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Chapter-4 Pricing and Valuation of Forward and Futures Contracts V2.0, April 2010For Associates

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Chapter 4 Pricing and Valuation of Forward and

Futures Contracts

Introduction

Forward contracts are generally easier to analyze when compared to that of Futures

contracts, because there is no daily settlement. In case of Forward Contracts, only a

single payment is made at the time of maturity. But one can easily observe that,

Forward Prices and Futures Prices are very close when the maturities of the two

contracts are one and the same.

Learning Objective

After reading this chapter you will:

Understand about Investment and Consumption Assets

Understand about Short Selling

Understand the pricing and valuation of Forward Contracts

Understand the pricing and valuation of Futures Contracts

Understand about Futures on Commodities

Topics Covered

Chapter 4 Pricing and Valuation of Forward and Futures Contracts .................................... 34.1. Introduction:................................................................................................................................... 44.2. Short Selling:................................................................................................................................... 44.3. Assumptions and Notations involved:................................................................................... 64.4. Forward Price for an investment asset:.................................................................................. 74.5. Forward Contracts with a known income:............................................................................ 94.6. Forward Contracts with a known Yield:............................................................................... 104.7. Valuing Forward Contracts: ..................................................................................................... 114.8. Future Prices of Stock Indices: ................................................................................................ 134.9. Forward and Futures contracts on currencies:.................................................................. 144.10. Futures on Commodities: ...................................................................................................... 17


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4.1. Introduction:

Before discussing about Forward and Futures prices and valuation, its mandatory to

understand or rather distinguish between the Investment Assets and Consumption

Assets

Investment Assets:

An investment asset is an asset that is held for investment purposes by significant

number of investors.

Example: Stocks, Bonds, Gold, Silver etc.

Consumption Assets:

A consumption Asset is an asset that is held for the purpose of consumption. These

assets are not usually held for investment purposes.

Example: Copper, Oil etc.

It is important to understand that, investment assets are not to be held exclusively for

investment purpose, rather than can also be used for industrial purposes.

4.2. Short Selling:

Short selling is one of the arbitrage strategies that is commonly used. This trade is also

known as Shorting, usually involves selling an asset that is not owned.

Process: When an investor short sell a stock, the broker will lend it to investor. The stock

will come from the brokerage's own inventory, from another one of the firm's

customers, or from another brokerage firm. The shares are sold and the proceeds are

credited to investors account. Sooner or later, the investor must "close" the short by

buying back the same number of shares (called covering) and returning them to the

broker. If the price drops, he can buy back the stock at the lower price and make a profit

on the difference.

Example:

Let us consider an investor who wants to short 500 Microsoft Shares. The broker of this

particular instructor will carry out the instructions by borrowing the above mentioned

shares from another client and selling them in the market in the usual way. The

investor can maintain the short position as long as possible for which he can borrow

the shares. If the investor closes out the position by buying the shares, these are then


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replaced into the client account. The investor makes a profit if there is a decline in the

price of a share, or has to face loss when there is a hike in the market value of the share.

The investor with a short position must pay to the broker any income, such as dividends

or interest that would normally be received on the securities that have been shorted.

The broker will transfer the above income to the account of the client from whom the

securities have been borrowed.

In the above example, if it happens that, the investor shorts 500 Microsoft Shares in

June at $150 per share and closes out the position in October by buying 500 shares at

$130 per share. And also assume that a dividend of $1.5 is paid in the month of

September.

Therefore, the net gain made by the investor is

Gain = (500*150) (500*130) (500*1.5)

= 75000 65000 750

= $9, 250.

Therefore, the investor made a net profit of $9,250 by shorting the Microsoft shares in

the month of April and thereby closing out the position in the month of October.

Let us compare the profits made by an investor, when purchase of shares is made and

when shorting is made.

