pricing of f&f

7/25/2019 pricing of f&f

1/39

Determination of Forward

and Futures Prices


2/39

Consumption vs Investment Assets

Investment Assets:

That is held for investment purposes by significantnumbers of investors.

!"amples: stoc#s$ bonds$ gold$ silver% Consumption Assets:

That is held by primarily for consumption.

!"amples: copper$ oil$ por#%


3/39

&hort &elling

&hort selling involves selling securitiesyou do not own

'our bro#er borrows the securities

from another client and sells them inthe mar#et in the usual way (e)uired to maintain a margin account

with the bro#er 'ou must pay dividends and other

benefits the owner of the securitiesreceives


4/39

Cash flows form short sale and

purchase of shares

Purchase of shares

April : Purchase *++ shares for ,-+ /,0+$+++

1ay : (eceive dividend 2,*++

3uly : &ell *++ shares for ,-++ per share 2,*+$+++

4et profit5 /,6$*++

/////////////////////////////////////////////////////////////////////////////

&hort sale of shares

April : 7orrow *++ shares and sell them for ,-+ 2,0+$+++ 1ay : Pay dividend / ,*++

3uly : 7uy *++ shares for ,-++ per share /,*+$+++

(eplace borrowed shared to short position

4et profit5 2,6$*++


5/39

Assumption and 4otation

Assumption:

-.4o transaction costs when they trade.

.The same ta" rate on all net trading profits. 8.7orrow money at the same ris#/free rate

of as they can lend money.

9.Ta#e advantage of arbitrage opportunitiesas they occur.


6/39

Assumption and 4otation

S+: Price of the asset underlying theforward or futures contract today

F+: Futures or forward price today

T: Time until delivery date

r: (is#/free interest rate for maturity T

4TATI4:


7/39

Forward Price For an Investment Asset

Assume : &+ 5 ,9+$r 5 *;$t 5 8 months

a%If F+ 5,98 < &+ert

-.7orrow ,9+ at ris#/free interest rate of *; per annum.

.&hort a forward contract to sell one share in 8/months.

,9+e+.+*"8=-

5 ,9+.* ,98 / ,9+.* 5 ,.*


8/39

b%If F+5,86 > &+ert

-.&hort one share$ invest the proceeds of the short sale

at *; per annum for 8 months.

.Ta#e a long position in a 8/months forward contract.

,9+e+.+*"8=- 5 ,9+.*

,9+.* / ,86 5 ,-.*

?e deduce that for there to be no arbitrage the forward

price must be e"actly ,9+.*.


9/39

F+= S+erT

This e)uation relates the forward price and the

spot price for any investment asset that provides

no income


10/39

?hat If &hort &ale Are 4ot Possible@

a%If F+< &+ert

-.7orrow &+dollars at an interest rate r for T years.

.7uy - ounce of gold.

8.&hort a forward contract on - ounce of gold.

The investor ma#e a profit of F+ / &+ert.


11/39

b%If F+> &+ert

-.&ell the gold for &+.

.Invest the proceeds at interest rate r for time T.

8.Ta#e a long position in a forward contract on

- ounce of gold.

The investor ma#e a profit of &+ert/ F+.


12/39

?hen an Investment Asset Provides a

nown Dollar Income

F+5 S+BI %erT

whereI is the present value of the incomeduring life of forward contract


13/39

nown Income

Assume : &+ 5 ,6++ I 5 9+e/+.+8"9=- 5 ,86.0

r 5+.+9 T 5 +.*6=-%

I: @ ,9+

+ 9 6

F+ 5 6++.++ B 86.0%e+.+9"+.C* 5 ,0.0+


14/39

a%If F+ 5 ,6-+ < &+ / I%ert 5 ,0.0+

-.7orrow ,6++ to buy the bond.

.&hort a forward contract.

6++.++ / 86.0 5 ,0+.9+

0+.9+e+.+9"+.C*

5 ,0.0+ 6-+.++ /0.0+ 5 ,8.9+


15/39

b%If F+ 5 ,+ > &+ / I%ert 5 ,0.0+

-.&hort the bond.

