Determination of Forward
and Futures Prices
Chapter 5
5.1
The Goals of Chapter 5

Background knowledge
– Investment vs. consumption assets, short selling
(賣空), and assumptions for market participants

Futures prices for investment assets
–
–
–
–

Adjustment for known dollar incomes or yields
Futures on stock indices and foreign currencies
Futures vs. forward prices
Valuing forward or futures contracts
Futures prices for consumption assets
– Convenience yield (便利殖利率) and cost of carry
theory (持有成本理論)

Futures price vs. expected spot price
5.2
5.1 Background Knowledge
5.3
Consumption vs. Investment
Assets


Investment assets are assets held for
investment purposes, e.g., stock shares,
bonds, currencies, gold, silver
Consumption assets are assets held for
consumption, e.g., copper, oil, pork, corn
※The no-arbitrage argument can (cannot) be used to
fully determine the forward and futures prices of
investment (consumption) assets
※Some investment assets, like gold or silver, have a
number of industrial uses and thus can be
consumed, so they are consumption assets as well
5.4
Short Selling

Short selling (賣空) involves selling securities
you do not own
– Your broker borrows securities from another clients
and sells them in a market on behalf of you
– Earn positive payoffs if the security price declines
– At some stage you must buy the securities and
return them back to the accounts of the clients who
lend you these securities
– You must pay dividends and any incomes that the
owners of the securities should receive in this short
selling period (The security owners feel as if they
continuously held these securities)
– There may be a small fee for borrowing securities
5.5
Assumptions for Market
Participants

Four assumptions associated with market
participants
– They are subject to no transaction costs when they
trade
– They are subject to the same tax rate on their net
trading profits
– They can borrow or lend money at the risk-free rate
with unlimited amount
– They take advantage of any arbitrage opportunity as
it occurs
5.6
5.2 Futures Prices for
Investment Assets
5.7
Theoretical Futures Price for
Investment Assets

The effect of the daily settlement of futures is
ignored and suppose the interest rate is
constant
– Under this simplified assumption, the forward and
futures prices are identical and used interchangeably

Suppose there is no income or storage costs for
the underlying asset of futures
– The spot price today is 𝑆0 , and the futures price
today for delivery in 𝑇 years is 𝐹0 . Chapter 1 shows
𝐹0 = 𝑆0 (1 + 𝑟)𝑇,
where 𝑟 is the risk-free interest rate expressed with
annual compounding
5.8
Arbitrage Example for Gold
Futures

Suppose that
–
–
–
–

Spot price of gold today is $1000
1-year gold futures price today is $1100 ($990)
The interest rate is 5% per annum
No income or storage costs for gold
The theoretical value of the futures price on
gold is $1,000×(1+5%)=$1,050
– Futures price > $1050  Buy the gold spot and take
a short position of the 1-year futures on gold
– Futures price < $1050  (Short) sell the gold spot
and take a long position of the 1-year futures on gold
5.9
When Interest Rates are Measured
with Continuous Compounding

The theoretical futures price expressed with
continuous compounding is
𝐹0 = 𝑆0 𝑒 𝑟𝑇 ,
where 𝑟 is the risk-free zero rate, with
continuous compounding, for the time to
maturity 𝑇
※This equation holds for any investment asset that
provides no income and has no storage costs, e.g.,
non-dividend-paying stocks
※In this course, we always use the formulae
expressed with continuous compounding
5.10
Consider a Known Dollar
Income of Investment Assets

When an investment asset provides a known
dollar income 𝐼𝑡 at time point 𝑡 ∈ (0, 𝑇], then
𝐹0 = (𝑆0 − 𝐼0 )𝑒 𝑟𝑇 ,
where 𝐼0 is the present value of the income 𝐼𝑡
※If there are multiple dollar incomes in (0, 𝑇], 𝐼0 is the
sum of the present values of them
– An intuitive way to understand this formula



You can treat 𝑆𝑡 as the stock price and 𝐼𝑡 as the cash
dividend payment at time 𝑡
It is known that after the payment of the cash dividend at 𝑡,
an identical amount is deducted from the stock price 𝑆𝑡
To reflect the above situation today, the PV of the cash
dividend payment 𝐼𝑡 , i.e., 𝐼0 , should be deducted from the
5.11
current stock price
Consider a Known Dollar
Income of Investment Assets

