Ch 05 Hull Fundamentals 8 the d

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Fundamentals of Futures and Options Markets, 8th Ed, Ch 5, Copyright John C. Hull !"#

Determination of Forwardand Futures Prices

Chapter 5

1


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Consumption vs Investment Assets

Investment assets are assets held bysignificant numbers of people purely for

investment purposes (Examples: gold,silver)Consumption assets are assets held

primarily for consumption (Examples:copper, oil)

2


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Short Selling (Page 10510!"

Short selling involves sellingsecurities you do not on

!our bro"er borros thesecurities from another client andsells them in the mar"et in the

usual ay

3


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Short Selling (continued"

#t some stage you must buy thesecurities so they can be replacedin the account of the client

!ou must pay dividends and otherbenefits the oner of the securitiesreceives

$here may be a small fee forborroing the securities

Fundamentals of Futures and Options Markets, 8th Ed, Ch 5, Copyright John C. Hull !"#4


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#$ample

!ou short %&& shares hen the price is'%&& and close out the short position

three months later hen the price is '&uring the three months a dividend of '*

per share is paid

+hat is your profit+hat ould be your loss if you had bought

%&& shares



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%otation for &aluing Futures and

Forward Contracts


S&: Spot price today

F&: -utures or forard price today

T: $ime until delivery date

r: .is"/free interest rate formaturity T


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An Ar'itrage pportunit)*

Suppose that: $he spot price of a non/dividend/

paying stoc" is '0& $he */month forard price is '0* $he */month 1S' interest rate is 52

per annum

Is there an arbitrage opportunity



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Another Ar'itrage pportunit)*

Suppose that: $he spot price of nondividend/paying

stoc" is '0& $he */month forard price is 1S'* $he %/year 1S' interest rate is 52

per annum Is there an arbitrage opportunity



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+he Forward Price

If the spot price of an investment asset isS0and the futures price for a contract

deliverable in Tyears isF0, then

F0= S0(1+r)T

here ris the T/year ris"/free rate ofinterest3

In our examples, S0=40, T=0.25, and r=0.05

so that

F0

= 40(1.05)0.25 =40.5Fundamentals of Futures and Options Markets, 8th Ed, Ch 5, Copyright John C. Hull !"#9


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,hen Interest -ates are .easured

with Continuous Compounding

F0= S0erT

$his e4uation relates the forard priceand the spot price for any investment

asset that provides no income and hasno storage costs

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If Short Sales Are %ot Possi'le//

-ormula still or"s for an investment assetbecause investors ho hold the asset ill sell itand buy forard contracts hen the forard

price is too lo



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,hen an Investment Asset

Provides a nown Dollar Income

(page 111 e2uation 5/3"

F0= (S0I )erT

hereI is the present value of theincome during life of forard contract

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,hen an Investment Asset

Provides a nown 4ield(Page 113 e2uation 5/"

F0= S

0e(rq )T

here qis the average yield during thelife of the contract (expressed ith

continuous compounding)

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&aluing a Forward Contract

# forard contract is orth ero (exceptfor bid/offer spread effects) hen it is firstnegotiated

6ater it may have a positive or negativevalue

Suppose thatKis the delivery price andF&is the forard price for a contract thatould be negotiated today



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&aluing a Forward ContractPage 113115

7y considering the difference beteen acontract ith delivery priceKand acontract ith delivery priceF&e candeduce that: $he value,f, of a long forard contract is

(F& 8K)e8rT

the value of a short forard contract is

(KF0)erT



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Forward vs Futures Prices

+hen the maturity and asset price are the same, forardand futures prices are usually assumed to be e4ual3

(Eurodollar futures are an exception) +hen interest rates are uncertain they are, in theory,

slightly different: # strong positive correlation beteen interest rates and the asset

price implies the futures price is slightly higher than the forard

price # strong negative correlation implies the reverse



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Stoc6 Inde$ (Page 11!"

Can be vieed as an investment assetpaying a dividend yield

$he futures price and spot price

relationship is thereforeF0= S0e

(rq )T

here qis the dividend yield on the

portfolio represented by the indexduring life of contract

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Stoc6 Inde$ (continued"

-or the formula to be true it is importantthat the index represent an investmentasset

In other ords, changes in the index mustcorrespond to changes in the value of atradable portfolio

$he 9i""ei index vieed as a dollar numberdoes not represent an investment asset(See 7usiness Snapshot 53*, page %%)



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Inde$ Ar'itrage

+henF0 > S0e(r-q)Tan arbitrageur buys the

stoc"s underlying the index and sells

futures+henF0 < S0e

(r-q)Tan arbitrageur buys

futures and shorts or sells the stoc"s

underlying the index

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Inde$ Ar'itrage(continued"

Index arbitrage involves simultaneous trades infutures and many different stoc"s

;ery often a computer is used to generate the

trades


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Futures and Forwards on

Currencies (Page 117131"

# foreign currency is analogous to asecurity providing a yield

$he yield is the foreign ris"/freeinterest rate

It follos that if rf is the foreign ris"/

free interest rate


F S e r r Tf

0 0=( )


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#$planation of the -elationship

8etween Spot and Forward (Figure

5/1 page 119"


%&&& units offoreign currency

(time ero)

%&&&S&dollars

at time ero

%&&&S&erT

dollars at time T

T

e Trf

timeat

currencyforeign

ofunits%&&&

T

eF Trf

timeatdollars

%&&& &


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Consumption Assets: Storage is %egative Income

F0S0e(r+u )T

here uis the storage cost per unittime as a percent of the asset value3

#lternatively,

F0(S0+U )erThere Uis the present value of thestorage costs3



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+he Cost of Carr) (Page 13;"

$he cost of carry, c, is the storage cost plus theinterest costs less the income earned

-or an investment asset F0= S0ecT

-or a consumption asset F0S0ecT

$he convenience yield on the consumptionasset,y, is defined so that

F0= S0e(cy )T

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Futures Prices < #$pected Future

Spot Prices (Page 135137"

Suppose kis the expected return re4uired by investors inan asset

+e can investF0er T at the ris"/free rate and enter into a

long futures contract to create a cash inflo of STatmaturity

$his shos that


Tkr

T

T

kTrT

eSEF

SEeeF

)(

0

0

)(

or)(

=

=


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Futures Prices < Future Spot Prices (continued"


9o Systematic .is" k = r F& >E(ST)

?ositive Systematic .is" k > r F& @E(ST)

9egative Systematic.is"

k < r F& AE(ST)

?ositive systematic ris": stoc" indices9egative systematic ris": gold (at least for some periods)

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Ch 05 Hull Fundamentals 8 the d