by which the forward (futures) price changes in response to an infinitesimal change in the time to expiration of the contract, ceteris paribus

Investment asset: ∂F/∂T = SecT× c

c > 0 implies ∂F/∂T > 0: normal or contango market

c < 0 implies ∂F/∂T < 0: inverted market, backwardation

Consumption asset: ∂F/∂T= Se(c-y)T× (c – y)

c > y implies ∂F/∂T > 0: normal or contango market

c < y implies ∂F/∂T < 0: inverted market, backwardation

Normal market

- c > 0 (investment asset)
- c > y (consumption asset)
- F > S
- ∂F/∂T > 0, that is, forward (futures) contracts with longer times to expiration trade at higher prices than forward (futures) contracts with shorter times to expiration.

Inverted market

- c < 0 (investment asset)
- c < y (consumption asset)
- F < S
- ∂F/∂T < 0, that is, forward (futures) contracts with longer times to expiration trade at lower prices than forward (futures) contracts with shorter times to expiration.

Amaranth Advisors LLC

- The Connecticut-based hedge fund lost about $6 billion (40% of its value) in September 2006 trading natural gas derivatives.
- G&M, September 22, 2006: “The problem with oil and gas these days is that the market is morphing from backwardation, when spot prices are higher than prices for delivery in the future, to contango, when futures prices are higher than spot.”

Spread trades

- A spread trade provides exposure to the difference between two prices.
- It is a long-short futures position.
- Example:
- Calendar spread: go long long-term contract and short short-term contract on the same underlying asset, or vice versa.
- Intercommodity spread: go long futures on commodity A and short futures on commodity B
- Geographical spread: go long NYMEX oil futures and go short London’s ICE Brent oil futures
- For speculators, it offers reduced risk.

Amaranth Advisors LLC

- Traders expected the market to be in backwardation but it has moved into contango.
- They implemented spread trades based on this expectation.

Traders at Amaranth were betting:

. Expectation of a cold winter, active hurricane season, instability in oil and gas producing countries

What has happened:

.Winter of 2005-2006 was warm, the hurricane season was benign, supply of oil and gas was relatively high

F

F

T

T

Delivery options for a futures contract

- T≡ the time to expiration of a forward or futures contract.
- Forward contract: we know T.
- Futures contract: we must estimate T.
- Question: When during the delivery period of a futures contract will the short choose to make delivery?

0

DP

DP

T

Delivery options for a futures contractNormal Market:

Investment asset, c > 0

Consumption asset, c > y

Short should deliver as soon as possible.

0

DP

DP

T

Delivery options for a futures contractInverted Market:

Investment asset, c < 0

Consumption asset, c < y

Short should deliver as late as possible.

Next class

- Hedging with futures

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