Suppose 𝑆0 = $900, an income of $40 occurs
at 4 months, and 4-month and 9-month rates
are 3% and 4% per annum. If the 9-month
futures price is $910 (or $870), is there any
arbitrage opportunity?
– The PV of the income at 4 months is
$40𝑒 −0.03∙(4/12) = $39.6
– The theoretical futures price is
𝐹0 = $900 − $39.6 𝑒 0.04∙ 9/12 = $886.6
※As long as the futures price deviates from this
theoretical price, there is an arbitrage opportunity
5.12
Consider a Known Dollar
Income of Investment Assets

For 𝐹0 = $910, which is overvalued than its
theoretical value
– At 𝑡 = 0



Borrow $900: $39.6 for 4 months and $860.4 for 9 months
Buy one unit of asset at 𝑆0 = $900
Enter into a short position of the 9-month futures (𝐹0 =
$910)
– At 𝑡 = 4 months


Receive $40 of income from the asset
Use this $40 (= $39.6𝑒 0.03∙(4/12) ) to repay the first loan
– At 𝑡 = 9 months



Sell the asset through the futures for $910
Use $860.4𝑒 0.04∙(9/12) =$886.6 to repay the second loan
Profit realized = $910 – $886.6 = $23.4
5.13
Consider a Known Dollar
Income of Investment Assets

For 𝐹0 = $870, which is undervalued than its
theoretical value
– At 𝑡 = 0



Short sell one unit of asset at 𝑆0 = $900
Invest $39.6 for 4 months and $860.4 for 9 months
Enter into a long position of the 9-month futures (𝐹0 = $870)
– At 𝑡 = 4 months


Receive $39.6𝑒 0.03∙(4/12) =$40 from the 4-month investment
Use this $40 to pay the lender of the asset
– At 𝑡 = 9 months




Receive $860.4𝑒 0.04∙(9/12) =$886.6 from 9-month investment
Buy the asset through the futures for $870
Return the asset to the lender
5.14
Profit realized = $886.6 – $870 = $16.6
Consider a Known Yield Income
of Investment Assets

When an investment asset provides a known
yield income 𝑞 (with continuous compounding)
in the period (0, 𝑇], then
𝐹0 = 𝑆0 𝑒 (𝑟−𝑞)𝑇
– An intuitive way to understand the formula


A yield income means that the income is expressed as a
percent of the asset’s price at the time the income is paid
Similar to the dollar income, whenever the yield income is
paid to the asset holder, there is a negative impact on the
asset price
– Suppose the asset price is 𝑆0 today, the expected annual
growth rate of the asset price is 𝜇, and the asset provides an
annual yield income of 𝑞
5.15
Consider a Known Yield Income
of Investment Assets
– Annual compounding:
𝑆1 = 𝑆0 × (1 + 𝜇) × (1 − 𝑞)
– Semiannual compounding:
𝜇
2
𝑞
2
𝑆1 = 𝑆0 × (1 + )2 × (1 − )2
– When the compounding frequency approaches infinity
𝑆1 = 𝑆0 × 𝑒 𝜇∙1 × 𝑒 −𝑞∙1
– Thus, the term 𝑒 −𝑞∙1 reflects the negative impact of the yield
income on the asset price for a year


If 𝑞 = 3%, 𝑒 −𝑞∙1 = 0.97045, which means an amount of 𝑆0 (1 −
𝑒 −𝑞∙1 ) = 𝑆0 (1 − 0.97045) is paid to the asset holder
Based on the original formula 𝐹0 = 𝑆0 𝑒 𝑟𝑇 , if the negative
impact of the yield income is considered, the formula
should be adjusted as 𝐹0 = (𝑆0 𝑒 −𝑞𝑇 )𝑒 𝑟𝑇 = 𝑆0 𝑒 (𝑟−𝑞)𝑇
– Note that the role of 𝑆0 𝑒 −𝑞𝑇 is similar to the role of 𝑆0 − 𝐼0 in
the futures price formula on Slide 5.11
5.16
Consider a Known Yield Income
of Investment Assets

The no-arbitrage argument for the formula of
𝐹0 = 𝑆0 𝑒 (𝑟−𝑞)𝑇
– If 𝐹0 > 𝑆0 𝑒 (𝑟−𝑞)𝑇




