2_Forward and Futures

Lecture 2:Pricing Forwards and Futures

This lecture studies the pricing of forward and futurescontracts. We first focus on the similarities of thecontracts and derive pricing formulas from marketequilibrium and the no-arbitrage principle. We thenanalyze the differences between the contracts.

I. No-Arbitrage Principle

II. Forward and Futures Pricing

A. Stocks without DividendsB. Stocks with Discrete Fixed DividendsC. Stocks with Continuous Dividend YieldD. Foreign Currencies (FX)E. CommoditiesF. Treasury Bills

III. Summary of Pricing Formulas

A. Financial Forwards and FuturesB. Costs and Benefits to Holding the SpotC. All Forwards and FuturesD. Value of the Forward Contract

Pricing Forwards and Futures

The Big Picture

Lecture is about forwards and futures, but we’re reallylearning a way of thinking

� Pricing technique we’ll use over and over

– Payoff replication

– Price the replicating portfolio

� The nature of derivative securities

– Risk transfer devices

– How to trade risks!

� They usually payoff in cash

� But we often think like we’re trading the actualrisks

Bus 35100 Page 2 Robert Novy-Marx


I. No-Arbitrage Principle

An arbitrage opportunity is any trading strategy thatdoes not require a cash input but has some positiveprobability of making profits without risking a loss.

Examples:

� Zero price and strictly positive payoff

� Negative price and non-negative payoff

Arbitrage opportunities are free lunches.They don’t exist in today’s financial markets.We will call this property NA = no arbitrage.

Important: arbitrage strategy must be riskless. Azero-cost strategy that generates positive returns onaverage does not violate NA.

If such opportunities arise, some investors will tradeon them. As a result, the opportunity disappears.How?



The Law of One-Price

� NA implies the “law of one price”

– assets with the same payoff have the same price

� We’ll typically be employing the law of one price toget the NA price of the assets that interest us

– We’ll look for a portfolio of assets that has exactlythe same payoffs as the derivative of interest

– We’ll price the derivative by pricing the portfolioand invoking the law of one price

� We’ll call a portfolio of assets that has exactly thesame payoffs as the derivative

– The replicating portfolio

– Or the “synthetic” derivative

� This portfolio is most intuitive with forwards (orfutures)

– That’s the main reason we start with these



Note: To get “tight bounds” (i.e., exact prices) fromNA pricing, the following must hold for someone:

� No transaction costs

� No short-sale costs

� No borrowing or short-sale constraints

If the above don’t hold, we only get “NA bounds”

� For example, with transaction costs two assetswith the same payoffs can differ in price by asmuch as the cost of undertaking the arbitrage

– The cost of setting up the “arbitrage portfolio”

� The cost of buying the cheap asset and sellingthe expensive one

We’ll typically ignore these (important) real worldconcerns

� Things will be complicated enough as it is

– For a first pass, at least



II. Pricing Forward and Futures

Intuition: The forward price should be the price of“holding the spot” until maturity

� If the price is higher than the cost of holding thespot until delivery, no one will buy forward

– It’s cheaper to get the underlying today

� Understand this, and you’ll understand forwards!

A. An Example: Stocks without Dividends

� Suppose you hold 10,000 shares of stock pricedat $100/shares.

� You can either sell today or commit to selling insix months.

� The risk-free interest rate is 4%, and the termstructure is flat

– Annualized continuously compounded

– I.e., the CC risk-free rate between today andany other month in the future is 4%.

� What is the 6-month forward/futures price of thestock?



Forward price should be the cost of “holding the spot”

F(t, T ) DS(t) � er�(T �t)

D$1M � e0.04�0.5

D$1, 020, 200

Immediate Sale for $1,000,000

Delayed Sale for $1,020,200

$1,000,000

today

today

6 months

6 months

Earn 4% (ann., c.c.) in the bank $1,020,200

$1,040,000$1,000,000 $1,020,200

That is, the forward stock price is $102.02/share.



The No-Arbitrage Argument

The forward/futures price must equal $102.02/share.

� What if the price is $103.00/share?

Transaction Payoff at t Payoff at T



Therefore, if

F(t, T ) 6D S(t) � er�(T �t)

then there is an arbitrage opportunity.