Case A: Purchase of Shares

June: Purchase 500 Microsoft Shares @ $150 per share: (500*150) = -75000

September: Receive Dividend: (500*1.5) = +750

October: Sell 500 Microsoft shares @ $130 per share: (500*130) = +65000

________________________________________________________________________

Net Profit: -$9,250

________________________________________________________________________

Case B: Short Sale of Shares

June: Borrow 500 shares and sell them for $150 per share: (500*150) = +75,000


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September: Pay Dividend: (500*1.5) = -750

October: Close out short position by buying shares @$130: (500*130) = -65,000

_______________________________________________________________________

Net Profit: +$9,250

________________________________________________________________________

The investor has to maintain a Margin Account with the broker consisting of either

cash or marketable securities. The purpose behind maintaining margin account is to

guarantee that the investor will not default from the short position if there is a hike in

the price of the share.

4.3. Assumptions and Notations involved:

There are certain assumptions that are required to be made which hold true for some

market participants. The assumptions are:

The market participants are subjected to no transaction costs when they

trade.

The market participants can borrow money at the same risk-free rate of

interest as they can lend money.

The market participants are subjected to same tax rate on all net trading

profits.

The market participants take advantage of arbitrage opportunities as and

when they occur.

There are certain notations that are to be precisely followed:

S: Price of the asset underlying the forward or futures contract today

F: Forward or Futures price today

r: Zero-coupon risk-free rate of interest per annum, expressed with continuous

compounding for an investment maturing at the delivery date (in T years)

T: Time until delivery date in a forward or futures contract


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The risk-free rate, r, is the rate at which money is borrowed or lent when there is no

credit risk, so that the money is certain to be repaid. This rate of interest is a LIBOR rate

rather than the treasury rates.

4.4. Forward Price for an investment asset:

The easiest forward contract to value is one written on an investment asset that

provides the holder with no income.

Example: Non-dividend-paying stocks and zero coupon bonds

With the example that is going to be discussed, there could be a clear understanding

regarding this.

Consider a long forward contract to purchase a non-dividend paying stock in 3 months.

Assume the current stock price is $40 and the 3 months risk free interest rate is 5% p.a.

Case A: If Forward Price = $43

An arbitrageur can borrow $40 at the rate of 5% for 3 months, then buy one share, and

then forward a short contract to sell the share at $43 after 3 months.

At the end of the three months, the arbitrageur sells the share and receives $43.

The amount required to pay the loan is

= 40* e^ (0.005*3/12) = $40.5

With this, the arbitrageur can accumulate a net profit of

Net Profit = $43 - $40.5

= $2.5

Case B: If Forward Price = $39

In this case, an arbitrageur can short a share for $40 and then invest it for a period of 3

months at the risk free rate of 5% and then purchase the share at $39 after three

months

Amount earned by investing = 40* e^ (0.005*3/12)

= $40.5

With this, the arbitrageur can accumulate a net profit of

Net Profit = $40.5 $39

= $1.5


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The above example can be presented in a lucid in the form of table as shown below.

These are primarily the two alternatives that are available depending on the forward

price fluctuations when compared to the spot price.

The first arbitrage works when the forward price is greater than $40.50 and the second

arbitrage works when the forward price is less than $40.50.

Let us summarize the above example into the following table.

Table 4.1. Arbitrage opportunities when forward price is out of line with the spot price for asset providing no income

(Asset price = $40; Interest rate = 5% and maturity of forward contract is 3 months.)

If Forward Price = $43 If Forward Price = $39

Action Now:

Borrow $40 at 5% for 3 months

Buy one unit of Asset

Enter into forward contract to sell

Asset in 3months for $43

Action Now:

Short one unit of asset to realize

$40

Invest $40 at 5% for 3 months

Enter into a forward contract to

Buy asset in 3 months for $39

Action in 3 months:

Sell Asset for $43

Use $40.50 to repay the loan with

interest

Action in 3 months:

Buy asset for $39

Close Short position

Receive $40.50 from investment

Profit Realized = $2.50 Profit Realized = $1.50

Mathematical Representation:

The example that was discussed above can be considered as a forward contract on an

investment asset with Price S, which provides no income. If T is the time to maturity

and r is the risk free rate and F is the forward price, the relationship between F and S can

be represented as

F = S e^(r*T)

If F>Se^(r*T):


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This case resembles to the case A of the above example where F = $43 and Se^(r*T) =

$40.5. In this scenario, arbitrageurs can buy the asset and short forward contracts on

the asset.