Buy 𝑁 units of the asset at 𝑆0 today by borrowing 𝑁𝑆0
dollars
Invest the continuously generated yield income in the
same asset, and thus by time 𝑇, it is expected to have
𝑁𝑒 𝑞𝑇 units of the asset
Enter into a futures to sell 𝑁𝑒 𝑞𝑇 units at 𝐹0
The final sales proceeds from the futures position, 𝑁𝑒 𝑞𝑇 𝐹0 ,
minus the repayment amount of the debt, 𝑁𝑆0 𝑒 𝑟𝑇 , can
generate a positive payoff, i.e.,
𝑁𝑒 𝑞𝑇 𝐹0 − 𝑁𝑆0 𝑒 𝑟𝑇 > 0 due to 𝐹0 > 𝑆0 𝑒 (𝑟−𝑞)𝑇
5.17
Consider a Known Yield Income
of Investment Assets
– If 𝐹0 < 𝑆0 𝑒 (𝑟−𝑞)𝑇




Short sell 𝑁 units of the asset at 𝑆0 today and deposit the
proceeds 𝑁𝑆0 in a bank to earn the interest rate 𝑟
Enter into a futures to buy 𝑁𝑒 𝑞𝑇 units at 𝐹0
When continuously generated yield income is paid on the
asset, the arbitrageur owes more on the short position. As
a result, the short selling position grows at the rate 𝑞 and
thus the arbitrageur needs to return 𝑁𝑒 𝑞𝑇 units at time 𝑇
The interest and principal of the deposit, 𝑁𝑆0 𝑒 𝑟𝑇 , minus the
fund to buy 𝑁𝑒 𝑞𝑇 units at 𝐹0 , 𝑁𝑒 𝑞𝑇 𝐹0 , can generate a
positive payoff, i.e.,
𝑁𝑆0 𝑒 𝑟𝑇 − 𝑁𝑒 𝑞𝑇 𝐹0 > 0 due to 𝐹0 < 𝑆0 𝑒 (𝑟−𝑞)𝑇
– To eliminate the above two arbitrage opportunities,
we can derive 𝐹0 = 𝑆0 𝑒 (𝑟−𝑞)𝑇 theoretically
5.18
Stock Index Futures

The underlying asset of stock index futures is a
stock index level, which can be viewed as an
investment asset paying a yield income
– A stock index reflects the performance of a virtual
portfolio of stocks
– It is infeasible to take the PVs of all cash dividends
of all stocks in this portfolio into account
– In practice, the continuous compounding dividend
yield is estimated for this virtual portfolio
– The futures and spot prices of a stock index futures
follows 𝐹0 = 𝑆0 𝑒 (𝑟−𝑞)𝑇 , where 𝑞 is the continuous
compounding dividend yields on the stock index
portfolio during the life of the futures contract
5.19
Index Arbitrage
When 𝐹0 > 𝑆0 𝑒 (𝑟−𝑞)𝑇 , an arbitrageur buys the
portfolio of stocks underlying the index and
takes a short position of stock index futures
(𝑟−𝑞)𝑇
 When 𝐹0 < 𝑆0 𝑒
, an arbitrageur takes a
long position of futures and (short) sells the
portfolio of stocks underlying the index
※The above two strategies are known as index
arbitrage and the details are similar to the
trading strategies introduced on Slides 5.17
and 5.18

5.20
Index Arbitrage

Index arbitrage involves simultaneous trades in
futures and many different stocks
– Very often a computer program is used to generate
the trades, which is known as program trading
– During a financial crisis, simultaneous trades could
be not possible and the no-arbitrage relationship
between 𝐹0 and 𝑆0 does not hold


On Oct. 19, 1987 (Black Monday), the S&P 500 index was
225.06 (down 57.88 on that day) and the futures price for
the Dec. S&P 500 index futures was 201.5 (down 80.75 on
that day)
The overloaded system on exchanges delays the execution
of orders and thus the index arbitrage becomes infeasible 5.21
Stock Index Futures

The futures contract on the Nikkei 225 Index
in CME views 5 times the Nikkei 225 Index,
which is measured in yen, as a dollar number
– Suppose you take a long position of the Nikki 225
index futures with 𝐹0 to be 1000, and on the
delivery date, the Nikki 225 index is 1100

Your payoff is USD$5×(1100 – 1000) = USD$500
– Note that traders cannot trade the stock index
portfolio underlying the Nikkei 225 Index in USD

The formula of 𝐹0 = 𝑆0 𝑒 (𝑟−𝑞)𝑇 cannot apply to Nikkei 225
index futures, which is a “quanto” futures (匯率連動期貨)
where the underlying asset is measured in one currency
and the payoff is in another currency
5.22
Futures and Forwards on
Currencies