B. Stocks with Discrete Fixed Dividends

Most stocks pay dividends. Assume dividends, orcosts (exceptional cash leaving the firm), are fixedand are paid at known times.

Example Continued ...

The stock pays a $1 dividend in 3 months. What isthe 6-month forward/futures price?

Now, two factors determine the relationship betweenthe spot and the forward/futures price:

� The cost (including financing) of the immediatepurchase.

� Benefits or costs of holding the security betweennow and the delivery date.

Both affect the cost of “holding the spot”



Equilibrium Argument

The stock holder will capture the dividend:


$1,000,000

today

today

6 months

6 months

$1,020,200

$1,010,100

$1,000,00010,100

$1,020,200(earn interest )

3 months

$10,000

Delay Sale and Receive Dividend

Earn 4% (ann., c.c.) in the bank


Forward/Futures Price =

FV [ Current Price - PV(Dividends) ]

F(t, T ) D [S(t ) � PV (D)] er�(T �t)

D�

S(t) � D � e�r�(tD�t)�

� er�(T �t)

D�

$1M � $10, 000 � e�0.04�0.25�

� e0.04�0.5

D $1, 010, 100

D $101.01=share



No-Arbitrage Argument

The forward/futures price must equal $101.01/share.


Transaction Payoff at t Payoff at tD Payoff at T


Therefore, if

F(t, T ) 6D�

S(t) � PV�

D��

� er�(T �t)




Comments on Interest Rates

- On page 9, we made the assumption that the termstructure is flat:

r(t, T ) D r for any T

Usually, however, interest rates depend on thematurity of the bond.

- The convention is to measure time in years. Forexample, T � t D 1=12 is 1-month maturity.

T � t 1/12 2/12 3/12 4/12

r(t, T ) 3% 3.5% 4% 4.5%

- When the term structure is not flat, the formulas forthe forward/futures price become

F(t, T ) D S(t)er (t,T )(T �t)

F(t, T ) D�

S(t) � D(tD)e�r (t,tD)(tD�t)�

er (t,T )(T �t)

- Bond prices are

B(t, T ) D e�r (t,T )(T �t)



C. Stocks with Continuous Dividend Yield

Assume dividends, or costs, are proportional to theprice of the security and are paid continuously.

Example: 6-month forward/futures on the S&P 500

� Suppose:

– The spot price is $900

– The C.C. risk-free rate is rrr D 4%

– The annualized C.C. dividend yield is ııı D 3%

What is the price of the 6 month future if the and ?


The “underlying security” is not 1 share of the index.If you buy 1 share of the index, hold it, and reinvestthe dividends proportionally in the index, you willhave

eı�(T �t) D e0.03�0.5 D 1.01511

shares of the index at maturity.



Instead, the “underlying security” is e�ı�(T �t) sharesbecause at maturity e�ı�(T �t) shares yield:

e�ı�(T �t) � eı�(T �t) D 1 share of the index

The cost of the underlying security is:

S(t) � e�ı�(T �t) D $900 � e�0.03�0.5

D $886.60

Therefore, the index forward/futures price is:

F(t, T ) De�ı�(T �t) � S(t) � er�(T �t)

DS(t) � e(r�ı)�(T �t)

D$904.51356




The forward/futures price must equal $904.51.

� What if the price is $905.00?



Therefore, if

F(t, T ) 6D S(t) � e(r�ı)�(T �t)




D. Foreign Currencies (FX)

Foreign currency forwards/futures are very similar toindex forwards/futures. A unit of foreign currency canbe viewed as a stock that pays a continuous dividendyield equal to the foreign interest rate.

Covered Interest Rate Parity:

After adjusting for exchange rate differentials, interestrates must be equal across countries.

Example: Suppose rUS D 7.41%, rSwiss D 8.87%, and

the exchange rate is $0.6667/SF, or SF 1.5/$.

What is the price of a 4-month SF forward/futures?

The “underlying security” is not 1 Swiss Franc.Instead, it is SF e�rSwiss�(T �t).




