Lecture 2

Forward and Futures contracts

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 Spot markets and outright purchases

Suppose you wish to buy a share of firm XYZ ’s stock. This involves at least three
steps:

▶ set the price to be paid

▶ transfer cash from the buyer to the seller

▶ transfer the share from the seller to the buyer

In an outright purchase all three steps occur simultaneously

▶ Examples: cash purchase of coffee, sandwiches, books, ...

The markets for outright purchases are called spot markets, and the price at which
we trade in such market is called a spot price.

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 Forward contracts: exchange cash and assets at a later date
Definition: A forward contract initiated at time 0 is an agreement between two
counterparties to buy or sell certain amount (N0 units) of an underlying asset on a future
date T for a unit price F0,T .

Convention: it costs nothing to enter into a forward contract when initiated.

(However, the parties to an agreement can certainly choose any other arrangement, if
they all prefer it.)

Forward price F0,T : Under the convention, this price is set to ensure that the value
of the forward contract equals zero for both parties at the inception of the contract

Expiration (maturity) date T : date of the final purchase and payment

Notional value N0 × S0 : total market value of the underlying today, where S0 is the
spot price of the underlying (we saw some very big notional numbers in Lecture 1)

Recall from Lecture 1: Forward contracts are traded on OTC markets, not on
exchanges
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 Forward contracts: counterparties
Two counterparties:
▶ long position: the buyer of the underlying at expiration date T
▶ short position: the seller of the underlying at expiration date T

If St is the spot price of one unit of the underlying at t, and N0 is the size of the
contract (# of units), then the payoff to a contract at the maturity T is:
▶ payoff to long forward contract = N0 × (ST − F0,T )
▶ payoff to short forward contract = N0 × (F0,T − ST )

The forward is then a zero-sum game:

▶ If ST > F0,T , the buyer (long position) makes a profit (pays only F0,T instead of ST )
▶ If ST < F0,T , the seller (the short position) makes a profit (gets F0,T instead of ST )

Recall: in Lecture 1 we saw also a payoff of a stock. This was the cashflow from
selling the stock. It did not account for how much we paid to get the stock (we may
as well have stolen it). That payoff was always a non-negative number
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 Example: short selling
A share of firm XYZ’s stock is trading at S0 = $100, no dividends.

A hedge fund expects that the stock price will drop within the next year

Problem: XYZ cannot be shorted (no one lends, regulations, costs...)

Solution: short a forward contract which delivers one share of XYZ at T = 1. Let’s
say that the forward price is F0,T = $100.

If the hedge fund needs to put up $20 in cash as collateral, then:

Table 1: Payoff and return to hedge fund (note the leverage)

Two scenarios

Spot price ST $80 $120

Stock return −20% 20%

Payoff to the short forward $20 −$20

Return to hedge fund on the short forward 100% −100%

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 Example: synthetic zero-coupon bond
A hedge fund owns 10,000 shares of firm ZYX, which account for 10% of the total
equity of ZYX (that’s a lot!). Each share is trading at S0 = $100

The hedge fund wants to clear the entire position in ZYX and replace it with
zero-coupon bonds with maturity T = 1.8 years. The simple interest rate today is 5%.

Problem: large transaction costs, market liquidation issues, maturity mismatch with
standardized bonds traded on exchanges

Solution: short a forward contract which delivers N0 = 10, 000 shares of ZYX’s stock
at T = 1.8, with forward price F0,T = $109. (Assume no collateral is required)
Table 2: Payoffs

Value at time 0 Payoff time T

Stock N0 × S0 = $1 Mil. N 0 × ST

Short forward 0 N0 × (F0,T − ST )

Total: synthetic zero-coupon bond N0 × S0 = $1 Mil. N0 × F0,T = $1.09 Mil.

(We still don’t know where does the $109 come from. Will find it later.)
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 Example: synthetic zero-coupon bond (cont’d)
The future stock price ST is unknown today. But it is canceled out! So, the outcome
(i.e., the payoff N0 × F0,T ) is independent of ST and is risk-free (because we know it
at time 0). Therefore, we can also derive from it a risk-free rate (see this slide below).

