Derivatives 2

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FORWARD AND FUTURES PRICING

Week 2MSc in Finance and InvestmentSemester 2: 2009-10Peter Moles

Week 2: Derivatives – Forward & Futures Pricing 2

Today’s key ideas

Mechanics of forwards and futures

Cost of carry

Pricing via replication

Arbitraging & other issues with forward and futures prices

Remember, the price that people agree to in the pit is not the price that people think is going to exist in the future. It's the price that

both sides vehemently agree won't be there.- Jeffrey Silverman

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1. Forwards & Futures Mechanics


Forwards vs. futures

ForwardsAny amountAny maturity dateTerms and conditions as negotiatedAny counterpartyCancellation by mutual consentNo mark-to-marketNo margin

Traded OTC

FuturesSpecific contract amountSpecific dates (normally 4 times a year)

Standard terms and conditionsClearing houseHas to be sold (repurchased)Daily mark-to-marketInitial and variation margin required

Traded on an organized exchange

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Futures contracts

Available on a wide range of underliers

Exchange tradedMargin (performance bond)Clearing houseStandardization

– Means contracts are fungible, hence liquid

Specifications need to be defined:What can be delivered,Where it can be delivered, & When it can be delivered

Settled dailyGains and losses paid in/out of margin account


Financial futures

Fixes the price, exchange rate, interest rate or stock index level at which a financial transaction will occur at a future date

currency futures

short-term interest rate futures (money markets)

medium and long-term interest rate futures (bond markets; swaps)

stock index futures

miscellaneous

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Exchange pricing mechanisms

Currency futuresExchange rate pair ($ per unit of foreign currency)€125 000 per contract

Short-term interest rate futuresInterest rate index = (100 – interest rate)Notional deposit = $1mm (e.g. eurodollar futures)Cash settled

Government bond futuresnotional bond with x% couponActual bonds used for delivery

– Conversion factor– Cheapest to deliver

Stock index futures$ value per index pointS&P futures $250 × indexCash settled

2. Pricing Forwards & Futures

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Cost of carry model [I]

BuyerDefers purchaseGains interest advantageLoses any income from underlier

SellerDefers saleGains any income from underlierLoses interest advantagePays storage, wastage, etc.

Cost of carry (CC) principle =equilibrium (fair value)

model which:(a) Incorporates allkey pricing variables(b) Equates benefits andcosts of buyer and seller

Cost of carry (CC) principle =equilibrium (fair value)

model which:(a) Incorporates allkey pricing variables(b) Equates benefits andcosts of buyer and seller


Interest rate calculations

Simple interest(money markets)

Compound interest(standard textbook)

Continuous interest*(Derivatives pricing)

( ) Ts FVTrPV =+1

( ) TT FVrPV =+1

TTr FVePV c =

( )[ ]rrc += 1ln*Note that

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Cost of carry model [II]

Model at its simplestNo income lost (value leakage)No wastageNo storage costs

S0 = spot price of underlierr = risk-free interest rate to time TT = time to maturity (expiration) of contractFT = forward price with maturity at time T

( )T

Tr FeS =0

With simplest CC model, only the time value of money (cost of borrowing) matters


Cost of carry model [III]

What of value leakage?1. Continuous dividends

– q = continuous dividend yield

2. Known dividend(s)– D = dividend paid at time t

(T > t)

Storage costs?– u = storage costs as a yield

Convenience yield?

– y = unobservable convenience yield

( )( )T

Tqr FeS =−0

( )( )T

Tyur FeS =−+0

( )( )T

Tur FeS =+0

( )[ ] ( )

( ) ( )T

trTr

TTrtr

FDeeS

FeDeS

=−

=−

∑

∑ −

0

0

Dividends are either present valued and subtracted

Or future valued and subtracted

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The Cost of carry [IV]

The cost of carry, c, is the storage cost plus the interest costs less the income earned

For an investment asset

F0 = S0ecT

For a consumption asset

F0 ≤ S0ecT

The convenience yield on the consumption asset, y, is defined so that :

