forwards futuresi

8/12/2019 Forwards FuturesI

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Forward and FuturesContracts

Marcela Valenzuela


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Forward Contract

A Forward Contract is a contract made today for delivery of an

asset at a pre-specified time in the future at a price agreed upontoday:

The buyer of a forward contract agrees to take delivery of an

underlying asset at a future time, T, at a price agreed upontoday.

No money changes hands until time T.

The seller agrees to deliver the underlying asset at a futuretime, T, at a price agreed upon today.

A forward contract, therefore, simply amounts to setting a price

today for a trade that will occur in the future.


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Forward Contract

Suppose that on February 1 you want to buy a book, go to the

bookstore but the book is sold out. The cashier tells you that hewill reorder the book for you. The cost of the book is $10.

If you agree on February 1 to pick up and pay $10 for the bookwhen called, you and the cashier have engaged in aforward

contract.

February 1 Date when Book Arrives

BuyerBuyer agrees to: Buyer:1. Pay the purchase price of$10 1. Pays purchase price of $10

2. Receive book when book arrives 2. Receives the bookSellerSeller agrees to: Seller:1. Give up book when book arrives 1. Gives up book2. Accept payment of$10 when book arrives 2. Accepts payment of$10


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Long and Short Positions

The buyer (long position) of a forward contract is obligated to:

take delivery of the asset at the maturity date.

pay the agreed-upon price at the maturity date.

The seller (short position) of a forward contract is obligated to:

deliver the asset at the maturity date.

accept the agreed-upon price at the maturity date.


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Hedging with Commodity Forwards

A farmer is currently growing 1 ton of wheat for harvest and

sale in September.

The farmer needs to plan his September budget now, becauseof upcoming expenses.

Wheat currently sells for $100 per ton.

There is no current trend (expected increase or decrease) inwheat prices.

Wheat prices can change unexpectedly.


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A cereal maker plans to purchase 1 ton of wheat in Septemberto make cereal.

The cereal maker would like to set its sales price for September

now (e.g. for marketing purposes).

The farmer and cereal maker canbothhedge their Septembercash flows by signing a forward contract.


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Commodity forward

Cereal maker promises to purchase 1 ton of wheat fromthe farmer on September 1st at $100.

This is called going long or buying a forwardcontract.

Farmer promises to sell the same amount and price. This is called going short or selling the forwardcontract.

No money changes hands at contract signing

The delivery price is set so that the contract has a valueof zero at time 0.


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Cash Settlement vs. Delivery

On September 1st the spot price of wheat is $110 per ton.

The cereal maker and farmer have a forward contract to tradeat $100.

Cash settlement saves transport costs:

Farmer sells his wheat for $110 to his local grainstore.

Cereal maker purchases his wheat for $110 from his local

depot.

The farmer sends the cereal maker a check for $10.

Both parties effectively traded at $100, as agreed.


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Payoffs from Forward Contracts

The payoff from a long position in a forward contract on one

unit of an asset is:

STK

The payoff from a short position in a forward contract on oneunit of an asset is:

K ST

where K is the delivery price and STis the spot price of the assetat maturity of the contract.

As it costs nothing to enter into a forward contract,the payoffis also the traders total gain or loss from the contract.


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Forward Prices

How forward prices are related to the spot price of the

underlying asset?

First, it is important to distinguish between investment assetsand consumption assets.

An investment asset is an asset that is held for investmentpurposes, for instance stocks and bonds.

A consumption asset is an asset that is held primarily forconsumption. It is not usually held for investment. For instance

oil, pork bellies, etc.


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Forward Prices

Notation:

T: Time until delivery date in a forward contract (in years).

S0: Price of the asset underlying the forward contract today.

F0: Forward price today.

r: Risk-free rate of interest per annum for an investmentmaturing at the delivery date (i.e., in T years).


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Forward ContractsNo Income

Forward contract written on an investment asset that providesthe holder with no income. For example: non-dividend-payingstocks and zero-coupon bonds.

Consider a long forward contract to purchase anon-dividend-paying stock in three months. Assume that thecurrent stock price is $40 and the 3-month risk-free interest rate is5% per annum.

Suppose that the forward price is $43. This price is too high,why? How can an arbitrageur make profit?


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An arbitrageur can:

borrow $40 at the risk-free interest rate of 5% per annum.

buy one share.

short a forward contract to sell one share in 3 months.

At the end of three months, the arbitrageur:

delivers the share and receives $43.

the sum of the money required to pay off the loan is:

40e0.05 x 3/12 = $40.5

Profit: $43 $40.5= $2.50


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If the forward price is instead $39, an arbitrageur can:

short one share. invest the proceeds at 5% per annum for 3 months.

take a long position in a 3-month forward contract.