If F


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Case B: If Forward Price = $900

An investor can short the bond and enter into long forward contract. Of the $1000

realized from shorting the bond, $48.76 is invested for a period of 5 months at an

interest rate of 6% p.a. so that it grows into an amount sufficient to pay the coupon on

the bond. The remaining $951.24 is invested at the rate of 9% for the period of 12

months and grows to an amount of $1040.82. Under the terms of the contract $850 is

paid to buy the bond and the short position is closed out.

Therefore, the net profit made by the arbitrageur is

= $1040.82 - $900

=$140.82

Mathematical Representation:

The above discussed example can be mathematically represented as,

F = (S-I) e^(r*T)

Where I indicates an income with a present value.

In the case of an above example

S = 1000

I= 50*e^ (-0.06*5/12)

= $48.76

r= 0.09

T= 1 year

Therefore, F = (1000-48.76) * e^ (0.09)

= $1040.82

This is how the above concept can be explained.

4.6. Forward Contracts with a known Yield:

This is the case where the asset underlying a forward contract provides a known yield

rather than a known income. Yields are also normally measured on the basis of a

continuous compounding.

Let q be the average yield per annum on an asset during the life of a forward contract

with a continuous compounding. This can be represented as


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F = S e^ [(r-q)*T]

4.7. Valuing Forward Contracts:

Here let us discuss in detail about valuation of forward contracts.

The value of a forward contract is zero at the time it is first

entered. Later on, the value may become either positive or negative. Banks and other

financial institutions value the contract on a daily basis. This is referred to as Marking

to Market.

Let us again understand the following notations:

K: Delivery Price for a contract that was negotiated sometime ago

T: Delivery date in years

r: Risk free Interest rate

F: Forward Price that would be applicable if the contract is to be negotiated today.

f: Value of forward contract today

As discussed above, the value of a forward contract is zero at the time it is first entered,

because, the delivery price K will be equal to the forward price, F.

As time passes by, the value of K remains the same as per the definition of a contract,

but the forward price changes and hence the value of a forward contract can either

become positive or negative.

Generalization can be made to all long forward contracts by using the mathematical

formula,

This can be explained by comparing a long forward contract that has a delivery price of

F with an otherwise identical long forward contract that has a delivery price of K. The

difference between the two is only in the amount that will be paid for the underlying

f = (F K) e^ (-r*T)

f = (F K) e^(-r*T)


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asset at time T. Under the first contract, let us assume it is F, and in the second contract

it is K. A cash outflow equal to (F K) will be equal to (F K) e^ (-r*T) today. Therefore,

the contract with a delivery price F is less valuable than a contract with a delivery price

K and is equal to (F K) e^ (-r*T).

Similarly, generalization can be made to all short forward contracts by using the

mathematical formula,

f = (K - F) e^ (-r*T)

Let us see, how the above formulae changes for the three cases:

Case A - Forward contract with no income:

In this case, as we have discussed above F = S e^(r*T)

Substituting the value of F in the equation f = (F K) e^ (-r*T) we get,

f = [S e^(r*T) K] e^ (-r*T)

Therefore,

Case B Forward contract with known income:

In this case, as we have discussed above F = (S I) e^(r*T)


f = [(S I) e^(r*T) K] e^ (-r*T)

Therefore,

f = S - K e^ (-r*T)

f = (K - F) e^ (-r*T)

f = S I K e^ (-r*T)


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Case C Forward Contract with a known yield:

In this case, as we have discussed above F = S e^[(r-q) * T]


f = [(S e^ [(r-q) * T]) K] e^ (-r*T)

Therefore,

4.8. Future Prices of Stock Indices:

A stock index is generally considered as the price of an investment asset that pays

dividends. The investment asset is the portfolio of stocks underlying the index and the

dividend paid by the investment asset is received by the holder of the portfolio. It is

always assumed that, dividends pay a known yield rather than a known income. If q is

the dividend yield rate, then the futures price F is given as

F = S e^ [(r-q)*T]

The value of q that is to be chosen should represent the average annualized dividend

yield during the life of the contract. The dividend used for estimating q should be those

for which the ex-dividend date is during the life of the futures contract.

