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Original Example Revisited

Since p is a risk-neutral probabilitye/ * / = p + ( - p) p = /

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Delta

Delta () is the ratio of the change in the price of a stock option to the change in the price of the underlying stock.

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Mofid Brokerage co.

/

: F = Se(r-rf)T C = e-rT [FN(d) - KN(d)]

P = e-rT [KN(-d) - FN(-d)]

ln d

T

K F T

v

v

= + c m

ln d

T K F T

d T

v

v

v = -

= - c m

Mofid Brokerage co.

/

21

.oc egarekorB difoM

/

.oc egarekorB difoM

/

:. .

. .

.oc egarekorB difoM

/

:

. .. .

.oc egarekorB difoM

/

: = K - F

= F - KF .

.oc egarekorB difoM

/

. .

)tip( .

.

.oc egarekorB difoM

/

-

:. Tr-eK

. eF

Tr- T . :

eF + p = Tr-eK + cTr-

.oc egarekorB difoM

/

Fe-rT - K < C - P < F - Ke-rT

c > (F - K)e-rT

p > (F - K)e-rT

Mofid Brokerage co.

/

.

)( = )( =

)( = )( =

)( = =

.oc egarekorB difoM

/

: /

= / = .

-

-

.oc egarekorB difoM

/

) =(

: /

: /-

: * /-e /-

/- =

.oc egarekorB difoM

/

: /

/- .

. / .

.oc egarekorB difoM

/

T .

0F

u0Fu

d0Fd

.oc egarekorB difoM

/

)(

.

:

d F u F

f f

d u

- =

-

u - 0F - u0F

d -0F - d0F

.oc egarekorB difoM

/

)(

T :uf - F - uF

: f- :

Tr-e ]uf - F - uF[

.oc egarekorB difoM

/

)(

:]df )p - ( + ufp[Tr-e = f

:

p d u

- = d -

.oc egarekorB difoM

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:

S = )F(q = )r(

r = q F .

.oc egarekorB difoM

/

. .

.

r .

.oc egarekorB difoM

/

.])d(NK - )d(NF[Tr-e = c])d-(NF - )d-(NK[Tr-e = p

:erehw

nl d

T K T F

v

v

= m c +

nl d

T K T F

T d

v

v

= v -

- = m c

.oc egarekorB difoM

/

) ( .

. )

( .

.oc egarekorB difoM

/

-

Indices:

c + Ke-rT = p + Se-qT

Foreign exchange:

c + Ke-rT = p + Se-rf T

Futures:

c + Ke-rT = p + Fe-rT

Mofid Brokerage co.

/

q .

: = q : fr = q

: r = q

.oc egarekorB difoM

/

31

.oc egarekorB difoM

/

.oc egarekorB difoM

/

-

-:eS + p (

)Tr-eK + c = Tq-

.

-

.

.oc egarekorB difoM

/

. .

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/

.

.

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) - (

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)-( . .

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)-(

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.(

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-:

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//

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///

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/

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)sretteL keerG(

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. = n , = v , = r , = K , = S

T . - .

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- .

-

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: .

.

.

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)( .

B

A

=

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. q

:Tq-e )d(N =

:] - )d(N[ Tq-e =

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)(

.

- .

)-( )-( .

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T)q-r(e . T)q-r(e

.

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) ( .

)-( .

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)( )( .

)-( .

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C

CC

S S

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:

S /+ tH . P

( . ( .

SS

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/

q :

/ + S )q - r( + H Pr = Sv

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)o( .

)-( o .

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/

.

.

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)ohR( .

ohR .

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.

.

.

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o .

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51

.oc egarekorB difoM

/

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/

RaV

X N

.oc egarekorB difoM

/

:

:

.

.oc egarekorB difoM

/

RaV

X N

.oc egarekorB difoM

/

RaV

RaV

. K RaV K

.

.oc egarekorB difoM

/

RaV RaV-C

RaV .

RaV-C :

RaV-C .

.oc egarekorB difoM

/

RaV

.

. : ) (

?teg sgniht nac dab woH

.oc egarekorB difoM

/

.

. )(

... .

.oc egarekorB difoM

/

)(

i iv . m .

- m . i

: ) + m(

.oc egarekorB difoM

/

-

. - .

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. RaV .

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/

)(

yadv .

.