At the end of three months, the arbitrageur: pays $39 (forward contract).

takes delivery of the share, and uses it to close put the shortposition.

He gets from the savings: 40e0.05 x 3/12

= $40.5

Profit: $40.5 $39= $1.5

Under what circumstances do arbitrage opportunities do not

exist?


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GeneralizationNo Income

Forward contract on an investment asset with price S0 that

provides no income. Then:

F0=S0erT

IfF0 > S0erT, arbitrageurs can buy the asset and short forward

contracts on the asset.

IfF0 < S0erT, they can short the asset and enter into long

forward contracts on it.

In our example:

F0 =40e0.05 x 3/12 = $40.5

E l


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Example

Consider a 4-month forward contract to buy a zero-coupon

bond that will mature 1 year from today. The current price of thebond is $930. We assume that the 4-month risk-free rate ofinterest (continuously compounded) is 6% per annum. What is theforward price, F0?

In this case T =4/12, r=0.06, and S0= $930

F0 is given by:

F0=S0erT =930e0.06 x 4/12 = $948.79

This would be the delivery price in a contract negotiatedtoday.

F d C K I


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Forward ContractsKnown Income

Forward contract written on an investment asset that providesthe holder with a predictable cash income.

Consider a long forward contract to purchase a coupon-bond

whose current price is $900. The forward contract matures in 9months. The coupon payment of$40 is expected after 4 months.We assume that the 4-month and 9-month risk-free rates are,respectively, 3% and 4% per annum (continuously compounded).

Suppose that the forward price is $910. This price is too high,why? How can an arbitrageur make profit?

F d C t t K I


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The coupon payment has a present value of:

PV($40) =40e0.034/12 = $39.6

An arbitrageur can:

borrow $900 to buy the bond. $900 comes from:

borrowing $39.6 at 3% per annum for 4 months so thatit can be repaid with the coupon payment.

borrowing $860.4 at 4% per annum for 9 months.

enter into a short forward contract.

At the end of nine months, the arbitrageur:

pays the loan: 860.4 e0.04 x 9/12 = $886.6.

delivers the bond and receives $910 from the forward contract.

He gets: $910 $886.6= $23.4.

F d C t t K I


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Suppose instead that the forward price is $870. This price is

too low, an arbitrageur can: short the bond

enter into a long forward contract.

Att=0 the arbitrageur: receives $900 from the bond.

invest PV($40)=$39.6 to pay the coupon on the bond.

The remaining $900-$39.6=$860.4 is invested for 9 months at

4% per annum.

At the end of nine months, the arbitrageur gets:

860.4e0.04 x 9/12 870= $886.6 $870= $16.6

G li ti K I


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GeneralizationKnown Income

Forward contract on an investment asset with price S0 that

provides an income with a present value of I. Then:

F0= (S0 I)erT

IfF0 > (S0 I)erT, arbitrageurs can buy the asset and shortforward contracts on the asset.

IfF0 < (S0 I)erT, they can short the asset and enter into

long forward contracts on it.

Exercise


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Exercise

Consider a 10 month forward contract on a stock when the

stock price is $50. We assume that the risk-free rate of interest(continuously compounded) is 8% per annum for all maturities. Wealso assume that dividends of$0.75 per share are expected after 3months, 6 months, and 9 months. What is the forward price, F0?

Forward Contracts Known Yield


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Forward ContractsKnown Yield

Here we consider the situation where the asset underlying a

forward contract provides a known yield rather than a known cashincome.

Define qas the average yield per annum on an asset during thelife of a forward contract with continuous compounding. Then:

F0=S0e(rq)T

Example


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Example

Consider a 6-month forward contract on an asset. The asset

price is$

25. The risk-free rate of interest (with continuouscompounding) is 10% per annum. The yield is 3.96% per annum.What is the forward price F0?

In this case S0 =25, r=0.1, T =0.5 and q=0.0396.

The price of the forward is given by:

F0=S0e(rq)T =25e(0.10.0396)0.5 = $25.77

What is a Forward Price?


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What is a Forward Price?

The Forward Price (Ft) is the price such that the present value

of the contract payoff at time tequals zero.

At the beginning of the life of the forward contract, the deliveryprice, K, is set equal to the forward price F0.

As time passes, Kstays the same (because it is part of thedefinition of the contract), but the forward price changes and thevalue of the contract becomes either positive or negative.

Ft=Ste(rq)(Tt)

The price of the forward today is:

F0=S0e(rq)T

Valuing Forward Contracts


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The value of a forward fcontract at the time it is first entered

into is zero.

At a later stage, it may prove to have a positive or negativevalue.

