FNCE 30007
Derivative Securities
Lecture – Forward & futures pricing
Outline
Introduction
Arbitrage
Investment v consumption assets
Short selling
Investment assets
Simple relation (no income)
Known income (bond futures)
Known yield (stock index futures, currency futures)
Consumption assets
Valuation
Environmental, social and governance movement
Futures & expected spot prices
Reading: Chapter 5
Introduction: No arbitrage pricing
Lecture 1 considered speculators, lecture 2 considered
hedgers.
We now focus on the other major participant –
arbitrageurs
Arbitrage takes advantage of a price differential between
two or more markets
No up front payment and a positive cash flow later
Arbitrage possible if
Same asset does not trade at the same price on all
markets (the law of one price); or
2 assets with identical cash flows do not trade at the
same price; or
An asset with a known price in the future does not
today trade at its future price discounted at the risk-
free interest rate
Introduction: Forward v Futures prices
Forward and futures prices are usually equal.
We therefore price futures contracts as if they are
forwards
Greatly simplifies the problem
Slight differences may arise from
Taxes
Transaction costs
Margins
Futures more liquid & have no counterparty risk
Introduction: Investment v Consumption assets
Investment assets: held by significant numbers of
people purely for investment purposes (gold, silver,
stocks, bonds).
Can price forwards & futures off spot via
arbitrage.
Consumption assets: held primarily for consumption
& not usually for investment purposes (copper, oil,
pork bellies, soybeans).
Not able to price forwards & futures off spot via
arbitrage.
Introduction: Short selling
Short selling: sell securities you do not own
Your broker borrows the securities from another
client and sells them in the spot market
At some stage you buy the securities back & return
them
If the price falls: sell high, buy low.
You pay dividends & other benefits to the owner of
the securities.
Diagram
Introduction: Assumptions & notation
Assume:
No transaction costs
Same tax rate
Borrowing/lending at risk-free rate
No-arbitrage opportunities (arbitrage opportunities
taken advantage of immediately).
S0: Spot price today
F0: Futures or forward price today
T: Time until delivery date (years)
r: Risk-free interest rate for maturity T
(continuously compounded rate p.a).
Investment assets - Simple pricing
The relationship between F0 and S0 is
If F0 > S0erT, arbitrageurs buy the asset
and short forward contracts
If F0 < S0erT, arbitrageurs short the asset
and buy forward contracts.
rTeSF 00 =
Investment assets - Example 1
Suppose:
The spot price of gold is US$600
The quoted 1-year futures price of gold is
US$650
1-year US$ interest rate is 5% p.a
No income or storage costs for gold
Is there an arbitrage opportunity?
Investment assets - Example 2
Suppose:
The spot price of gold is US$600
The quoted 1-year futures price of gold is
US$590
1-year US$ interest rate is 5% p.a
No income or storage costs for gold
Is there an arbitrage opportunity?
Short sale constraint
What if short sales are not possible for all
investment assets?
Short selling not needed for the forward
price equation.
Only require that at least some people hold
the asset only for investment purposes.
If the forward price is low, they sell the asset &
take a long forward position.
Investment assets - known $ income
Investment asset has income during the life of a
forward contract:
F0 = (S0 – I)erT
where I = present value of the income during life of
forward contract
If F0 > (S0 – I )erT, arbitrageurs buy the asset and
short the forward contract.
If F0 < (S0 – I )erT, arbitrageurs short the asset and
buy the forward contract.
Example: known $ income
Consider a long forward contract to buy a coupon
bond with a current price of $900.
-The forward contract has a 9 month maturity & a
coupon payment of $40 in 4 months.
-The continuously compounded 4 and 9 month risk-
free rates are 3% and 4% p.a
The forward contract price should be $886.60.
To show why we will consider 2 situations
i) where the futures is overvalued at $910.
ii) where the futures is undervalued at $870.
Example: known $ income (i)
The forward price is $910. Arbitrageur will
Today:
Borrow $900. The coupon payment has a present
value of 40e(-0.03)(4/12) = $39.60. So, $39.60 is
borrowed for four months and the rest ($860.40) is
borrowed for nine months (at 4%).
Buy the bond
Enter into a forward contract to sell the asset for $910.
In four months:
Receive $40 coupon payment
Use $40 to repay first loan with interest
In nine months:
Sell bond & receive $910 under the terms of the
forward contract.
Use $886.60 to repay second loan with interest.
Profit: $910 – $886.60 = $23.40
Example: known $ income (ii)
Forward price is $870. Arbitrageur will
Today:
Short the bond ($900)
Enter into a forward contract to buy the bond for $870
in nine months.