U.S. Deposit

$1,000,000

August December

e 0.0741/3=1.025 $1,025,000

SFr 1,500,000

Swiss Deposit

SFr 1,545,000

$1,000,000 $1,025,000

$0.6667 / SFr

or

SFr 1.50 / $

$0.6634 / SFr

August

August

December

December

e 0.0887/3=1.03

$0.6667 / SFr

or

SFr 1.50 / $

Forward/Futures Price =

Spot Exchange Rate �

�

US FV factorSwiss FV factor

�

F(t, T ) D S(t) �

�

erUS�(T �t)

erSwiss�(T �t)

�

D S(t) � e(rUS�rSwiss)�(T �t)

D $0.6667 � e�0.0146=3 D $0.6634




The forward/futures price must equal $0.6634.





Therefore, if

F(t, T ) 6D S(t) � e(rUS�rSwiss)�(T �t)




� Does this mean we expect the price four monthsfrom now to be $0.6634?

� No!

– Covered interest parity is enforced by NA

– It doesn’t tell us where we expect prices to go

� In fact, in practice if tomorrow’s rate is lower thantoday’s it is just as likely the rate will rise

– And if tomorrow’s rate is higher than today’s,then it’s just as likely the rate will fall

� This fact partly drives the “carry trade”



E. Commodities Cost-of-carry is the physical storage

cost of a commodity – the cost of spoilage, the costof insurance, etc.

The cost-of-carry increases the forward/futures price,relative to the spot price.

� If we know the cost-of-carry until maturity U , wetreat it as a negative dividend:

F(t, T ) D�

S(t) C PV (U )�

� er�(T �t)

� Otherwise, we just assume that the cost-of-carry isproportional to the value of the commodity and ispaid continuously at a rate u. That is, we treat it asa negative dividend yield:

F(t, T ) D S(t) � e(rCu)�(T �t)



Question: Does the cost-of-carry have to bepositive?

� Yes, but only semantically.

� We call it a “Convenience yield” if it’s “negative”

– The convenience yield can only be positive.

Owning gold provides a convenience yield:

� In February 1998

– Spot price of gold in May 1997: $343.10

– 8 month interest rate (annualized): 5.7%

– You could get a 0.63% annualized yield rentingout gold

� To the shorts!

) futures price D 343.1(1 C 0.057 � 0.0063)8=12

D 354.60

� A convenience yield lowers the futures price

– It makes owning the spot relatively attractive



In General:

� Costs

– financing costs (r )

– transportation costs

– storage costs (“cost-of-carry”)

� Benefits

– dividends, coupons, etc.

– lending fees (“convenience yield”)

� Benefits to holding the spot can outweigh the costs

– Then you’ll see futures prices below spot rates

� The futures price lower than the price forimmediate delivery

� A downward sloping forward price curve

– In this case prices are said to be “backwardated”

� When the forward price curves slopes up, wesay it’s “in contango”



F. Treasury Bills

Pricing forwards or futures on T-Bills is like pricingforwards or futures on a non-dividend paying stocks.

� The underlying is a T-Bill that matures at TD C TMTD C TMTD C TM ,where

– TD is the delivery date in the future, and

– TM is the time-to-maturity for the delivered T-Billspecified in the contract.

� Why is this the underlying?

– It’s a bill that matures in TM at delivery.

– That is, if you buy it today and hold it for TD it’s abill maturing in TM .

So what’s the forward price?

� It’s the cost of buying and holding to delivery

– as always!

� But what’s the spot price?

– It’s the cost of the underlying:

B(t, t C (TD C TM)) D $100 � e�(TDCTM )r

t,tC(TDCTM )



Example:

What is the 4-month forward price of a 3-monthTreasury bill?

Assume that:

� the 4-month interest rate is 4%, and

� the 7-month interest rate is 5%?.

Both rates are annualized, c.c. rates.


T-Bill purchase for $97.13

$100 • e-0.05•7/12 =

$97.13

t

(=August)

T+3

(=March)

$100

Delayed Sale (June Futures Contract) for 98.42

$100

T

(=December)

t

(=August)

F(t,T)

-F(t,T)

F(t,T) • e-0.04•4/12 =

$98.42 • 0.9868 =$97.13

T

(=December)

T+3

(=March)



That is, the standard forward/futures vs spotrelationship can be used to price a Treasury billforward/futures.

