1 Finance School of Management Chapter 14: Forward & Futures Prices Objective How to price forward...

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FinanceFinance School of Management School of Management

Chapter 14: Forward & Chapter 14: Forward & Futures PricesFutures Prices

Objective• How to price forward and futures

• Storage of commodities• Cost of carry

• Understanding financial futures

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Chapter 14: ContentsChapter 14: Contents Distinction Between Forward &

Futures Contracts The Economic Function of

Futures Markets The Role of Speculators Relationship Between

Commodity Spot & Futures Prices

Extracting Information from Commodity Futures Prices

Spot-Futures Price Parity for Gold

Financial Futures

The “Implied” Risk-Free Rate The Forward Price is not a

Forecast of the Spot Price Forward-Spot Parity with Cash

Payouts “Implied” Dividends The Foreign Exchange Parity

Relation The Role of Expectations in

Determining Exchange Rates

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Two parties agree to exchange some item on a specified future date at a delivery price specified now.

The forward price is defined as the delivery price which makes the current market value of the contract zero.

No money is paid in the present by either party to the other.

The face value of the contract is the quantity of the item specified in the contract times the forward price.

Features of Forward ContractsFeatures of Forward Contracts

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Features of Forward ContractsFeatures of Forward Contracts

The party who agrees to buy the specified item is said to take a long position, and the party who agrees to sell the item is said to take a short position.

“Customization”, difficulty of “closing out” positions, low liquidity

The risk of contract default, credit risk

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Characteristics of FuturesCharacteristics of Futures

Futures are– standard contracts– immune from the credit worthiness of buyer

and seller because exchange stands between traders contracts marked to market daily margin requirements (enough collateral)

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TermsTerms Futures List

Open High Low Settle Change Lifetime High Lifetime Low Open Interest

292 2941/2 289 2943/4 -71/4 326 258 16.168

Monday Aug.5,2007WHEAT(CBT)5,000bu; cents per bu

Mark-to-market Margin requirement Margin call

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An Illustration An Illustration You place an order

to take a long position in a September wheat futures contract on August 4, 1991.

The broker requires you to deposit an initial margin of $1,500 in your account.

On August 5, the futures price closes 71/4 cents per bushel lower.

– You have lost 71/4 cents*5,000 bushels = $362.50 that day.

– Marking to Market: the broker takes that amount out of your account and transfers it to the future exchange, which transfers it to one of the parties who was on the short side of the contract.

– If you do not have enough money in your account to meet the margin requirement (variation / maintenance margin), you’ll receive a margin call from the broker asking you to add money.

– If you do not respond immediately, then the broker liquidates your position at the prevailing market price.

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Spot-Futures Price Parity for GoldSpot-Futures Price Parity for Gold There are two ways to invest in gold.

– buy an ounce of gold at S0, store it for a year at a storage cost of $hS0, and sell it for S1.

– invest S0 in a 1-year T-bill with return rf , and purchase a

1-ounce of gold forward, F, for delivery in 1-year.

fsyngoldgold rS

FSrrh

S

SS

0

1)(

0

01

0)1( ShrF f

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The spot price of gold is $300, the storage costs is 2% per year, and the risk-free rate is 8%.

– If the forward price is $340 (too high)

Sell a forward contract 0 $340-S1

Borrow $300 $300 -$324Buy an ounce of gold -$300 S1

Pay storage costs -$6Net cash flows 0 $340-$330=$10

Arbitrage PositionImmediate Cash

FlowCash Flow 1 Year

from Now

Arbitrage Opportunities of Forwards: Arbitrage Opportunities of Forwards: An IllustrationAn Illustration

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Sell short an ounce of gold $300 0-S1

Buy a forward contract 0 S1-$320

Receive storage costs $6Net cash flows 0 $330-$320=$10



from Now

Invest $300 in 1-year purediscount bond

-$300 $324

– If the forward price is $320 (too low)

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Spot-Futures Price Parity for GoldSpot-Futures Price Parity for Gold

A contract with life T:

01 ShrF Tf

– This is not a causal relationship, but the forward and current spot are jointly determined in the market.

– If we know one, then the law of one price determines that we know the other.

