Ch 20 Hull Fundamentals 8 the d

7/25/2019 Ch 20 Hull Fundamentals 8 the d

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Fundamentals of Futures and Options Markets, 8th Ed, Ch 20, Copyright John C. ull 20!"

Value at Risk

Chapter 20

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Fundamentals of Futures and Options Markets, 8th Ed, Ch 20, Copyright John C. ull20!"

The Question Being Asked in VaR

What loss level is such that we areX%

confident it will not be exceeded inNbusiness days?

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VaR and Regulatory Capital

Regulators base the capital they reuireban!s to !eep on "aR

#or $ar!et ris! they use a 0&day ti$ehori'on and a ((% confidence level

#or credit ris! they use a (()(%

confidence level and a year ti$ehori'on

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VaR vs. Expected ShortfallSee !igures "#.$ and "#."% page &'()

"aR is the loss level that will not beexceeded with a specified probability

*xpected shortfall is the expected lossgiven that the loss is greater than the "aRlevel

+lthough expected shortfall is theoretically$ore appealing than "aR, it is not aswidely used

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Advantages of VaR

-t captures an i$portant aspect of ris!

in a single nu$ber

-t is easy to understand -t as!s the si$ple uestion. /ow bad can

things get?

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*istorical Si+ulation

Create a database of the daily $ove$ents in all$ar!et variables)

he first si$ulation trial assu$es that the

percentage changes in all $ar!et variables areas on the first day

he second si$ulation trial assu$es that the

percentage changes in all $ar!et variables areas on the second day and so on

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*istorical Si+ulation continued

1uppose we use 0 days of historical data34ay 0 to 4ay 005

6et vibe the value of a $ar!et variable on day i

here are 00 si$ulation trials he ith trial assu$es that the value of the

$ar!et variable to$orrow is

7

1

500

i

i

v

vv


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*istorical Si+ulation continued

he portfolio7s value to$orrow iscalculated for each si$ulation trial

he loss between today and to$orrow isthen calculated for each trial 3gains arenegative losses5

he losses are ran!ed and the one&day((% "aR is set eual to the thworst loss


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Exa+ple , Calculation of $-day% /

VaR for a 0ortfolio on Sept "(% "##1 Ta2le"$.$% page &'3)


9

Index Value ($000s)

48-+ 9,000

#1* 00 :,000

C+C 90 ,000

;i!!ei 22 2,000


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4ata After Ad5usting for Exchange

Rates Ta2le "#."% page &'1)


10

Day Date DJIA FTSE 100 CAC 40 Niei !!"

0 +ug ,:)>9 =,:( :), 200= , =,:)= :9):>

2 +ug (, 200= ,0 ,>): =,9)> =,0::)(: 9)2=

00 1ep 2, 200> ,022)0= (,(()(0 =,200)90 2)>2


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Scenarios 6enerated Ta2le "#.'% page &'1)


11

S#enai% DJIA FTSE 100 CAC 40 Niei !!" &%t'%li%Value ($000s)

%ss($000s)

0,( (,>


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Ranked 7osses Ta2le "#.&% page &')


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S#enai% Nu*e %ss ($000s)

9(9 92)209

:2( 2

22< 2


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The 8-day VaR

heN&day "aR for $ar!et ris! is usuallyassu$ed to be ti$es the one&day "aR

-n our exa$ple the 0&day "aR would becalculated as

his assu$ption is in theory only perfectlycorrect if daily changes are nor$allydistributed and independent