Index Arbitrage:

If F > S e^ ((r-q)*T)

Then, profits can be made by buying the stocks underlying the index at the spot price

and then shorting the futures contracts.

If F < S e^ ((r-q)*T)

Then, shorting or selling the stocks underlying the index needs to be done and then

taking a long position in the futures contracts will result in profits.

f = [(S e^ (-q*T)) K e^ (-r*T)]


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Often index arbitrage is implemented through Program Trading. This involves using a

computer system to generate the trades.

4.9. Forward and Futures contracts on currencies:

Here we shall discuss in detail about forward and futures contracts from the perspective

of an US investor. The underlying asset in this case is one unit of the foreign currency.

Let us again define the following notations:

S: Current spot price in dollars of one unit of the foreign currency.

F: Forward or Futures price in dollars of one unit of the foreign currency.

For a major exchange reates, a spot or forward exchange rate is normally quoted as the

number of units of the currency that are equivalent to one US dollar.

The peculiarity of a foreign currency is that the holder of the currency can earn interest

at the risk free rate of interest prevailing in the foreign country. Hence,

rf: Value of the foreign risk free interest rate, when money is invested for time period T

r: US dollar risk free rate when money is invested for the same period T

The relationship between F and S can be shown as:

F = S e^ [(r - rf ) * T]

Example 1:

Suppose that an individual starts with 1,000 units of the foreign currency. There are

two ways it can be converted into dollars at time T

1. Investing the 1,000 units of foreign currency for T years at rf and entering into a

forward contract to sell the underlying asset for dollars at time T. This results in

[1,000 e^ (rf * T)]* F dollars.

2. Exchanging the foreign currency for dollars in the spot market and investing the

proceeds for T years at the rate r. This results in 1,000 S e^(r*T) dollars.

In the absence of the arbitrage opportunities, both of them will be equal to each other.


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Therefore,

[1,000 e^ (rf * T)]* F = 1,000 S e^(r*T)

So that,

F = S e^ [(r - rf ) * T]

The above example can be explained diagrammatically as,

Fig. 6.1 Ways of converting the 1,000 units of foreign currency to dollars at time T

Example 2:

Suppose that the two year interest rates in Australia and United States are 5% and 7%

respectively. The spot exchange rate between the Australian Dollar and United States

dollar is 0.62 US dollar per Australian Dollar.

In this case,

r = 0.07

rf = 0.05

The two year forward exchange rate will be

0.62 e^ [(0.07 0.05) * 2]

= 0.6453

Let us again take two cases,

1 0 0 0 u n its o f fo re ig n c u rre n c y

a t t im e z e ro

u n its o f fo re ig n c u rre n c y a t t im e T

Tr fe1 0 0 0

d o lla rs a t t im e T

Tr feF 01 0 0 0

1 0 0 0 S 0 d o lla rs a t t im e z e ro


rTeS 01 0 0 0

1 0 0 0 u n its o f fo re ig n c u rre n c y

a t t im e z e ro

u n its o f fo re ig n c u rre n c y a t t im e T

Tr fe1 0 0 0


Tr feF 01 0 0 0

1 0 0 0 S 0 d o lla rs a t t im e z e ro


rTeS 01 0 0 0


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Case A: The 2 year forward exchange rate is less than 0.6453, lets say it is 0.63:

In this case, the arbitrageur can

Borrow 1,000 AUD at 5% pa for the period of two years and then convert to 620

USD and the invest these USD at 7%

Then enter into a forward contract to buy 1,105.17 AUD for 692.6 USD

No. of AUD that should be returned after the period of two years

= 1,000 * e^ (0.05*2)