.!nter into a long forward contract.

6++ / 86.0 5 , 0+.9

0+.9+e+.+9"+.C* 5 ,0.0+

0.0+ / + 5 ,-0.0+

The forward price must be ,0.0+

Options, Futures, and Other Derivatives 6thEdition,

Copyright John C. Hull 20055.5


16/39

?hen an Investment Asset

Provides a nown 'ield

F+= S+erBq %T

where qis the average yield during the life of the

contract e"pressed with continuous

compounding%


17/39

nown 'ield

Assume : &+ 5 *$r 5 +.-$and T 5 +.*$

the yield is 9; per annum with semiannual

compounding.

-2+.+9 5 -2)=% ) 5 8.60;

F+5 *e+.-+ B +.+860%"+.* 5 ,*.


18/39

Ealuing a Forward Contract

Kis delivery price in a forward contract

F+is forwardpricetoday

:Ealueof forward contract today

The value of a long forward contract$ $ is 5 F0 K%eBrT

&imilarly$ the value of a short forward contract is KBF+%eBrT


19/39

The value of a forward contract on an investment asset thatprovides no income:

5 F+B%e/rt

!)uation shows that F+ 5 &+ert 5 &+ertB%e/rt

5 &+B e/rt

The value of a long forward contract on an investment asset thatprovides a #nown income with present value I:

5 &+B I B e/rt

The value of a long forward contract on an investment asset thatprovides a #nown yield at rate ):

5 &+e/)tB e/rt


20/39

Forward vs Futures Prices

A strong positive correlation between interest

rates and the asset price implies the futures price

is slightly higher than the forward price

A strong negative correlation implies the reverse Gast only a few months are in most circumstances

sufficiently small to be ignored

Forward and futures prices are usually assumed

to be the same. ?hen interest rates are uncertain

they are$ in theory$ slightly different


21/39

Futures Prices f &toc# Inde"

Can be viewed as an investment asset paying adividend yield

The futures price and spot price relationship is

thereforeF+5 S+erBq %T

where qis the average dividend yield on the

portfolio represented by the inde" during lifeof contract


22/39

Futures Prices f &toc# Inde"

F+ 5 &+er/)%T

):The dividend yield

!"ample:

r 5 +.+* &+ 5 -$8++ T 5 8=- +.*% ) 5 +.+-

F+ 5 -$8++e+.+*/+.+-%"+.* 5 ,-$8-8.+


23/39

Inde" Arbitrage

If F+ < &+er/)%T

-.7uying the stoc#s underlying the inde" at

the spot price

.&horting futures contracts

7y a corporation holding short/term moneymar#et investment.


24/39

Inde" Arbitrage

If F+ > &+er/)%T

-.&horting or selling the stoc#s underlying

the inde"

.Ta#ing a long position in futures contracts

7y a pension fund that owns an inde"edportfolio of stoc#s


25/39

Inde" Arbitrage

Program trading

ccasionally e.g.$ on 7lac# 1onday%

simultaneous trades are not possible andthe theoretical no/arbitrage relationship

betweenF+and S+does not hold
http://zh.wikipedia.org/zh-tw/%E9%BB%91%E8%89%B2%E6%98%9F%E6%9C%9F%E4%B8%80http://zh.wikipedia.org/zh-tw/%E9%BB%91%E8%89%B2%E6%98%9F%E6%9C%9F%E4%B8%80


26/39

Futures and Forwards on Currencies

-+++ units of

foreign currency

at time Hero

units of foreign

currency at timeT

Trfe-+++

dollars at time T

TrfeF+-+++

-+++S+ dollarsat time Hero

dollars at time T

rTeS+-+++

-+++ units of

foreign currency

at time Hero

units of foreign

currency at timeT

Trfe-+++

dollars at time T

TrfeF+-+++

-+++S+ dollarsat time Hero

dollars at time T

rTeS+-+++

Two ways of converting -$+++ units of a foreigncurrency to dollars at time T. ere$ &+is spot e"change

rate$ F+is forward e"change rate$ and r and rfare the

dollar and foreign ris#/free rates.