13

N

274,801385,25310 =


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The 9odel-Building Approach

he $ain alternative to historicalsi$ulation is to $a!e assu$ptions aboutthe probability distributions of the return on

the $ar!et variables and calculate theprobability distribution of the change in thevalue of the portfolio analytically

his is !nown as the $odel buildingapproach or the variance&covarianceapproach


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4aily Volatilities

-n option pricing we express volatility asvolatility per year

-n "aR calculations we express volatilityas volatility per day

15

day

yea$

= 252


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4aily Volatility continued

1trictly spea!ing we should define dayas

the standard deviation of the continuously

co$pounded return in one day -n practice we assu$e that it is the

standard deviation of the percentagechange in one day

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9icrosoft Exa+ple

We have a position worth A0 $illion inBicrosoft shares

he volatility of Bicrosoft is 2% per day3about :2% per year5We useN 0 andX ((

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9icrosoft Exa+ple continued

he standard deviation of the change inthe portfolio in day is A200,000

he standard deviation of the change in0 days is

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200 000 10 456, $632,=


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9icrosoft Exa+ple continued

We assu$e that the expected change inthe value of the portfolio is 'ero 3his isDE for short ti$e periods5

We assu$e that the change in the valueof the portfolio is nor$ally distributed

1ince ;3F2):2=50)0, the "aR is

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300,471,1$456,632326.2 =


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AT:T Exa+ple

Consider a position of A $illion in+Ghe daily volatility of +G is %

3approx =% per year5he 1)4 per 0 days is

he "aR is

20

114,158$10000,50 =

800,367$326.2114,158 =


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0ortfolio

;ow consider a portfolio consisting of bothBicrosoft and +G

+ssu$e that the returns of +G andBicrosoft are bivariate nor$al and that thecorrelation between the returns is 0):


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S.4. of 0ortfolio

+ standard result in statistics states that

-n this case X 200,000 and+ 0,000

and 0):) he standard deviation of the

change in the portfolio value in one day istherefore 220,200

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+X+X+X

++= +

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VaR for 0ortfolio

he 0&day ((% "aR for the portfolio is

he benefits of diversification are

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The 7inear 9odel

We assu$ehe daily change in the value of a portfolio

is linearly related to the daily returns fro$$ar!et variableshe returns fro$ the $ar!et variables are

nor$ally distributed

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9arko;it< Result for Variance of

Return on 0ortfolio


25

sinstru$entthand

thofreturnsbetweenncorrelatiois

portfolioin

instru$entthonreturnofvarianceis

portfolioininstru$entthofweightis

ReturnIortfolioof"ariance

iJ

2

i

,

i

i

i-

--

i

n

i

n

,

,i,ii,

= = =1 1


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VaR Result for Variance of

0ortfolio Value i= w

iP)

26

))&i=<


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Covariance 9atrix vari= cov

ii)

Ta2le "#.>% page &&()

Fundamentals of Futures and Options Markets, 8th Ed, Ch 20, Copyright John C. ull20!" 27

=

nnnn

n

n

n

C

varcovcovcov

covvarcovcovcovcovvarcov

covcovcovvar

321

333231

223221

113121


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Alternative Expressions for P

"

page &&>

Fundamentals of Futures and Options Markets, 8th Ed, Ch 20, Copyright John C. ull20!" 28

transposeitsisandisele$ent

thwhosevectorcolu$ntheiswhereT

T

?

?

??

i

&

,

n

i

n

,

ii,&

.

i

C=

= = =

2

1 1

2 cov


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*andling @nterest Rates

We do not want to define every bond as adifferent $ar!et variable

We therefore choose as assets 'ero&coupon

bonds with standard $aturities. &$onth, :$onths, year, 2 years, years, < years, 0years, and :0 years

Cash flows fro$ instru$ents in the portfolio are$apped to bonds with the standard $aturities

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hen 7inear 9odel Can 2e sed

Iortfolio of stoc!sIortfolio of bonds

#orward contract on foreign currency -nterest&rate swap

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The 7inear 9odel and ptions

Consider a portfolio of options dependenton a single stoc! price, S) 4efine

and

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S&

=

S

Sx

=


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7inear 9odel and ptions

continued eDuations "#.& and "#.(% page &&3)

+s an approxi$ation

1i$ilar when there are $any underlying$ar!et variables

where iis the delta of the portfolio withrespect to the ith asset

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xSS& ==

=i

iii xS&


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Exa+ple

Consider an invest$ent in options onBicrosoft and +G) 1uppose the stoc!prices are 20 and :0 respectively and the

deltas of the portfolio with respect to the twostoc! prices are ,000 and 20,000respectively