A foreign currency is analogous to a security
providing a yield income
– The foreign risk-free interest rate is the yield income
an investor can earn if he holds that currency
– It follows that if the dividend yield is replaced with
the foreign risk-free interest rate, we can derive the
futures price as
𝐹0 = 𝑆0 𝑒 (𝑟−𝑟𝑓)𝑇


𝑆0 (𝐹0 ) is the spot (futures) price of the foreign currency in
terms of the domestic currency, i.e., the current exchange
rate (the exchange rate applied on the delivery date)
The 𝑟 and 𝑟𝑓 are the domestic and foreign risk-free interest
5.23
rates, respectively
Why the Relation Must Be True
(US$ is the Domestic Currency)
Enter into a foreign
currency futures to
sell 1000𝑒 𝑟𝑓 𝑇 units of
foreign currency at 𝐹0

※If 1000𝑒 𝑟𝑓 𝑇 𝐹0 ≠ 1000𝑆0 𝑒 𝑟𝑇 , an arbitrage opportunity occurs
※To eliminate all arbitrage opportunities, we can derive 𝐹0 = 𝑆0 𝑒 (𝑟−𝑟𝑓 )𝑇 5.24
Forward vs. Futures Prices


Forward and futures prices are usually
assumed to be the same
When interest rates are stochastic (隨機),
forward and futures prices could be different
due to the enforcement of daily settlement for
trading futures
– The difference between forward and futures prices
can be significant if there exists a relationship
between the interest rate and the underlying
variable, e.g., Eurodollar futures introduced in Ch. 6
5.25
Forward vs. Futures Prices
– A positive correlation between interest rates and
the asset price



With the increase of the asset price, the futures holder
can earn doubly from the increase of the balance of the
margin account and the higher interest rate
With the decrease of the asset price, the futures holder
losses, but the opportunity cost of fund for these losses is
low due to the lower interest rate
Thus, the futures contract is more attractive and
demanded so that futures price > forward price
5.26
Forward vs. Futures Prices
– A negative correlation between interest rates and
the asset price



With the increase of the asset price, the benefit of the
increase of the balance of the margin account will be
offset by the lower interest rate
With the decrease of the asset price, the balance of the
margin account decreases such that the futures holder
cannot fully enjoy the higher interest rate
Thus, the futures contract is relatively not attractive so
that futures price < forward price
5.27
Valuing a Futures or a Forward
Contract

Suppose that 𝐾 is the delivery price specified
in a futures contract and at time point 𝑡, 𝐹𝑡 is
the current futures price that would apply to
the futures contracts
– The value of a long futures contract at 𝑡 is
𝑓𝑡 = (𝐹𝑡 − 𝐾)𝑒 −𝑟(𝑇−𝑡) ,
and the value of a short futures contract at 𝑡 is
𝑓𝑡 = (𝐾 − 𝐹𝑡 )𝑒 −𝑟(𝑇−𝑡) ,
where 𝑟 is the risk-free interest rate corresponding
to the maturity of 𝑇 − 𝑡
※Note that the values of the long and short positions
are with the same magnitude but with opposite signs
5.28
Valuing a Futures or a Forward
Contract (for a long position)

Suppose that at 𝑡 = 0, 𝑆0 = 21.72, 𝑟 = 0.1, 𝑇 = 1,
and 𝐾 = 𝐹0 = 21.72𝑒 0.1∙1 = 24
⇒ 𝑓0 = 24 − 24 𝑒 −0.1∙1 = 0
※ In practice, when a futures is initiated, the delivery price is
set to the current futures price and thus the initial value of
the futures equals 0

