LECTURE WEEK 3 Deriv
LECTURE WEEK 3 Deriv
LECTURE WEEK 3 Deriv
DETERMINATION
OF FORWARD
AND FUTURES
PRICES
Learning structure
Concept review: compounding interest rates, short selling, price vs. value
Different versions of cost-of-carry models to determine fair forward price
Valuation of forward contracts
FIN3074 RISK MANAGEMENT APPLICATIONS OF DERIVATIVES BY RAFF
CONCEPT REVIEW ON INTEREST RATE
FREQUENCIES
To be unambiguous, an interest rate must be quoted with a compounding frequency, 10% annually compounded rate
gives a different outcome than 10% quarterly compounded /or continuously compounded frequency
For 10% annually compounded, $100 bank deposit grows to 100(1+10%)= $110 in 1 year’s time.
For 10% semi-annually compounded, $100 bank deposit grows to 100(1+10/2)2 = $110.25 in 1 year’s time.
For 10% quarterly compounded, $100 bank deposit grows to 100(1+10/4)4= $110.38 in 1 year’s time
When the limit of the compounding frequency m becomes infinity, it is known as continuous
compounding.
The Future value (FV) and Present value (PV) formula then becomes:
or
For the purpose of forward contract price determination (unless it is stated otherwise) we always use
continuous compounding or discounting rate to calculate the fair forward price.
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THE CONCEPT OF “PRICE” VERSUS “VALUE”
Price is what you pay, value is what you get, which is often computed as the Present Value (PV) of future cash-
flows / payoffs
Normally in an efficient market, price = value.
However, the markets are not efficient for various reasons, therefore, price ≠ value
In forward contracts, the forward price (F0) is the price fixed (locked in) today at T0 for a deferred date purchase
(F0 is not equal to current spot price S0 or the forward date spot price ST ).
Value of a forward contract is the value in the normal sense, i.e. PV of future payoff (Time value of money is
involved). Based on price concept F0 = FT as the forward price is fixed today and stays same until the delivery
date. Based on value concept S0 = PV (FT)
A forward contract is designed such that its VALUE is zero at the time the contracted is entered (because the
underlying asset/product is not consumed at time T0).
REVIEW ON SOME NOTATIONS
AND
ASSET CONDITIONS
TIME NOTATIONS AND PRICE NOTATION
INVESTMENT ASSET AND CONSUMPTION ASSET
E.g. Stocks and Stock Indices, Bonds, Gold and Silver etc.
We can use arbitrage arguments to determine the forward prices from its current spot price and other observable
market variables.
Consumption Assets: Assets held primarily for consumption purpose (not for investment)
E.g. Copper, Crude oil, Corn, Wheat….etc. used as raw material in production process.
We cannot use arbitrage arguments to determine the forward prices.
NOTATIONS
Time notations:
0 : Current time / today
T : Delivery (maturity) date of forward contract, expiry date of an option contract, expressed in terms of
decimalized fraction of the number of years (e.g. 3 month contract has T=0.25, 6 month contract has T = 0.5)
t : some time between 0 and T: 0 ≤ t ≤ T
0 t T
Price Notations:
S0 = PV (FT ) t1 t2 T = F0
We use the Cost-of-Carry model to identify the theoretical (no-arbitrage) forward price on an asset.
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COST-OF-CARRY MODEL
Cost-of-carry model is used to determine the fair (theoretical) forward price the holder in long position must pay to
the holder in short position at delivery date to compensate for:
Forward price = Spot price + cost-of-carry (i.e. cost of holding the asset minus any income
derived through holding the asset)
Depending on the nature of the underlying asset, the form of cost-of-carry may include some, or all of the following:
Financing cost (risk free interest rate)
Storage cost (space rental, insurance and other maintenance)
Less income ( dividend or benefits received) during the holding period.
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COST- OF- CARRY MODELS (ASSUMPTIONS)
Main assumption: Market does not permit arbitrage.
What is "arbitrage?"
A profit opportunity which guarantees net cash inflows with no net cash outflows.
Assumption is not that arbitrage opportunities can never arise, but that they cannot persist.
