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مبانی مهندسی مالی و مدیریت ریسکTRANSCRIPT
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GBP
OfferBid
1.51221.5118Spot
1.51321.51271-Month Forward
1.51491.51443-Month Forward
1.51781.51726-Month Forward
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Chicago Board Options Exchange American Stock Exchange Philadelphia Stock Exchange Pacific Exchange LIFFOE (London) Eurex (Europe) And many more (see list at end of Book)
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Forward Rate fornth Year
(% per annum)
Zero Rate for annth Year Investment
(% per annum)Year (n)
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Mofid Brokerage co.
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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.
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11
.oc egarekorB difoM
-
/.oc egarekorB difoM
-
/
21
.oc egarekorB difoM
-
/
T :
. S q .
. Tq-eS .
.oc egarekorB difoM
-
/
)(
.
.oc egarekorB difoM
-
/
eS H C
Tr-eK - Tq-
eS - Tr-eK H p
Tq-
-eS + p = Tr-eK + c
Tq-
.oc egarekorB difoM
-
/
c = Se-qT N(d) - Ke
-rT N(d)
p = Ke-rT N(-d) - Se-qT N(-d)
where:
ln d
T K S
r q T
v
v
= + - + c c m m
ln d
T K S
r q T d T
v
v
v = + - -
= - c c m m
Mofid Brokerage co.
-
/
]df )p - ( + ufp[ Tr-e = f
0S
u0Su
d0Sd
p
)p - (
.oc egarekorB difoM
-
/
)(
q r q - r .
p . .eS = dS )p - ( + uSp
T)q - r(
: p
d u - = T q r ) ( d e
- -
.oc egarekorB difoM
-
/
:
)XJD( 10/0 semit xednI senoJ woD ehT)XDN( xednI 001 qadsaN ehT)TUR( xednI 0002 llessuR ehT)XEO( xednI 001 P&S ehT)XPS( xednI 005 P&S ehT
XEO
.
.oc egarekorB difoM
-
/
SPAEL
spaeL .
. .
spaeL .
.oc egarekorB difoM
-
/
.
.
.oc egarekorB difoM
-
/
S X . S
. S
. X
.
.oc egarekorB difoM
-
/
. .
.
.oc egarekorB difoM
-
/
.
. .
.
.oc egarekorB difoM
-
/
.
) ( . .
. : = - +
: .
.oc egarekorB difoM
-
/
)( )(
.
.oc egarekorB difoM
-
/
.
S = q =
.oc egarekorB difoM
-
/
. .
XF .
.oc egarekorB difoM
-
/
fr .
S . S fr .
) ( fr .
.oc egarekorB difoM
-
/
)( fr .
.
S = fr = q
.oc egarekorB difoM
-
/
C = Se-rf TN(d) - Ke
-rTN(d)
P = Ke-rTN(-d) - Se-rf TN(-d)
where:
ln d
T K S
r r T f
v
v
= + - + c c m m
ln d
T K S
r r T d T
f
v
v
v = + - -
= - c c m m
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
-
/
:
. .. .
.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
-
/
:
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
-
/
.oc egarekorB difoM
-
/
-
-:eS + p (
)Tr-eK + c = Tq-
.
-
.
.oc egarekorB difoM
-
/
. .
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-
/
.
.
.oc egarekorB difoM
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) -(
) - (
.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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//
/////
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-
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-
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. = n , = v , = r , = K , = S
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=
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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
.oc egarekorB difoM
-
/
q :
/ + S )q - r( + H Pr = Sv
.oc egarekorB difoM
-
/
)o( .
)-( o .
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-
/
.
.
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-
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)ohR( .
ohR .
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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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-
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)(
: . . .
.oc egarekorB difoM
-
/
51
.oc egarekorB difoM
-
/.oc egarekorB difoM
-
/
.oc egarekorB difoM
-
/
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 .
.oc egarekorB difoM
-
/
)(
yadv .
.
.oc egarekorB difoM
-
/
. ) (
= X = N
.oc egarekorB difoM
-
/
)(
. :
=
.oc egarekorB difoM
-
/
)(
. ) .(
: / = )/-(N RaV :
= * /
.oc egarekorB difoM
-
/
T & TA
T & TA . . ) (
: =
RaV : / = / *
.oc egarekorB difoM
-
/
) -(
T & TA .
/ .
.oc egarekorB difoM
-
/
:
= 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
-
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-
/
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
-
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q :
td)q - r(e = a
- : q .
q . r = q
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-
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:-
.-
.
- .
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( ) (
.oc egarekorB difoM
-
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/ .
e u t t r
= n d + - d v d v
e d t t r
= n d - - d v d v
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-
/
. .
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-
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71
.oc egarekorB difoM
-
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-
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. :
v . T
.
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T : XR:
r : TF :
v:
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:
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T : * T : XR:
*r : T
F: v :
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.
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r : TF: v :
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:
ro
D .
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)(
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yv v .
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telpaC
telpaC .
XR L t kd + t . kd + t :
) , XR - kR( xam kdL
.oc egarekorB difoM
-
/
spaC
stelpac . telpac
XR .
.
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-
/
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 .
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-
/
.
.
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-
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.
XR n T .
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L m R T.
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91
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. .
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.
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.
.
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DDC ) ( :) - A , ( xam = DDC
A .
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:
DDC DDH .
.
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:
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:Y =
P = T =
e = b- c- .
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TAC .
.
.
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-
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02
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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
-
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.oc egarekorB difoM
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