Suppose that at 𝑡 = 0.5 (after half a year), 𝑆0.5 = 25,
𝑟 = 0.1, 𝑇 − 𝑡 = 0.5, and 𝐹0.5 = 25𝑒 0.1∙0.5 = 26.28
⇒ 𝑓0.5 = 26.28 − 24 𝑒 −0.1∙0.5 = 2.17
※ With the passage of time, as long as the futures price
(determined by the demand and supply of futures) does not
equal the delivery price, the value of the futures emerges 5.29
Theoretical Futures/Forward
Prices and Futures Values
Theoretical
futures/forward
prices (𝑭𝒕 )
Theoretical value of the
long position of a
futures/forward
(𝒇𝒕 = (𝑭𝒕 − 𝑲)𝒆−𝒓(𝑻−𝒕) )
𝐹𝑡 = 𝑆𝑡 𝑒 𝑟(𝑇−𝑡)
𝑓𝑡 = 𝑆𝑡 − 𝐾𝑒 −𝑟(𝑇−𝑡)
With known income whose
present value is 𝐼𝑡 at 𝑡
𝐹𝑡 = (𝑆𝑡 − 𝐼𝑡 )𝑒 𝑟(𝑇−𝑡)
𝑓𝑡 = 𝑆𝑡 − 𝐼𝑡 − 𝐾𝑒 −𝑟(𝑇−𝑡)
With known yield income 𝑞
𝐹𝑡 = 𝑆𝑡 𝑒 (𝑟−𝑞)(𝑇−𝑡)
𝑓𝑡 = 𝑆𝑡 𝑒 −𝑞(𝑇−𝑡) − 𝐾𝑒 −𝑟(𝑇−𝑡)
Asset
Without any income
※ The current time point is 𝑡, the maturity time point is 𝑇, and 𝐾 is the
delivery price of futures/forwards
※ If 𝑡 is 0 (at the beginning of a contract), the theoretical futures/forward
prices (𝐹𝑡 ) are identical to those introduced on Slides 5.10, 5.11, and 5.15
※ The formulae of the theoretical values of the futures/forward can be
obtained by replacing the futures/forward prices 𝐹𝑡 with the formulae in
the middle column
5.30
5.3 Futures Prices for
Consumption Assets
5.31
Futures Prices for Consumption
Assets

Commodities that are consumption assets rather
than investment assets usually provide no
income, but are subject to significant storage
costs
– The first way to model the storage cost:
𝐹0 = 𝑆0 𝑒 𝑟+𝑢 𝑇 ,
where 𝑢 is the storage cost per unit time as a percent
of the asset value (The arbitrage strategies on gold
leading to this formula are on Slides 1.35 to 1.37)
– Alternative way (for the one-time payment of storage
costs):
𝐹0 = (𝑆0 + 𝑈0 )𝑒 𝑟𝑇 ,
where 𝑈0 is the present value of the storage costs 5.32
Futures Prices for Consumption
Assets
– If 𝐹0 > 𝑆0 + 𝑈0 𝑒 𝑟𝑇
Borrow 𝑆0 + 𝑈0 at the risk-free rate and use it to purchase
one unit of the commodity and to pay storage costs
 Short a futures contract on one unit of the commodity
Lead to a positive payoff of 𝐹0 − 𝑆0 + 𝑈0 𝑒 𝑟𝑇  𝐹0 >
𝑆0 + 𝑈0 𝑒 𝑟𝑇 cannot hold

– If 𝐹0 < 𝑆0 + 𝑈0 𝑒 𝑟𝑇
Sell the commodity at 𝑆0 , save the PV of the storage costs,
𝑈0 , and invest the proceeds (𝑆0 + 𝑈0 ) to earn 𝑟 for 𝑇 years
 Take a long position in a futures contract
Lead to a positive payoff of 𝑆0 + 𝑈0 𝑒 𝑟𝑇 − 𝐹0
※Owners of a commodity are reluctant to do so because they
can consume the commodity but cannot consume the long
position of a futures or forward contract
5.33

Futures on Consumption Assets

Due to the concern for the need of using or
consuming commodities, the relationship
between the futures and spot prices of a
consumption commodity is
𝐹0 ≤ 𝑆0 𝑒 𝑟+𝑢 𝑇 ,
where 𝑢 is the storage cost per unit time as a
percent of the asset value, or is
𝐹0 ≤ (𝑆0 + 𝑈0 )𝑒 𝑟𝑇 ,
where 𝑈0 is the present value of the storage
costs
5.34
Convenience Yield and Cost of
Carry

The benefits from holding the physical asset are
referred to as the convenience yield (便利殖利率)
provided by the commodity. Denote the
convenience yield as 𝑦, then we can derive
𝐹0 𝑒 𝑦𝑇 = (𝑆0 + 𝑈0 )𝑒 𝑟𝑇 (⇒ 𝐹0 = (𝑆0 + 𝑈0 )𝑒
𝐹0 𝑒 𝑦𝑇 = 𝑆0 𝑒 𝑟+𝑢 𝑇 (⇒ 𝐹0 = 𝑆0 𝑒 𝑟+𝑢−𝑦 𝑇 )
𝑟−𝑦 𝑇 )
– The convenience yield reflects the concern of the future
availability of the commodity