This is a minimal market rationality condition: it is impossible to say anything sensible about a market where such
(3) Investors can borrow and lend at the same rate which is risk-free rate
COST-OF-CARRY MODEL(1):
FORWARD PRICE FOR A NON-INCOME EARNING ASSET OR (ZERO COUPON BOND)
Assuming that the forward contract is written on a security that pays no income during the life time of the contract or
on a Zero coupon bond:
We use the Cost-of-carry model to determine the fair / theoretical / no-arbitrage forward price at T0
Cost of carry = ONLY financing cost = S0erT – F0
Then the link between the spot and fair forward price is: F0 = S0erT
Cost-of-carry is based on an arbitrage relationship: if the above relationship does not hold, subject to assumptions
discussed in previous slide, arbitrageurs will be able to make profit from the differences between F0 and FT prices.
FT is the forward price quoted by the counter party (seller / buyer) to transact at the maturity time.
If the F0 ≠F there is an arbitrage opportunity to be exploited
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THE FAIR FORWARD PRICING FORMULA
Fair-Forward price is the forward price that should be the rightful price or theoretical price and it should not create any arbitrage
possibilities.
To determine the no-arbitrage forward price through Cost-of-Carry model (1).
F0 = S0 erT
Suppose the current gold price is S0 = $350/oz., 3-month risk free interest rate is 8% p.a.; no holding costs involved. What should be
the 3-month forward price of gold (or the no-arbitrage forward price)?
F0 = 350e(0.08)(0.25) = $357.07
Suppose the quoted F > 357.07, lets say $370, then the forward is said to be overpriced (OP) relative to the spot price today by $370 –
T
357.07 = $12.93 (short the forward and long the spot)
Or suppose quoted the F0 < 357.07, lets say $340, then the forward is said to be underpriced (UP) relative to spot price today by $357.07 – 340
= $17.07 (long forward and short spot)
REMEMBER IN THIS CASE THE ASSET (GOLD) HAS ZERO INCOME. THEREFORE, THE ONLY COST IS
COMPOUNDED INTEREST COST
ARBITRAGE STRATEGY IN FORWARD CONTRACTS
When the FT is overpriced (i.e. FT > S0erT), the arbitrageur can make profit by
selling (short) the forward and buying (long) the spot.
When the FT is underpriced (i.e. FT < S0erT), the arbitrager can make profit by
buying (long)the forward and selling (short) the spot.
S0 FT
$370.00 (OVERPRICED)
$340.00 (UNDERPRICED)
Action Now: (Buy Spot, Sell Forward) Action Now: (Buy Forward, Sell Spot )
1) Borrow $40 at 8% for 6 months 1) Sell one unit of asset to realize $40 (spot)
2) Buy one unit of the asset for $40 (spot) 2) Invest $40 at 8% for 6 months
3) Enter into 6-month forward contract 3) Enter into a 6-month forward contract
to sell this asset for $43 to buy asset in for $38 (as fixed at inception)
Action in 6 months: Action in 6 months:
1) Sell asset for $43 (as fixed at inception) 1) Buy asset for - $38 (forward)
2) Use $41.63 to repay the loan with interest 2) Receive $41.63 from investment
( $40 e 0.08 x 0.5 = $41.63) ($40 e 0.08 x 0.5 = $41.63)
Profit realized = $1.37 = (43 – 41.63) Profit realized = $3.63 = (41.63 – 38)
Step 1: determine the PV of the dividend $5 that will be received in 3 months ( let the PV of the div. = I)
I = 5e -(0.10)(0.25) = $4.88 [ it is important to observe the timelines of the Div. payments]
Step 2: Thus, the arbitrage-free forward price F0 must satisfy
F0 = (S0 - I)erT = (95 – 4.88)e (0.10 x 0.5) = $94.74
Any other quoted forward price (either higher or lower than $94.74) leads to an arbitrage.
FIN3074 RISK MANAGEMENT APPLICATIONS OF DERIVATIVES BY RAFF
(given that the fair-forward price as calculated is $94.74) and the S0 = $95
Then, the forward is termed as overpriced ($98 > $94.74, relative to spot, so we should to sell
forward, borrow to buy spot).
Buying and holding the spot asset leads to a cash outflow of $95 today, but we will receive a coupon
of $5 in 3 months.