27/39

Futures and Forwards on Currencies

-$+++ erfT F+ 5 -$+++ &+ erT

F+ 5 &+ erT= erfT

The relationship between F+and &+

F S e

r r Tf

+ +=

%


28/39

Futures on Commodities

Income and &torage Costs

a%In the absence of storage costs and income$

the forward price of a commodity that is an

investment asset is give by:

F+ 5 &+erT

b%If J is the present value of all the storage costs$

net of income$ during the life of a forward contract:

F+ 5 &+ 2 J%erT


29/39

c%If the storage costs net of income incurred at anytime are proportional to the price of the commodity$

they can be treated as negative:

F+5&+er2u%T

?here u denotes the storage costs per annum asproportion of the spot price net of any yield earned on

the asset.


30/39

a% F+ < &+ 2 J%erT

-. 7orrow an amount &+ 2 J at the ris#/free rate and

use it to purchase one unit of the commodity and

to pay storage costs.

. &hort a forward contract on one unit of the

commodity.

Futures on Consumption Assets


31/39

b% F+ > &+ 2 J%erT

-. &ell the commodity$ save the storage costs$

and invest the proceeds at the ris#/free

interest rate.

. Ta#e a long position in a forward contract.


32/39

Futures on Consumption Assets

F+S+er+u %T

where uis the storage cost per unit time as a

percent of the asset value. Alternatively$

F+S++U %erT

where Uis the present value of the storagecosts.


33/39

Convenience 'ield

K The benefits from holding the physical asset are

sometimes referred to as the convenience yield.

If the dollar amount of storage costs is #nown and has a

present value J$ that the convenience yield y is defined

such that:

F+eyT5 &+ 2 J %erT

If the storage costs per unit are a constant proportion$ u$ of thespot price$ then y is defined so that:

F+eyT5 &+er2u%T or F+ 5 &+er2u/y%T


34/39

The Cost of Carry

The cost of carry$ c$ is the storage cost plus

the interest costs less the income earned

For a non/dividend/paying stoc#$ it is r.

For a stoc# inde"$ it is r / ).

For a currency$ it is r / rf.

For a commodity that provide income at rate ) and

re)uire storage costs at rate u$ it is r / ) 2 u.


35/39

Define the cost of carry as c.

For an investment asset $ the futures price is

F+= S+ecT

For a consumption asset$ The convenience yield

on the consumption asset$y$ is defined

so thatF+5 S+ecBy %T

The Cost of Carry


36/39

Delivery ptions

Form e)uation F+ 5 S+ ecBy %T % that c < y$ the benefitsfrom holding the asset including convenience yield andnet of storage costs% are less than the ris#/free rate .

If futures prices are decreasing as time to maturityincrease c > y%.It is then usually optimal for the partywith the short position to deliver as late as possible$ and

futures prices should$ as a rule$ be calculated on thisassumption


37/39

The (is# in a Futures Position

The cash flow to the speculator are as follow :

Today : / F+e/rT

!nd of futures contract : 2&T The futures prices today : F+

The prices of the asset at time T : &T

The ris#/free return on funds invested for time : T

The investorLs re)uired return : k

The e"pected value : !

The PE of this investment : / F+e/rT2 !&T%e/kT

Assume net present value 5 +

/ F+e/rT2 !&T%e/kT 5 + F+5!&T%er/k%T


38/39

The (is# in a Futures Position

If the asset has

nosystematic ris#$ then k5 r F+5!&T%

and F+is an unbiased estimate of STpositive systematic ris#$ then k< rand

F+>E ST %

negative systematic ris#$ then k> randF+


39/39

4ormal 7ac#wardation and Contango

4ormal bac#wardation:

?hen the futures price is below the e"pected

future spot price.

Contango:

?hen the futures price is above the e"pected

future spot price.

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pricing of f&f