F0 = S0 e(c–y )T


Valuing a forward contract

Suppose that K is delivery (contracted) price in a forward contract & F0 is forward price that would apply to the

contract todayThe value of a long forward contract, ƒ, is

ƒL = (F0 – K )e–rT

Similarly, the value of a short forward contract is

fS = (K – F0 )e–rT

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3. Arbitrage & Other Issues


Principles of arbitrage

Classical arbitrage involves simultaneous purchase and sale of asset in two markets (see Week 1 class)

Rule is:Buy the underpriced assetSell overpriced one

Cash-futures arbitrage operates on same principleCash and carry = sell futures, buy cash instrumentReverse cash and carry = buy futures, sell cash instrument

Profit is made when prices come into line or contract expires

Scalping involves arbitraging (exploiting) very short-term price differences

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Index arbitrage

When F0>S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futures

When F0<S0e(r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index

Index arbitrage involves simultaneous trades in futures & many different stocks

Very often a computer is used to generate the trades (program trading)

Occasionally (e.g., on Black Monday) simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0 and S0 may not hold


Factors affecting arbitrage profits

Transaction costs

Short selling restrictions

Limits on borrowing funds

Unequal borrowing and lending rates

Interest received or paid in marking-to-market futures contracts

Note: these are either ‘imperfections’ in the standard model or ‘real-world frictions’ which the model does not take into account

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Basis risk

Basis is the difference between spot & futuresS – F = Basis

Simple basis = S – FCarry (theoretical) basis = S – F* [ F* = S0erT ]Value basis = F – F*

Basis risk arises because of the uncertainty about the basis when the futures contract is closed out

NB mostly affects hedging transactions

Note there is no basis risk if futures contract held to expiration

(as futures contract price converges to cash market price)


Effect of basis change on hedge performance

Hedge Position and Return

Type of Hedge Basis Weakens Basis Strengthens

Short Returns < 0 Returns > 0

Long Returns > 0 Returns < 0

Weaker basis: cash price increases less or falls more than futures price

Stronger basis: cash price increases more or falls less than futures price

T – futures expiration

Value ofhedgeposition

Positive basis = S – F > 0

Negative basis = S – F < 0

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Variation in basis

If the spot price = 100, cost of carry value is 105, and actual value of the futures contract is 105.5 then:Simple basis (S – F) = 100 – 105.5 = –5.5Carry basis (S – F*) = 100 – 105 = – 5.0Value basis (F – F*) = 105.5 – 105 = 0.5

TB

asis

Variation in actual basisto the (theoretical) carry basis

Actual basis

Carry basis


Risks from hedging with futures

Cross-asset positions

Underlying position and futures contract are not the same leading to potential differences in performance

Rounding error

Requirement to transact in whole contract amounts leads to slight over (under) hedging

Variation margin

Margin flows on futures position can cause uncertainty in hedging

Timing mismatches

Gains and losses on the two sides may not match

Basis risk

Problem of variable convergence

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Case 3: The basis in futures contracts

1 Briefly explain using an appropriate example what is meant by a cross-asset or cross-market spread in the futures market and why market participants would want to establish such a position.

2. The cash market price of an index is 4625 and the 3-month futures price is 4595. The index has a dividend yield of 4.20 per cent. The risk-free interest rate is 3.95 per cent. What is the ‘raw basis’, the ‘carry basis’, and the ‘value basis’? Is it possible to undertake an arbitrage transaction if transaction costs are one per cent of the cash market value of the index?


Case 4: Arbitrage in futures markets

1 Using an appropriate example, explain the steps required by a market arbitrageur to exploit mispricing between the cash market and the futures market. What are the difficulties involved?

2 The party with a short position in a futures contract sometimes has options as to the precise asset or underlier that can be delivered, where the delivery will take place, when the deliverywill take place, and so on. How do these delivery options, as they are known, affect the futures price? Explain your reasoning.

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Let’s design a new futures contract on the weather

Let’s see if we can put together a contract that allows us to hedge or speculate on the weather!

(Talking about the weather is a very common topic of conversation in the UK.)

Documents

Derivatives 2