Forward and Futures Pricing
Forward and Futures Pricing
Forward and Futures Pricing
Week 2
MSc in Finance and Investment
Semester 2: 2009-10
Peter Moles
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
Week 2: Derivatives Forward & Futures Pricing
Futures
Any amount
Specific dates
Traded OTC
(normally 4 times a
year)
Futures contracts
Available on a wide range of underliers
Exchange traded
Margin (performance bond)
Clearing house
Standardization
Means contracts are fungible, hence liquid
Settled daily
Gains 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
Conversion factor
Cheapest to deliver
Stock index futures
$ value per index point
S&P futures $250 index
Cash settled
Week 2: Derivatives Forward & Futures Pricing
Seller
Defers sale
Gains any income from underlier
Loses interest advantage
Pays storage, wastage, etc.
Cost
Costofofcarry
carry(CC)
(CC)principle
principle==
equilibrium
equilibrium(fair
(fairvalue)
value)
model
modelwhich:
which:
(a)
(a)Incorporates
Incorporatesall
all
key
keypricing
pricingvariables
variables
(b)
(b)Equates
Equatesbenefits
benefitsand
and
costs
of
buyer
costs of buyerand
andseller
seller
Simple interest
(money markets)
PV (1 + rsT ) = FV T
PV (1 + r ) = FV T
T
Compound interest
(standard textbook)
Continuous interest*
PV e rcT = FVT
(Derivatives pricing)
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S0
r
T
FT
S 0 e r (T ) = FT
With simplest CC model, only the time value of money
(cost of borrowing) matters
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S 0 e (r q )(T ) = FT
1. Continuous dividends
q = continuous dividend yield
2. Known dividend(s)
D = dividend paid at time t
(T > t)
[S De ( ) ]e ( ) = F
r t
r T
Storage costs?
u = storage costs as a yield
Convenience yield?
y = unobservable convenience yield
Week 2: Derivatives Forward & Futures Pricing
S0e r (T ) De r (t ) = FT
S0e (r +u )(T ) = FT
S0e (r +u y )(T ) = FT
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F0 = S0ecT
For a consumption asset
F0 S0ecT
The convenience yield on the consumption asset, y, is defined so that :
F0 = S0 e(cy )T
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Suppose that
K is delivery (contracted) price in a forward contract &
F0 is forward price that would apply to the
contract today
The value of a long forward contract, , is
L = (F0 K )erT
Similarly, the value of a short forward contract is
fS = (K F0 )erT
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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 asset
Sell overpriced one
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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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Basis risk
Basis is the difference between spot & futures
S F = Basis
Simple basis =
Carry (theoretical) basis =
Value basis =
SF
S F*
F F*
[ F* = S0erT ]
Basis risk arises because of the uncertainty about the basis when the futures
contract is closed out
NB mostly affects hedging transactions
Basis Weakens
Basis Strengthens
Short
Returns < 0
Returns > 0
Long
Returns > 0
Returns < 0
Weaker basis:
Stronger basis:
Value of
hedge
position
T futures expiration
Week 2: Derivatives Forward & Futures Pricing
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Variation in basis
T
Actual basis
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 = 5.5
Carry basis (S F*) = 100 105 = 5.0
Value basis (F F*) = 105.5 105 = 0.5
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Cross-asset positions
Rounding error
Variation margin
Timing mismatches
Basis risk
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11
Briefly explain using an appropriate example what is meant by a crossasset 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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