We say ”synthetic”, because the outcome will be exactly the same if at time 0 the
asset manager (ignoring the liquidity issues, etc.) does the following
▶ sells N0 shares of stock ZYX at S0 = $100, and
▶ uses the proceeds to buy a zero-coupon bond at a simple interest rate r0S = 5%

Definition: the simple interest rate r0S corresponding to F0,T = 109 is

 
1 F0,T 1
r0S = −1 = (109/100 − 1) = 5% (comes from FV = S(1 + r0S T ))
T S0 1.8

Definition: the continuously-compounded rate r0 corresponding to F0,T = 109 is

 
1 F0,T 1
r0 = ln = ln (109/100) = 4.79% (comes from FV = Se r0 T )
T S0 1.8

Slide 13 below will clarify further, if you need it

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 Example: synthetic zero-coupon bond (cont’d)

Original Stock Exposure

Payoff

Total Position:
0
∙ 𝑆𝑇
Payoff Synthetic Zero-Coupon Bond

+ = 𝐹0,𝑇
∙
Short Forward Position
Payoff
𝐹0,𝑇
∙ 0
∙ 𝑆𝑇

0
∙ 𝑆𝑇

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 Finance 101: The no arbitrage principle
The idea: two securities with identical payoffs should have the same price in
well-functioning markets
▶ otherwise arbitrageurs will buy the cheap one and sell the expensive one. They will
keep doing that until the two prices are equal. In well-functioning markets this will
happen very fast.
▶ this is the Law of One Price which at the heart of derivative pricing

Definition: An arbitrage opportunity is a trading strategy that

▶ either costs nothing today and yields a profit in the future (this profit may not happen
always, but at least should be possible, and no loss should ever be possible),
▶ or yields a profit today, and has a guaranteed zero cash flows in the future

The definition says nothing about the magnitude of the profit. Why?

The value of derivative securities is determined by assuming no arbitrage

Is no arbitrage the same as saying that ”No risk-free profit should exist”?
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 The forward price F0,T
How to determine the forward price F0,T ? (or, how to get the $109 on slide 6?)

Replicating portfolio: replicates the payoff of the forward contract

Suppose there is a forward contract that buys N0 units of the underlying with forward
price F0,T , when the underlying has spot price S0

Investors can replicate the forward by borrowing zero-coupon bonds with face value
N0 × F0,T and using the proceeds to buy N0 units of the underlying

Table 3: Payoffs vs. market value at time 0 (present value)

Portfolio holdings Value at time 0 Payoffs at time T

Portfolio #1 Long forward 0 N0 × (ST − F0,T )

Long underlying N0 × S0 N0 × ST

Portfolio #2 Short bonds −N0 × F0,T × e −r0 ×T −N0 × F0,T

Total N0 × (S0 − F0,T × e −r0 ×T ) N0 × (ST − F0,T )

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 Replicating portfolio and no arbitrage

Deriving the forward price F0,T :

Portfolio #1 and portfolio #2 have identical payoffs at T

Therefore, they should have the same values today (no arbitrage)

Therefore, 0 = N0 × (S0 − F0,T × e −r0 ×T ) =⇒ F0,T = S0 × e r0 ×T

If F0,T > S0 × e r0 ×T :
Portfolio #2 is relatively underpriced, we go long portfolio #2 and short portfolio #1

Get a risk-free profit −N0 × (S0 − F0,T × e −r0 ×T ) > 0

If F0,T < S0 × e r0 ×T :
Portfolio #2 is relatively overpriced, so we short portfolio #2 and long portfolio #1

Get a risk-free profit N0 × (S0 − F0,T × e −r0 ×T ) > 0

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 Does the forward price predict the future spot price?

The forward price depends only on the interest rate (r0 ) and the spot price (S0 )

So it only uses information from the current spot markets

By definition, the expected log-return of the underlying (µ0 ) can be found from:

E0 [ST ] = S0 × e µ0 ×T

However, from the previous slide, the forward price is S0 × e r0 ×T

Therefore, the forward price typically differs from the expected future price

The difference is µ0 − r0 and is called the “risk premium”. It is usually positive if the
underlying is risky. (This is discussed at length in an Investments class.)