There are a few ways to structure the arbitrage strategy.
Here is one. We split the loan repayment of $95 into two parts, with
the balance repaid in six months with the delivery price received on the forward contract.
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THE ARBITRAGE STRATEGY ON EXAMPLE 2
Trades CF @ T0 CF @ 3mts. CF @ T
Net CF 0 0 $3.26
If the observed / quoted forward price of this stock is anything (higher or lower) than $51.39 will lead to an
arbitrage
Net cash flow + 1.30 0.00 Net cash flow 0.00 1.39
Either an arbitrage profit of $1.30 is realized now or 1.30e(0.08x 10/12) of $1.39 is realized at the maturity.
Either way the same arbitrage value can be exploited!
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COST-OF-CARRY MODEL (2A):
FORWARD PRICE FOR ASSET THAT PAYS YIELD (IN %)
If the security pays a known income which is expressed in terms of yield q (instead of dollar amount), the
cost of carry model becomes:
Example: Suppose that the risk-free interest rate is 10% per annum with continuous compounding and that the
dividend yield on a stock index is 4% per annum. Currently, the index is trading at 400 points, calculate the forward
price for a contract deliverable in four months
F0 = S0e(r – q)T
F0 = 400e(0.1 – 0.04)0.25 = 408.8
Any other index point quoted for FT (either higher or lower) than 408.8 will lead to an arbitrage.
Two points to note:
(1) the q % should always be lower than the r % in order to determine the F 0 in this model.
(2) The trading index is always given in points, to convert it to $ value you must multiply it with the multiplier $ of each point.
COST-OF-CARRY MODEL (3):
FORWARD PRICE FOR ASSET THAT INCURS STORAGE COSTS
If the asset incurs known storage cost, the storage costs can be treated as negative income that can add to the
current spot price. If U is the present value of all storage costs during the life of a forward contract, the cost-
of-carry model becomes:
The storage cost, insurance and other cost of holding the asset to the forward date could incur at different timelines
within the holding period: it may incur at the beginning or at the end of the contract or periodically during the holding
period, therefore it is important to calculate the PV(U) appropriately to the timelines of incurrence.
If the costs are expressed as percentage of spot price, the cost of carry model is:
CF @T0 CF at T CF @T0 CF at T
Save storage cost + 18.65 0.00 Save storage cost + 18.65 0.00
Invest PV (Ft) @ r - 484.84 + 520 Invest @ r - 518.65 + 556.26
Net cash flow $33.81 0 Net cash flow 0.00 $36.26
Either you can realize a profit of $33.81 now or 33.81e0.07 = $36.26 at the maturity
Either way the same arbitrage value to be exploited
There are cases where consumption assets are stored to have an immediate use in production process rather that holding an
equivalent derivative that may be delivered in a deferred date, e.g. oil refiners hold crude oil for production, heating oil is
stored to use in winter season, Christmas tree to be decorated on 25 th December or bunch of Roses for valentines day.
The benefit of owning the actual physical asset at the most needed time is translated as Convenience Yield.
Just like the dividend benefit in the Cost-of Carry model, Convenience Yield is a benefit that can reduce the forward price
F0.
▪ Convenience yield measures the extent to which there are benefits obtained from ownership of the physical asset that are not
obtained by owners of long forward contracts. The cost of carry is the interest cost plus storage cost less the income earned.
In this case the convenience yield is an income earned.
Let y be the convenience yield %. The cost-of-carry model then becomes:
F0 = (S0 + U)e(r – y)T
To be expressed as % yield, the equation becomes : F0 = S0 e(r + u – y)T
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Or
COST OF CARRY MODEL (4): CONCEPT CHECK
Assuming your firm is keeping (storing) a barrel of Heating oil to consume in the current winter
season. The current price per barrel of heating oil is $100. The storage and other associated costs
are 5% of the S0. The risk-free interest rate is 4% p.a. and the Convenience yield is 3.5% of the
S0. You are required to calculate the theoretical six-month forward price of this heating oil.
For all long forward contracts: based on the PV of the K and the S0 at the time of determination:
The value of a forward contract f will be: f = (F0 – K)e-rT that can be directly interpreted as: f = S0 – Ke
-rT