The greater the possibility that shortages will occur, the higher the
convenience yield
– Use 𝐹0 = 𝑆0 𝑒 𝑟+𝑢−𝑦 𝑇 to explain normal and inverted
markets on Slides 2.28 and 2.29


If 𝑟 + 𝑢 > 𝑦  𝐹0 increases with 𝑇  normal market
If 𝑟 + 𝑢 < 𝑦  𝐹0 decreases with 𝑇  inverted market
5.35
Convenience Yield and Cost of
Carry

The relationship between futures and spot
prices can be unified in terms of cost of carry
(持有成本)
– The cost of carry, 𝑐, is the interest costs plus the
storage cost less the income earned, i.e., 𝑐 = 𝑟 +
𝑢−𝑞
– For an investment asset, 𝐹0 = 𝑆0 𝑒 𝑐𝑇
– For a consumption asset, 𝐹0 ≤ 𝑆0 𝑒 𝑐𝑇
– The convenience yield on the consumption asset,
𝑦, is defined so that 𝐹0 = 𝑆0 𝑒 (𝑐−𝑦)𝑇
※Note that for investment assets, the convenience
yield must be zero to eliminate arbitrage
opportunities
5.36
5.4 Futures Price vs.
Expected Spot Price
5.37
Relationship Between Futures Prices
and Expected Spot Prices

Is the futures price an unbiased estimate of
the expected spot prices on the delivery date?
1. Keynes and Hicks: Hedgers prepare to lose, but
speculators hope to make money

If hedgers hold short positions and speculators hold long
positions of futures, then 𝐹0 < 𝐸[𝑆𝑇 ]
– Speculators buy undervalued futures, i.e., 𝐹0 < 𝐸[𝑆𝑇 ], and
hedgers would sacrifice (to accept undervalued futures price)
in order to reduce risks

If hedgers hold long positions and speculators hold short
positions of futures, then 𝐹0 > 𝐸[𝑆𝑇 ]
– Speculators sell overvalued futures, i.e., 𝐹0 > 𝐸[𝑆𝑇 ], and
hedgers would sacrifice (to accept overvalued futures price)
in order to reduce risks
5.38
Relationship Between Futures Prices
and Expected Spot Prices
2. Analysis of the systematic risk based on the CAPM




Suppose 𝑘 is the expected return required by investors on
an asset with the price 𝑆𝑡
We can invest the amount of 𝐹0 𝑒 −𝑟𝑇 now (to earn the riskfree rate 𝑟) to get 𝑆𝑇 back at maturity via a futures contract
with the futures price 𝐹0
The present value of the expectation of 𝑆𝑇 (discounted by
𝑘) should be the initial investment 𝐹0 𝑒 −𝑟𝑇
𝐹0 𝑒 −𝑟𝑇 = 𝐸[𝑆𝑇 ]𝑒 −𝑘𝑇 ⇒ 𝐹0 = 𝐸[𝑆𝑇 ]𝑒 (𝑟−𝑘)𝑇
According to the CAPM, if the asset has
– no systematic risk (𝛽 = 0), then 𝑘 = 𝑟 and 𝐹0 = 𝐸[𝑆𝑇 ]
⇒ 𝐹0 is an unbiased estimate of 𝑆𝑇
– positive systematic risk (𝛽 > 0), then 𝑘 > 𝑟 and 𝐹0 < 𝐸[𝑆𝑇 ]
– negative systematic risk (𝛽 < 0), then 𝑘 < 𝑟 and 𝐹0 > 𝐸[𝑆𝑇 ]
5.39
Relationship Between Futures Prices
and Expected Spot Prices
※ (Normal) backwardation (逆價差) is the market condition
where the price of a forward or futures contract is trading
below the expected spot price, i.e., 𝐹0 < 𝐸[𝑆𝑇 ]
(Backwardation is the normal case in practice since for
most assets, they have positive systematic risk)
※ Contango (正價差) is the market condition where the price
of a forward or futures contract is trading above the
expected spot price, i.e., 𝐹0 > 𝐸[𝑆𝑇 ]
※ Note that in practice, the backwardation and contango are
sometimes used to refer to whether the futures prices is
below or above the current spot price, i.e., the
backwardation is for 𝐹0 < 𝑆0 and the contago is for 𝐹0 > 𝑆0
5.40