Of the $900 realized from shorting the bond, $39.60 is
invested for four months at 3% per annum (grows to
$40). The remaining $860.40 is invested for nine
months at 4% per annum (grows to $886.60)
In four months:
Receive $40 from four-month investment
Use $40 to pay coupon on the bond
In nine months:
Receive $886.60 from nine-month investment
Buy the bond for $870 under the terms of the forward
contract
Close out short position in the bond
Profit: $886.60 – $870 = $16.60
Investment assets - known yield
If the asset underlying a forward contract
has a known yield
F0 = S0e(r–q)T
where q = average yield over the life of the contract
Two examples
Stock index futures
Currency futures
Known yield – stock index futures
Investment asset paying a dividend yield
The price relationship is
F0 = S0 e(r–q )T
where q = dividend yield (over life of contract) on
the portfolio represented by the index
Index must represent an investment asset
changes in the index must correspond to changes
in the value of a tradable portfolio
Known yield – stock index futures
F0 > S0e(r-q)T arbitrageur buys the stocks
underlying the index and sells futures
F0 < S0e(r-q)T an arbitrageur buys futures
and sells the stocks underlying the index
Index arbitrage involves simultaneous
trades in futures & many different stocks
Very often computer used to generate
trades
Known yield – stock index futures
Lets consider the E-mini S&P500 futures contract
S&P500 close Feb 3/2020 3225.52
4 week US T-bill rate 1.53%p.a
Dividend yield 1.79%p.a
Contract specs reveal expires 3rd
Friday of the month. Therefore
34 trading days prior to expiration
T=34/252
[ ]3225.52exp (0.0153 0.0179) 34 252 3224.39oF = − × =
Known yield – currency
Foreign currency provides a continuous
yield - the foreign risk-free interest rate rf
Underlying asset is 1 unit of foreign
currency.
F S e r r Tf0 0=
−( )
Known yield – currency example
2year interest rates in Australia and the U.S. are 5%
and 7%. Spot exchange rate is 0.62 USD per AUD.
Find the two year forward rate.
Given that fx rate is 1 unit of foreign currency,
we consider from perspective of US investor.
Can also redo from the perspective of an
Australian investor.
Consider when 2year forward rate is 0.63 & 0.66.
Consumption assets
F0 ≤ S0 e(r+u )T
where u is the storage cost per unit time as
a percent of the asset value.
Alternatively,
F0 ≤ (S0+U )erT
where U = present value of the storage
costs.
Consumption assets - example
Consider a 1 year copper futures contract.
Assume no income and that it costs $2 per ounce
per year to store copper, with payment being
made at the end of the year.
The spot price is $600 and the risk-free rate is
5% p.a for all maturities.
Find the futures price.
Consider when futures price is $700 & $610
Consumption assets – convenience yield
Convenience yield is the benefit from holding
the physical asset.
Reflects market expectations of future availability
High inventory expectations low conv yield
Low inventory expectations high conv yield
Convenience yield is
F0eyT = (S0+U )erT
If U is proportional to the spot price
F0eyT = S0e(r + u)T
Example
Cost of carry
Cost of carry, c, is the storage cost plus the
interest costs less the income earned
Investment asset F0 = S0ecT
Consumption asset F0 ≤ S0ecT
Convenience yield on consumption asset, y,
is defined so that
F0 = S0 e(c–y )T
Forward valuation
The value of a futures contract is zero - value reflected in
the margin account.
K is delivery price in a forward contract entered into
previously. F0 is forward price that would apply to the
contract today.
Value of a long forward contract, ƒ, is
ƒ = (F0 – K )e–rT
Value of a short forward contract is
ƒ = (K – F0 )e–rT
Forwards have a value of zero at the time first entered
into because F0=K.
As time passes, the forward price and the value of the
contract change.
Forward valuation: example
Consider a long forward contract on a non-dividend
paying stock entered into some time ago that has
5 months left to maturity. The risk-free rate with
continuous compounding is 9% p.a. The current
stock price is $30 and the delivery price is $28.
The value of the contract is:
F0 = 30e(0.09)(5/12) = $31.15
ƒ = (F0 – K )e–rT = (31.15 – 28)e(-0.09)(5/12) = $3.03
or
ƒ = S0 – Ke–rT = 30 - 28e(-0.09)(5/12) = $3.03
Will use this in the BSM lecture
ESG
Environmental, social and governance (ESG)
movement is now emerging in the derivatives
markets.
ESG companies typically less exposed to
environmental and regulatory tail risks (Value at
Risk lecture)
Eurex and Nasdaq have launched ESG futures
Similar plans for many other markets
China will soon open an exchange solely dedicated
to the trading of carbon finance futures
ESG
Eurex
Futures on the STOXX Europe 600 ESG-X index
Index screens out companies with low ESG rankings
Index enables investors to easily switch portfolio to an ESG
compliant benchmark with low cost and tracking error
Futures on the index now traded
Futures can be used for hedging and speculative purposes
This should add liquidity to the underlying index
Nasdaq
In October 2019 Nasdaq launched a futures based on the
OMXS30 responsible index
Also excludes companies with poor ESG standards
Futures Prices & Expected Spot Prices
So far we have focused on the contemporaneous
relation between F0 and S0.
What about the relation between F0 and E (ST )?
If
F0 = E (ST ) F0 unbiased estimate of ST
F0
F0>E (ST ) contango