Download - 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
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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
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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
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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
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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
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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
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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
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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)
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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
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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?
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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.)