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Implied Cost of CarryImplied Cost of Carry

Implied cost of carry = F − S0 = (rf + h) S0

The implied cost of storing the gold (per $spot) is

h = (F − S0)/S0 − rf

Suppose F = $330, S0 = $300, and rf = 8%, then

– Implied cost of carry = $330 − 300 = $30

– Implied storage cost = (330 − 300)/300 − 8% = 2%

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Financial FuturesFinancial Futures

Financial futures contracts are usually settled in cash.

With no storage cost, the relationship between the forward and the spot is

TfrSF 1

– Any deviation from this will result in an arbitrage opportunity.

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Replication of Non-Dividend-Paying Stock Replication of Non-Dividend-Paying Stock Using a pure Discount Bond and a Stock Using a pure Discount Bond and a Stock

Forward ContractForward Contract

Buy a share of stock – S S 1

Replicating Portfolio (Synthetic Stock)

Total replicating portfolio – F/(1+r f) S 1

Cash Flow 1 Yearfrom Now

Position

Go long a forward contracton stock

Buy a pure discount bondwith face value of F

Immediate Cash Flow

0

– F/(1+r f )

S 1 – F

F

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Arbitrage in Stock FuturesArbitrage in Stock Futures The spot price of a stock is $100, and the risk-free rate

is 8%. The forward price is $109.

Sell a forward contract 0 $109-S1

Borrow $100 $100.00 -$108.00Buy a share of stock -$100.00 S1

Net cash flows 0 $1



from Now

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The “Implied” Risk-Free RateThe “Implied” Risk-Free Rate

Rearranging the formula, the implied interest rate on a forward given the spot is

0

0

1

0

1, if;1S

SFrT

S

Fr

T

– This is reminiscent of the formula for the interest rate on a discount bond.

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Replication of a Pure Discount Bond Replication of a Pure Discount Bond Using a Stock and a Forward ContractUsing a Stock and a Forward Contract

Buy a share of stock – S S 1

Go short a forward contract 0 F – S 1

Total replicating portfolio – S F

– F/(1+r f ) F

Replicating Portfolio (Synthetic T-Bill)

Position

Buy a T-bill with face valueof F

Immediate CashFlow

Cash Flow 1 Yearfrom Now

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The Forward Price is not a Forecast The Forward Price is not a Forecast of the Spot Priceof the Spot Price

FrSpremiumriskrSS ff )1()1( 001

Following the diagrams in Chapter 13 we might suppose that the expected price of a stock is

If this is indeed correct, then the forward price is not an indicator of the expected spot price at the maturity of the forward.

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Forward-Spot Parity with Cash PayoutsForward-Spot Parity with Cash Payouts

The S − F relationship becomes

– Note: (forward price > the spot price) if (D < rf S)

– Because D is not known with certainty, this is a quasi-arbitrage situation.

DSrSFr

FDS f

f

1

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Replication of Dividend-Paying Stock Replication of Dividend-Paying Stock Using a Pure Discount Bond and a Stock Using a Pure Discount Bond and a Stock

Forward ContractForward Contract

Buy a share of stock - S D + S 1

S 1 – F

- (D + F)/(1+r f ) D + F

Go long a forward contracton stock

Buy a pure discount bondwith face value of D + F

Replicating Portfolio (Synthetic Stock)

PositionImmediate Cash


from Now

0

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““Implied” DividendsImplied” Dividends

From the last slide, we may obtain the implied dividend

FSrD f 1

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Exchange Rate ExampleExchange Rate Example

Time

15450 ¥ 15450 ¥(Repaid)

3% ¥/¥ (direct)

3% ¥/£/£/¥

£109(Matures)

9%£/£

Forward ¥/£

15000 ¥(Borrowed)

£100(Invested)

150 ¥/£

Japan U.K.

15450 ¥ 15450 ¥(Repaid)

3% ¥/¥ (direct)

3% ¥/£/£/¥

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The Foreign Exchange Parity RelationThe Foreign Exchange Parity Relation

We used the diagram to show that

tYt

$ r1

Yen for SpotdDenominate $

r1

Yen on Forward dDenominate $

1 Finance School of Management Chapter 14: Forward & Futures Prices Objective How to price forward...

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