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 Calculating returns and expected returns
The price of an asset today is S0 and its price at a future time T is ST . The (simple)
return of this asset from today till time T is ret = STS−S
0
0
= SST0 − 1, and its annualized
return is T1 ret (we measure time in years, so T means T years from today)

 
ST
The log-return of this asset from today till time T is logret = ln S0 , and its
1
annualized log-return is T logret

Similarly, the expected return of this asset (estimated today) is E0 [ret] = E0S[S0 T ] − 1

The expected log-return (estimated today) is E0 [logret] = ln E0S[S0 T ] , and the

 

1
expected annualized log-return is T E0 [logret].

If we denote µ0 = T1 E0 [logret], then from the previous point we get

µ0 × T = ln E0S[S0 T ] . If we take exponents on both sides of this expression, we get
 

E0 [ST ] = S0 × e µ0 ×T , exactly as in the previous slide. (See also slide 7, which does
similar things for the risk-free return.)

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 The value of a forward contract
It costs nothing to enter a forward, so its initial value is 0. At maturity T , its value is
equal to the payoff (e.g., F0,T − ST for a short). But what is its value at any point t
between 0 and T ?

Consider a forward on a stock with no dividend (and assume constant interest rate):
▶ Our hedge fund was hedging a position of N0 shares of stock ZYX by shorting a
forward contract with forward price F0,T

▶ Assume that at time t the stock price is St (0 < t < T )

▶ If at time t the fund wants to get out of the forward contract, how much does it have
to pay?

The fund could cancel the original contract if the original counterparty (say, a bank)
agrees. But the fund can also find some new counterparty and enter with it into the
reverse forward contract, i.e., long a forward contract which also matures at time T .
This reverse contract will have today (i.e., at time t) a forward price Ft,T .

Regardless of whether the fund deals with the original counterparty or a new one, the
outcome must be the same (otherwise there is arbitrage).
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 The value of a forward contract (cont’d)

The payoff at T of the reverse forward contract is N0 × (ST − Ft,T ) and its forward
price today (time t) is Ft,T = St × e r ×(T −t) (exactly as on slide 11).

The final payoff at time T from the original short forward combined with the reverse
forward (whih is held long) is:
N0 × (F0,T − ST ) + N0 × (ST − Ft,T ) = N0 × (F0,T − Ft,T )

So, at time T the fund gets a risk-free payoff equal to N0 × (F0,T − Ft,T ).

Therefore, the value of the original short forward contract today equals the present
value of this risk-free payoff at time T , which is e −r ×(T −t) × N0 × (F0,T − Ft,T )

This is the value today of the original forward contract to our hedge fund (who was
the short side). What is its value to the long side?

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 Quiz questions
It is an unfortunate fact of life that some people steal things. They steal various
things, including stocks or bonds. Do you think they also steal forwards? Explain.

A non-dividend paying security is currently trading at 100. The interest rate is

positive. What is the fair compensation that you would require to be paid today if
you agree to sell one year from now this security for 50? (”Fair” means that in
expectation you neither gain nor lose from this trade.)
▶ Zero
▶ 50
▶ More than 50
▶ Less than 50

What does it mean to sell short? What does it mean to sell long?

The S&P 500 index spot price is 1100, the risk-free rate is 5% (continuously
compounded), and the dividend yield on the index is zero.
▶ If someone quotes a 6-month forward price of 1135, what do you do?
▶ If someone quotes a 6-month forward price of 1115, what do you do?
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 Answers to the last question
6
1: The theoretical price of the six-month forward is $1100 × e 5%× 12 = $1127.85. If the
forward price is $1135, we borrow $1100 to buy one share of the index, and short the
six-month forward. After six months, we deliver the share, receive the forward price and
use it to repay the loan. If ST is the spot price after six months, the payoffs are:
t=0 t = 6M
Borrow $1100 $1100 -$1127.85
Buy one share -$1100 ST
Short the forward 0 $1135 − ST
Total Payoff 0 $7.15

2: If the price of a six month forward contract is $1115, we short-sell one share of the
index and deposit the proceeds at the risk-free rate. We also buy the forward. After six
months, we pay the forward price, receive the share and cover the short (i.e., give the
share back to its original owner). The payoffs are:
t=0 t = 6M
Short one share $1100 −ST
Deposit (lend) risk-free -$1100 $1127.85
Long the forward 0 ST − $1115
Total Payoff 0 $12.85

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 Forward on a stock with a known dividend yield
Continuously-compounded dividend yield q (like interest rate on the stock)
▶ If you have 1 share at t = 0, then you will have 1 × e q×T shares at T .