.oc egarekorB difoM

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. ) (

= X = N

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/

)(

. :

=

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)(

. ) .(

: / = )/-(N RaV :

= * /

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T & TA

T & TA . . ) (

: =

RaV : / = / *

.oc egarekorB difoM

/

) -(

T & TA .

/ .

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/

:

= xv = yv / = t /

.

.oc egarekorB difoM

/

RaV : = / * *

: = - ) + (

T & TA RaV

.oc egarekorB difoM

/

: .

.

.oc egarekorB difoM

/

Where vi is the volatility of variable i and vp is the portfolios

standard deviation.

Mofid Brokerage co.

/

.

.

.

.oc egarekorB difoM

/

.oc egarekorB difoM

/

.

.

.oc egarekorB difoM

/

)(

:xdS = Sd = Pd

.

i i .

.oc egarekorB difoM

/

.

. :xd * * + xd * * = Pd

xd xd .

.oc egarekorB difoM

/

Skewness (See Figures 16.3, 16.4, and 16.5)

The linear model fails to capture skewness in the

probability distribution of the portfolio value.

Mofid Brokerage co.

/

.

:

.oc egarekorB difoM

/

:

ixd . ixd .

.

.oc egarekorB difoM

/

pd . pd

. RaV N .

.

.oc egarekorB difoM

/

Define vn as the volatility per day btween day n - and day n, as estimated at end of day n -

Define Si as the value of market variable at end of day i

Define ui = (Si / Si-) The standard estimated of volatility from m ooservation is:

Mofid Brokerage co.

/

Define ui as:

Assume that the mean value of ui is zero.

Replace m - by m.

This gives:

Mofid Brokerage co.

/

Weighting Scheme

Instead of assigning equal weights to the observations we can set:

Where:

Mofid Brokerage co.

/

EWMA

In an exponentially weighted moving average model, the weights assigned to the u decline exponentially as we move

back through time.

This leads to:

vn = mvn - + ( - m) un -

Mofid Brokerage co.

/

EWMA

Relatively little data needs to be stored. We needs only remember the current estimate of the variance

rate and the most recent observation on the market variable.

Tracks volatility changes.

JP Morgan use m = / for daily volatility forecasting.

Mofid Brokerage co.

/

Dfine ui and vi as:

Alsovu,n: daily vol of U calculated on day n -

vv,n: daily vol of V calculated on day n -

covn = tnvu,nvv,nWhere tn on day n -

Mofid Brokerage co.

/

AMWE - nv - nu )m - ( + - nvoc m = nvoc

.oc egarekorB difoM

/

. / = m .

.oc egarekorB difoM

/

.

.

.oc egarekorB difoM

/

.

.oc egarekorB difoM

/

RaV . : RaV

.

.oc egarekorB difoM

/

.oc egarekorB difoM

/

71

.oc egarekorB difoM

/

.

.

.oc egarekorB difoM

/

S

uS

dS

P

P _ 1

.oc egarekorB difoM

/

p u d

.

.oc egarekorB difoM

/

p u d

.d )p - ( + up = tdre

]d )p - ( + up[ - d )p - ( + up = tdv

:

.oc egarekorB difoM

/

td :

tdre = a

.oc egarekorB difoM

/

S0

S0u

S0d

S0u2

S0d2

S0

S0u

S0d

S0u3

S0d3

S0u2

S0d2

S0

S0d4

S0u4

Mofid Brokerage co.

/

. .

.

.oc egarekorB difoM

/

:

S = , K = , r = , v =

T = month = /

dt = month = /

The parameters imply

u = /, d = /

a = /, p = /

Mofid Brokerage co.

/

)(

//

//

//

//

//

//

//

//

//

//

//

//

//

//

//

//

//

//

//

//

//

.oc egarekorB difoM

/

t .

/ - / - = / - / / = atleD

.oc egarekorB difoM

/

t .

= /

/ =

)/ - /( ) - /(

)/ - /( )/ - (

= /- = - /

/ =

)/ - /( ) - /(

)/ - /( )/ - (

/- = -

/ = ammaG

/

=

)/ - /( ) - /(

)/ - /( )/ - (

-

.oc egarekorB difoM

/

td .

= ammaG / /

- = / - /

yad radnelac rep /- ro

.oc egarekorB difoM

/

.

/ . :

/ = / - /

.oc egarekorB difoM

/

q :

td)q - r(e = a

- : q .

q . r = q

.oc egarekorB difoM

/

:-

.-

.