+s an approxi$ation

where xand x2are the percentagechanges in the two stoc! prices

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21 000,2030000,1120 xx& +=


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But the distri2ution of the daily

return on an option is not nor+al

See !igure "#.&% page &&1)


Iositive Ka$$a ;egative Ka$$a

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Translation of Asset 0rice Change

to 0rice Change for 7ong Call

!igure "#.(% page &&)


6ong Call

+sset Irice

35

T l ti f A t 0 i Ch


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Translation of Asset 0rice Change

to 0rice Change for Short Call!igure "#.>% page &&)


1hortCall

+sset Irice

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Quadratic 9odel

#or a portfolio dependent on a singlestoc! price

where is the ga$$a of the portfolio)his beco$es

37

2)(

2

1SS& +=

22 )(21 xSxS& +=


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se of Quadratic 9odel

+nalytic results are not as readily available/istorical si$ulation can be used in

conJunction with the uadratic $odel 3hisavoids the need to revalue the portfolio foreach si$ulation trial5

he uadratic $odel is also so$eti$esused in conJunction with a Bonte Carlosi$ulation


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Esti+ating Volatility for 9odel

Building Approach eDuation "#.3% page &($)

4efine nas the volatility per day between day n#

and day n, as esti$ated at end of dayn#

4efine Sias the value of $ar!et variable at end of

day i 4efine ui$ ln3Si/Si15

he usual esti$ate of volatility fro$ observationsis.

39

=

=

=

=

i

in

i

inn

u)

u

uu)

1

1

22

1

)(1

1


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Si+plifications eDuations "#.1 and "#.% page &($)

4efineuias (SiSi1)/Si1+ssu$e that the $ean value of uiis 'ero

Replace 1by

his gives

40

n n ii)

)u2 2

1

1= =


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eighting Sche+e

-nstead of assigning eual weights to theobservations we can set

41

n i n ii

)

ii

)

u2 21

1

1

=

=

=

=

where


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E9A 9odel eDuation "#.$$% page &(")

-n an exponentially weighted $ovingaverage $odel, the weights assigned tothe u2decline exponentially as we $ovebac! through ti$e

his leads to

42

21

21

2 )1( += nnn u


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Attractions of E9A

Relatively little data needs to be storedWe need only re$e$ber the current

esti$ate of the variance rate and the $ostrecent observation on the $ar!et variable

rac!s volatility changes 0)(9 is a popular choice for daily

volatility forecasting

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Correlations

4efine ui2(3i3i1)/3i1and vi2(ViVi1)/Vi1+lso

un5daily vol of 3calculated on day n&vn. daily vol of Vcalculated on day n&

covn. covariance calculated on day n&

covn$ nunvn

where nis the correlation between 3and V

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Correlations continuedeDuation "#.$'% page &(&)

Lsing the *WB+

covn covn&H3&5un&vn&

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9 d l B ildi *i t i l


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Fundamentals of Futures and Options Markets, 8th Ed, Ch 20, Copyright John C. ull

20!"

9odel Building vs *istorical

Si+ulation Approaches

Bodel building approach has thedisadvantage that it assu$es that $ar!etvariables have a $ultivariate nor$aldistribution

/istorical si$ulation is co$putationallyslower and cannot easily incorporate

volatility updating sche$es

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B k T ti


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Fundamentals of Futures and Options Markets, 8th Ed, Ch 20, Copyright John C. ull

20!"

Back-Testing

ests how well "aR esti$ates wouldhave perfor$ed in the past

We could as! the uestion. /ow oftenwas the loss greater than the "aRlevel

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St T ti


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Fundamentals of Futures and Options Markets 8th Ed Ch 20 Copyright John C ull

Stress Testing

his involves testing how well aportfolio would perfor$ under so$eof the $ost extre$e $ar!et $oves

seen in the last 0 to 20 years

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Ch 20 Hull Fundamentals 8 the d

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