▶ Or, if you have e −q×T shares at t = 0, then you will have 1 share at T .

▶ q is the “implied” continously-compounded dividend yield; does not mean that the
company is really paying dividend every day.

Holdings Value at time 0 Payoffs at T

Portfolio #1 Long forward 0 ST − F0,T

Long e −q×T share of stock S0 × e −q×T ST × 1

Portfolio #2 Short bonds with maturity T −F0,T × e −r ×T −F0,T

Total S0 × e −q×T − F0,T × e −r ×T ST − F0,T

Again from no arbitrage: S0 × e −q×T − F0,T × e −r ×T = 0

Thus, F0,T = S0 × e (r −q)×T .

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Forward on a stock with a known dividend yield (cont’d)
If F0,T > S0 × e (r −q)T :

▶ then short the forward with forward price F0,T and at the same time

▶ borrow S0 × N0 and buy N0 = e −q×T < 1 shares

Total number of shares at T : NT = N0 × e q×T = e −q×T × e q×T = 1

So, at T you have one share and no problem to cover your short forward position

Payoff at time T = F0,T − S0 × N0 × e r ×T = F0,T − S0 × e (r −q)×T > 0

What will you do if F0,T < S0 × e (r −q)T :

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 Futures contracts

Definition: Futures contracts are similar to forward contracts – they are agreements
between two counterparties to exchange a pre-specified amount of an asset at a
pre-specified time for a pre-determined price

The main differences between forwards and futures are:

▶ Futures are traded on an exchange
▶ Futures are standardized (with respect to size, maturity, underlying)
▶ The profits and losses of futures are marked to market (daily settlement)

Standardization is important for reducing liquidity risk. However, it may introduce a

possible mismatch between the futures’ contract size or maturity and the actual
needs of market participants

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 Futures contracts (cont’d)

Mark-to-market means that at the end of each trading day, the exchange calculates
traders’ losses, and if they are too large, the trader either receives a margin call which
asks him to post additional collateral, or his position is immediately closed. In fact,
no exchange in the U.S. has ever defaulted on its payments, and this is precisely
because they have controlled counterparty risk via mark-to-market.

Either counterparty can reverse its position at any time by closing out/reversing its
contract. Reversing means to take a position opposite to the original one

If some time in the past a trader took a long position in five S&P 500 contracts for
delivery on December 30, he can reverse today by taking a short position in five S&P
500 contracts for delivery on December 30

The exchange clearing house is the counterparty to both parties in the trade and the
clearinghouse itself does NOT take substantial risky positions. (Lehman Brothers’
bankruptcy would not have been so devastating if its various derivative positions had
been held at a clearinghouse.)

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 Various underlying assets
Futures contracts are available on:
▶ commodities, such as
⋆ softs: corn, wheat, soybean, cocoa, coffee, orange juice etc.
⋆ metals: gold, silver etc.
⋆ energies: crude oil, natural gas etc.

▶ currencies, such as
⋆ USD/EUR, USD/GBP, USD/AUD, EUR/AUD etc.

▶ equity indexes, such as

⋆ S&P 500, NASDAQ 100, RUSSELL 2000, etc.

▶ interest rate products, such as

⋆ Eurodollars, Treasury bonds and notes, etc.

▶ weather products, such as

⋆ heating degree in Atlanta, Chicago, New York, Las Vegas etc.

▶ energy, such as electricity etc.

For more information, see the CME website: http://www.cmegroup.com/

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 Futures prices
Futures price Fb0,T = price at which the trader who shorts the futures agrees to sell
the underlying asset to the trader who buys (is long) the futures. The agreement
happens at time 0, and the sale (delivery) happens at time T . On any subsequent
day (before T ), the futures price is Fbt,T .
▶ As the futures price moves, profits and losses accrue to both counterparties
▶ The party who agrees to buy the asset at maturity gains when price increases, and loses
otherwise. For any day t before the maturity T the daily P&L (Profit and Loss) of the
buyer is:  
Daily P&L = Contract Size × Fbt,T − Fbt−1,T

▶ The party who agrees to sell the asset at maturity (i.e., the short) gains when price
decreases, and their P&L is the negative of the above. (So what one party gains the
other loses – again a zero-sum game.)