- .

.oc egarekorB difoM

/

( ) (

.oc egarekorB difoM

/

/ .

e u t t r

= n d + - d v d v

e d t t r

= n d - - d v d v

.oc egarekorB difoM

/

. .

.oc egarekorB difoM

/

71

.oc egarekorB difoM

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)TOBC(

.oc egarekorB difoM

/

: .

. :

.

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- .

T

.

.oc egarekorB difoM

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)(

. :

v . T

.

.oc egarekorB difoM

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])d( N XR - )d( N F[ Tr-e = c])1d-( N F - )d-( N XR[Tr-e = p

T : XR:

r : TF :

v:

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/

:

])d( N XR - )d( N F[* T*r-e = c])d-( N F - )d-( N XR[* T*r-e = p

T : * T : XR:

*r : T

F: v :

.oc egarekorB difoM

/

.

.oc egarekorB difoM

/

)(

])d( N XR - )d( N F[ Tr-e = c])d-( N F - )d-( N XR[Tr-e = p

T : XR:

r : TF: v :

.oc egarekorB difoM

/

:

ro

D .

.oc egarekorB difoM

/

)(

:yvyD = v

yv v .

.oc egarekorB difoM

/

telpaC

telpaC .

XR L t kd + t . kd + t :

) , XR - kR( xam kdL

.oc egarekorB difoM

/

spaC

stelpac . telpac

XR .

.

.oc egarekorB difoM

/

Caps

[tk , tk+] caplet Ldk e-rk+

tk+ [Fk N (d) - RX N (d)] Where:

Fk: forward interest rate for (tk , tk+)vk: interest rate volatilityrk: interest rate for maturity tkL: principal RX: cap rate dk = tk+ - tk

Mofid Brokerage co.

/

spaC ...

) ( ) ( .

: telpac

pac .

.oc egarekorB difoM

/

.

.

.oc egarekorB difoM

/

.

XR n T .

:L) , XR - R( xam m

L m R T.

.oc egarekorB difoM

/

:])d(N XR - )d(NF[ AL

:erehW

F . v it i

= Ae m

it ir-

.oc egarekorB difoM

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.

.

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)(

.

.

.

.oc egarekorB difoM

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.

.

) ( .

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81

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1

.oc egarekorB difoM

/

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)(

.

.

.oc egarekorB difoM

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)sSBM(

.oc egarekorB difoM

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)(

SMC TMC

.oc egarekorB difoM

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)(

)TB G&P(

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) ( .

) SMC TMC ( .

.oc egarekorB difoM

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91

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02

.oc egarekorB difoM

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:

.oc egarekorB difoM

/

.

ytitne ecnerefeR . .

SDC ) (

. .

.oc egarekorB difoM

/

.

.

.oc egarekorB difoM

/

:

DDH ) ( :)A - , (xam = DDH

DDC ) ( :) - A , ( xam = DDC

A .

.oc egarekorB difoM

/

:

DDC DDH .

.

.oc egarekorB difoM

/

:

.oc egarekorB difoM

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. )XEMYN(

)EPI( .

.oc egarekorB difoM

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.

UTB .

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/

.

. .

.

.oc egarekorB difoM

/

)(

. :

x x x

.oc egarekorB difoM

/

:e + Tc + Pb + a = Y

:Y =

P = T =

e = b- c- .

.oc egarekorB difoM

/

TAC .

.

.

.oc egarekorB difoM

/

02

.oc egarekorB difoM

/

12

.oc egarekorB difoM

/

Barings ($1 billion) Chemical Bank ($33 million) Daiwa ($1 billion) Kidder Peabody ($350 million) LTCM ($4 billion) Midland Bank ($500 million) National Bank ($130 million) Sumitomo ($2 billion)

Mofid Brokerage co.

/

Allied Lyons ($150 million) Gibsons Greetings ($20 million) Hammersmith and Fulham ($600 million) Metallgesellschaft ($1.8 billion) Orange County ($2 billion) Procter and Gamble ($90 million)

Shell ($1 billion)

Mofid Brokerage co.

/

.

.

.

. .

.oc egarekorB difoM

/

. .

. . .

. .

.oc egarekorB difoM

/

.

. .

.oc egarekorB difoM

/

.oc egarekorB difoM

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