Notation: Let Kbe the delivery price for a contract that was negotiated

some time ago.

The delivery date is T years fromtoday.

r is the T-year risk-free rate. F0 is the forward price that would be applicable is we

negotiated the contract today.

f: value of the forwad contract today.



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The value of a long forward contract (f) is given by:

f = (F0 K)erT

The value of a short forward contract is given by:

(K F0)erT

At the beginning, the value of the contract is 0 since F0=K.

Exercise


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Exercise

A long forward contract on a non-dividend-paying stock was

entered into some time ago. It currently has 6 months to maturity.The risk-free rate of interest (with continuous compounding) is10% per annum, the stock price is $25, and the delivery price is$24.



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Value of a forward contract on an asset that provides no

income:

f =S0 KerT

Value of a forward contract on an asset that provides a knownincome with present value I:

f =S0 IKerT

Value of a forward contract on an asset that provides a knownyield at rate q:

f =S0eqT

KerT

Future Contracts


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Future Contracts

When a standardized forward contract is traded on an exchange,

it is called a futures contract same contract, but a different label.

The exchange is called a futures exchange.

The distinction between futures and forward does notapply to the contract, but to how the contract is traded.

Differences between Forwards and Futures


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Differences between Forwards and Futures

Forward Futures

Private contract between two parties Traded on an exchange

Not standarized Standarized contract

Usually one specified delivery date Range of delivery dates

Settled at end of contract Settled daily

Some credit risk Virtually no credit risk

The Mechanics of Futures Trading


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g

When you buy or sell a futures contract, the price is fixed today

but payment is not made until later.

You will, however, be asked to put up margin to demonstratethat you have the money to honor your side of the bargain.

Margin is typically set at an amount that is larger than a usualone-day moves in the futures price.

Futures contracts are marked to market. This means that eachday any profits or losses on the contract are calculated; you pay

the exchange any losses and receive any profits.

Margin


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g

The futures exchange guarantees the contracts and protectsitself by settling up profits or losses each day.

The margin ensures that both parties will have sufficient funds

available to mark to market.

Futures trading eliminates counterparty risk. The wholepurpose of the margining system is to eliminate the risk that a

trade who makes a profit will not be paid.

Margin


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Margin rules are stated in terms of:

Initial margin (which must be posted when entering thecontract).

Maintenance margin (which is the minimum acceptable

balance in the margin account).

If the balance of the account falls below the maintenance level,the exchange makes a margin call upon the individual, who must

then restore the account to the level of initial margin before thestart of trading the following day.

Futures Contracts - Marking to Market


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Futures Marking to Market:

Buy 200 Futures at $600.

Initial Margin = $4,000 per contract.

At the end of each trading day, the margin account isadjusted to reflect the investors gain or loss.

If the futures price dropped from$

600 to$

597, loss of$600(= 200 x $3). Margin account balance: $3,400.

If the futures prices dropped from $597 to $596.1, loss of$180(= 200 x $(597 596.1)). Cumulative loss:

600+180=780. Margin account balance: $

3,220. Assume that the maintenance margin is $3,000. On June 13,the margin in the balance account falls $340 below themaintenance margin level margin call from the broker for an

additional$

1,340.

Futures Contracts - Marking to Market


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Day Futures Daily gain Cumulative Margin Account MarginPrice (loss) gain (loss) Balance call

($) ($) ($) ($) ($)600.0 4,000

June 5 597.0 (600) (600) 3,400June 6 596.1 (180) (780) 3,220June 9 598.2 420 (360) 3,640June 10 597.1 (220) (580) 3,420

June 11 596.7 (80) (660) 3,340June 12 595.4 (260) (920) 3,080June 13 593.3 (420) (1,340) 2,660 1,340June 16 593.6 60 (1,280) 4,060June 17 591.8 (360) (1,640) 3,700June 18 592.7 180 (1,460) 3,880June 19 587.0 (1,140) (2,600) 2,740 1,260June 20 587.0 0 (2,600) 4,000June 23 588.1 220 (2,380) 4,220June 24 588.7 120 (2,260) 4,340June 25 591.0 460 (1,800) 4,800June 26 592.3 260 (1,540) 5,060

Futures Prices of Stock Indices


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Here we consider a situation where the asset underlying a

forward contract provides a known yield rather than a known cashincome:

F0=S0e(rq)T

Example


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Consider a 3-month futures contract on the S&P500. Suppose

that the stocks underlying the index provide a dividend yield of 1%per annum, that the current value of the index is $1,300, and thatthe continuously compounded risk-free interest rate is 5% perannum.

The futures price, F0, is given by:

F0=1, 300e(0.050.01) x 0.25 = $1, 317.07

forwards futuresi

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