= 1,105.17 AUD

Converting the 1,000 AUD to USD as per the spot exchange rate will result in 620 USD

Investing 620 USD for a period of two years at 7% interest rate will yield

= 620 * e^ (0.07*2)

=$713.17

1,105.17 AUD has to be returned. So, purchasing 1,105.17 AUD at the exchange rate of

0.63 will incur a cost of

= 1,105.17 * 0.63

= $696.26

Therefore, Net Profit (Risk less profit) is,

Net Profit = $713.7 - $696.26 = $16.91

Case A: The 2 year forward exchange rate is more than 0.6453, lets say it is 0.66:

In this case, the arbitrageur can

Borrow 1,000 USD at an interest rate of 7% for a period of two years and then

convert them into equivalent AUD and invest those AUD at 5%

Enter into a forward contract to sell 1782.53 AUD at an exchange rate of 0.66

1,000 USD when converted into equivalent AUD results in 1000/0.62 = 1,612.90 AUD

1,612.90 AUD are invested at an interest rate of 5% for a period of 2 years

Therefore, No. of AUD at the end of the 2 years is,


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= 1,612.90 * e^ (0.05*2)

=1, 782.53

Selling 1,782.53 AUD at the exchange rate of 0.66 will yield

= 1,782.53 * 0.66

= $1,176.5

No. of USD those are required to be returned

= 1,000 * e^ (0.07*2)

=$1,150.274

Therefore, Net profit (Risk less Profit) is,

Net Profit = $1,176.5 - $1,150.274

= $26.2

The above concept can be mapped to a foreign currency as an asset, providing a known

yield. A foreign currency can be regarded as an investment asset paying a known yield.

This yield is nothing but a risk free rate of interest in the foreign currency.

4.10. Futures on Commodities:

As discussed before, the assets are classified into investment assets and consumption

assets. When dealing about futures on commodities, lets first deal about the

investment assets such as Gold and Silver.

As a part of the hedging strategies of a gold producer, there is a requirement on the

part of the investment banks to borrow gold. Gold owners like central banks charges

interest in the form of Gold Lease Rate when they lend gold. Gold and Silver therefore

provide income to the holder.

In the absence of a storage costs and income, the forward price of a commodity that is

an investment asset is given by

F = S e^(r*T)


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Storage costs are always treated as a negative income. If U is the present value of all the

storage costs, during the life of a forward contract, then it is mathematically

represented as

F = (S + U) e^ (r*T)


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Summary:

An investment asset is an asset that is held for investment purposes by

significant number of investors.

A consumption Asset is an asset that is held for the purpose of consumption.

These assets are not usually held for investment purposes.

It is important to understand that, investment assets are not to be held

exclusively for investment purpose, rather than can also be used for industrial

purposes

Shorting usually involves selling an asset that is not owned

The investor has to maintain a Margin Account with the broker consisting of

either cash or marketable securities

The purpose behind maintaining margin account is to guarantee that the

investor will not default from the short position if there is a hike in the price of

the share.

The market participants are subjected to no transaction costs when they trade.

The market participants can borrow money at the same risk-free rate of interest

as they can lend money.

The market participants are subjected to same tax rate on all net trading profits.

The market participants take advantage of arbitrage opportunities as and when

they occur.

The risk-free rate, r, is the rate at which money is borrowed or lent when there is

no credit risk, so that the money is certain to be repaid

The easiest forward contract to value is one written on an investment asset that

provides the holder with no income.


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F = S e^(r*T) is the forward price of an investment asset that provides no income

F = (S-I) e^(r*T) is the forward price of an investment asset that provides a

known income

F = S e^ [(r-q)*T] is the forward price of an investment asset that provides a

known yield

f = S - K e^ (-r*T) is the value of forward contract with no income

f = S I K e^ (-r*T) is the value of forward contract with known income

f = [(S e^ (-q*T)) K e^ (-r*T)] is the value of the forward contract with known

yield.

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