Hull et al: Chapters 3 & 5

FINM7041 Applied DerivativesLecture 2 Forwards and Futures

Review of Previous Lecture

In last weeks lecture we went through abroad overview/revision of forward, futuresand options contracts.

In particular we focussed on themechanics of forward and futures markets.

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Lecture Overview In todays lecture we will discuss Forwards and

Futures contracts in greater detail, and how theyare related to the spot price of the underlyingasset. We will focus on the following topics: Determination of interest rates; What is short selling?; Determination of Forwards and Futures prices; and, Hedging strategies using Forwards and Futures

contracts.

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1. Consumption Vs. Investment Assets When considering forward and futures contracts, it is

important to distinguish between investment assets andconsumption assets. Investment assets are assets held by significant numbers of

people purely for investment purposes. Examples of investment assets are stocks, bonds, gold and

silver. Consumption assets are assets held primarily for consumption

and not usually for investment purposes. Examples of consumption assets are commodities such as

copper, oil and pork bellies. We can use arbitrage arguments to determine the forward and

futures price of an investment asset from its spot price and otherobservable market variables. We cannot do this for consumptionassets.

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2. Short Selling Short selling (also called shorting), involves selling an

asset that is not owned. It is not possible for all investment assets. Your broker borrows the securities from another client and sells them in

the market in the usual way. At some stage you must buy the securities back so they can be

replaced in the account of the client you originally borrowed them from. You must pay dividends and other benefits that would have accrued to

the client you borrowed from, if they had still held the shares. In otherwords, the client you borrowed from should be no worse off as a resultof lending you their shares.

Likewise, the client can be no better off. Therefore, if you borrow a physical asset such as gold off a client, the client

must pay you for the storage costs of the gold. The investor (the person who has shorted the asset) benefits if prices

fall, as they sell the asset for a higher price than what they buy it backfor.

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2. Short Selling

The investor is required to maintain a margin accountwith the broker.

The initial margin is required so that possible adversemovements (increases) in the price of the asset that isbeing shorted are covered.

The margin account consists of cash or marketablesecurities deposited by the investor with the broker toguarantee that the investor will not walk away from theshort position if the share price increases.

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2. Short Selling Regulators in the United States currently allow a stock to

be shorted only on an uptick that is, when the mostrecent movement in the price of the stock was an increase.

In Australia, only a limited number of stocks are allowed tobe short sold, called the ASX Approved Securities List.

Further, in 2008, we saw a ban on various forms of shortselling in markets around the world.

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3. Measuring Interest Rates The compounding frequency used for an interest rate is

the unit of measurement. From FINM7006, you are familiar with the need to

calculate the effective rate of interest. The difference between quarterly and annual

compounding is analogous to the difference betweenmiles and kilometres.

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3. Measuring Interest Rates In this course, we will mainly use continuous

compounding. As such you will need to ensure that your interest rates are

expressed as continually compounded interest rates. We will detail any exceptions to this rule as we progress through

the course. In the limit, as we compound more and more

frequently, we obtain continuously compoundedinterest rates.

For Example: $100 grows to $100eRT when invested at a

continuously compounded rate R for time T. $100 received at time T discounts to $100e-RT at time

zero when the continuously compounded discountrate is R.

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3. Measuring Interest Rates Example: If a nominal rate of 10% p.a. is

compounded continuously what is theeffective rate?

eR-1e0.10-1

2.718280.10-1= 10.51%

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4. Assumptions In this lecture we make the following assumptions

regarding market participants:1. They are subject to no transaction costs when they trade;2. They are subject to the same tax rate on all net trading

profits;3. They can borrow money at the same risk-free rate of

interest as they can lend money; and,4. They take advantage of arbitrage opportunities as they

occur.

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5. Forward and Futures Contract Prices

Remember from FINM7006: It is well known in practice that if interest rates are constant, a

futures contract has the same value as an otherwise identicalforward contract.

That is, although a futures contract has a complicated cashflow pattern (via the marking to market feature) it can bevalued as though it were a forward contract.

Since a forward contract has only a single cash flow, it is easyto value.

Consequently, it is industry practice to value futures contractsas though they were forward contracts.

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5. Forward and Futures Contract Prices

Remember that: Forward and futures contracts can be valued by

recognizing that, in many cases, forward and futuresmarkets are redundant. This occurs when the payofffrom a forward or futures contract can be replicatedby a position in:

1. The underlying asset; and,2. Riskless bonds.

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5. Forward and Futures Contract Prices

Before illustrating this concept, we define thecost of carry (q) of the underlying commodity. This is the cost of holding a physical quantity of the

commodity. For wheat the cost of carry is the storagecost; for live hogs it consists of storage and feedcosts; and for gold it consists of storage and securitycosts.

Some commodities have a negative cost of carry. Forexample, holding a stock index provides the benefit ofreceiving dividends.

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5. Forward and Futures Contract Prices Consider the strategy of:

Borrowing enough money to buy one unit of an investmentasset that provides the holder with no income, and has noholding costs. Non-dividend paying stocks and zero-couponbonds are examples of such investment assets. Theborrower incurs the obligation to pay for the associatedinterest through time T; and,

Entering into a forward or futures contract to sell thecommodity at time T.

The value of this position in terms of the initial (time 0)and terminal (time T) cash flows is tabulated in thefollowing table.

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5. Forward and Futures Contract Prices Arbitrage Relationship Between Spot and Forward

ContractsPosition Initial Cash Flow Terminal Cash

Flow

Borrow and incur

cost of carry

S0 -S0erT

Buy one unit of

commodity

-S0 ST

Enter 6-month

forward sale

0 F-ST

Net portfolio

value

0 F- S0erT

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5. Forward and Futures Contract Prices

Example: Consider a four-month forward contract to buy a zero-coupon

bond that will mature one year from today. Note: this means that the bond will have eight months to go

when the forward contract matures. The current price of the bond is $930. We assume that the

four-month risk-free rate of interest (continuouslycompounded) is 6% per annum. The forward price, F0, isgiven by:

40.0612

0 930 $948.79F e

= =

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6. Arbitrage In general if:

Arbitrageurs can make a riskless profit from buying the assetand entering into a short forward contract on the asset. Thisstrategy is financed by borrowing funds at the risk free-rate ifinterest.

Arbitrageurs can make a riskless profit by shorting the asset andentering into a long forward contract. The excess funds areinvested at the risk-free rate of interest until they are needed tobuy back the asset.

0 0rTF S e>

0 0rTF S e

( )0 0

r d TF S e