Forward/Futures Price = FV(Spot Price)

F(t, T ) D $100 � e�0.05�7=12 � e0.04�4=12

D $97.13 � 1.0134

D $98.42


If the forward price isn’t $98.42, there’s arbitrage;these two transactions both pay $100 in 7 months:

� Buy a $100 Treasury bill maturing in 7 months.

� Buy a Forward with delivery K, and K face valueT-Bills maturing at delivery (in four months).

If K 6D 98.42, then buy low and sell high.

In general, F(t, TD) D $100 � eTDr

t,TD�(TDCTM )r

t,TDCTM



Comments on Treasury Bill Quotes:

� Treasury bill prices are quoted as an annualizeddiscount based on a 360-day year:

Quoted Yield D (100 � Price) �360

M

where M is the number of days to maturity.

� Treasury bill futures prices are also quoted as anannualized discount:

Quoted Price D 100 � (100 � Price) �360

M

or

F(t, T ) Quote D 100 � [100 � F(t, T )] �360

M



III. Summary of Pricing Formulas

A. Financial Forwards and Futures

Market prices are given by:

F(t, T ) D V (t) � er (t,T )�(T �t)

where

F(t, T ) D Forward/futures price.

r(t, T ) D Continuously compounded risk-free

interest rate between times t and T .

V (t) D Value of the “right” underlying

D Amount of money required at time t

for a strategy that generates one unit

of the underlying security at time T .

� The trick is always to find V (t)



The “right underlying” or strategies V (t) are:

� Stock without dividends

Buy one share at time t and hold until time T :

V (t) D S(t)

� Stock with known dividends

Buy one share at time t , borrow the presentvalue of the dividends, pay off the debt with thedividends, and hold until time T :

V (t) D S(t) � PV(D)

� Stock with known dividend yield ı

Buy e�ı�(T �t) shares at time t , reinvest thedividends in stock, and hold until time T :

V (t) D S(t) � e�ı�(T �t)



� Foreign currency (FX)

Buy e�rforeign�(T �t) units of the currency at time t ,earn foreign interest, and hold until time T :

V (t) D S(t) � e�rforeign�(T �t)

� Treasury bill with maturity T �

Buy one Treasury bill with maturity T � at time t andhold until time T :

V (t) D S(t) D e�r (t,T �)�(T ��t)



B. Costs and Benefits to Holding the Spot

� Costs to holding the spot raise the forward/futuresprice relative to the spot price.

Intuition:

With a higher cost, investors would rather not holdthe spot. The forward/futures price must thereforeincrease until investors are indifferent.

� Benefits to holding the spot lower the forward/futures price relative to the spot price.

Intuition:

With a higher benefit, investors would rather holdthe spot. The forward/futures price must thereforedecrease until investors are indifferent.



C. All Forwards and Futures

Market prices are given by:

F(t, T ) D�

S(t)CPV (U )�PV (D)�

�e(r (t,T )Cu�ı�y)�(T �t)

where

S(t) D Current spot price.

r(t, T ) D Continuously compounded risk-free interest

rate between t and T .

PV (U ) D PV of known cost-of-carry between t and T .

PV (D) D PV of all known dividends between t and T .

u D Known cost-of-carry rate between t and T .

ı D Known dividend yield between t and T .

y D Known convenience yield between t and T .

Note: In practice (and on exams) we usually use

either U or u and either D or ı.



D. Value of Forward Contract

The “stock price” is the value of the stock; the“forward price” is not the value of forward contract.

What is the value of a forward contract?

� Assume no costs or benefits to holding the spot.

� To find the value of a forward contract afterinitiation (f (t )), compare the following:

(i) Long one forward contract worth f (t ) at time t .

(i i) Long one unit of the underlying security worth S(t)

at time t and K � e�r�(T �t) of borrowing.

Both positions pay off S(T ) � K at time T , and musthave the same value at time t . Therefore, the valueof a forward contract at time t is:

f (t ) D S(t) � K � e�r�(T �t)

� The value of the replicating portfolio.

Question: Is the forward/futures price of a stock agood predictor of the stock’s future spot price?


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2_Forward and Futures