Settlement and Delivery:

▶ Physical vs. cash settlement
▶ Futures are referred to by their delivery months, specified by the exchange
▶ Delivery of the asset can occur only during these months and only on specific dates
▶ Delivery must occur at a given location, also specified by the contract
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 Margin requirement
Initial margin: When entering a futures contract, both counterparties must set up
collateral with the exchange. This is what allows the exchange to control their risk!!

As P/L accrue on an account, the margin may erode over time

▶ If the amount in the trader’s account falls below a maintenance margin, the exchange
issues a margin call
▶ The trader must post additional collateral to restore the initial margin if he wants to
keep the position open. Otherwise the position is closed by the exchange

The initial and maintenance margins are set by the exchange. CME: “Margins are
adjusted frequently across all of our products based on market volatility. When daily price
moves become more volatile, we typically raise margins to account for the increased risk.
Likewise, when daily price moves become less volatile, margins typically go down because the
risk of the position also decreases.”

At the close of each day traders pay losses (out of their margin accounts) or receive
gains (credited to their margin accounts). This is the daily ”mark to market”.

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 Mark to market example

Today is 09/10/2023 and you just bought on the CME 10 contracts of September 2023
S&P 500 futures (which mature on 09/20/2023). The S&P 500 is at $4,000 and the
futures price is $4,200. The contract size is 250 units of the S&P 500 index (CME
convention)

Notional value (for 10 contracts):

$10, 000, 000 = 10 × 250 × $4, 000

On the expiration date (09/20/2023), you must buy 2,500 units of the S&P 500
index.

No money change hands today (same as for the forward contracts!)

Suppose your initial margin is $3,000,000 and the maintenance margin is $2,600,000

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 Mark to market (cont’d)
Suppose that tomorrow the futures price falls to $3,800

This brings a loss for the long position:

▶ the exchange takes out of your margin account:

($4, 200 − $3, 800) × 250 × 10 = $1, 000, 000

▶ the exchange deposits this amount to your counterparty’s margin account

Your margin account at the end of tomorrow’s trading has a balance of

$3, 000, 000 × e r ×∆t − $1, 000, 000 = $2, 000, 000
where the annualized interest rate r is assumed here to be zero for simplicity (i.e.
r = 0) and ∆t = 1/365 (one day)

Note: The margin account is lower than the maintenance margin requirement!

$2, 000, 000 (margin account) < $2, 600, 000 (maintenance margin)
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 Mark to market example (cont’d)

Sometime before the open on the following day you receive a margin call

The exchange asks you to deposit (immediately) new funds into your margin account.
They ask for $1,000,000

▶ Note: they ask you to restore the initial margin

You can choose to NOT meet margin call. If you do that, the exchange will close
your long position, and will refund to your account the remaining amount in your
margin account of $2,000,000

This is how, by using mark-to-market, the exchange ensures that the value of an
existing futures position does not become too negative at any trading day (and this is
the one major difference between futures and forward contracts).

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 Example

The XYZ index is today at 950, the initial margin is 10%, and you buy 10 futures
contracts. You earn a continuously-compounded interest rate of 6% on your margin
balance, your position is marked to market weekly, and the maintenance margin is 80% of
the initial margin. Also, today’s futures price is the same as the spot price.

(a) What is the notional value of your position? What is the initial margin in dollars?

(b) What is the highest XYZ index futures price 1 week from today at which you will
receive a margin call?

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 Example (answer)
(a) Since each contract is for 250 units of the XYZ index, the notional value of the
position is $950 × 10 × 250 = $2, 375, 000. The initial margin is $237, 500.

1
(b) Your margin account grows to $237, 500 × e 6%× 52 = $237, 774.20 after a week. A
margin call occurs when the futures price drops for the first time to X (we need to
find this X . In case the exchange does not call you at X , which is possible, then you
can get a margin call for any price below X ).
▶ The maintenance margin is $950 × 8% × 250 × 10.

▶ This must be higher or equal to $237, 774.20 − ($950 − X ) × 250 × 10, which is the
amount in the margin account after marking your loss to market.

▶ The X comes from solving the equation:

950 × 8% × 250 × 10 = $237, 774.20 − ($950 − X ) × 250 × 10. Therefore, X = $930.89.

Btw, can you find some inconsistency in this example? (it of course does not affect the
correct answer for X that we just got)

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