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Quantitative Methods for Finance (FEB23006)

QMF Final Exam - Summary of Slides and Webcasts Analysis

Part
I
I
Assel
Returns
sweighted
sam
⑧
Simple
R
+=
(P
+
-Pt-1)
/Pt-
1
st
.
portfolio
Rpt
=
&2
,
WiRit
,
where
wi=
niPit-1
&
niPi
,
t-1
②
Log
#
=
In
(P
+
/P
+
-1)
St
.
multiple-period
h
=
H
+
-
-
-
+
F
-
h
+
Stylized
facts
of
asset
returns
&
herizon
sum
⑧
distr
.
of
return
is
not
normal
Ke
free
,
excess
K:3
:
large
&
Small
R
occurs
more
often
st
.
fat-tailed
&
peaked
distr
.
neg
.
SK
<0
:
large
neg
.
R
occur
more
often
than
large
pos
.
ones
-
which
are
more
freq
②
almost
no
sign
.
autocar
in
R
③
small
,
but
very
slowly
declining
autocorr
in
sq
.
&
abs.
R
SK
:
3
-
6
e
-
>
Normality
:
SK
=
M3ln
o
k
=
Myla(JB
=
S
+
-3 Exice)
,
where
R
:
HEL
mj
=
E((r-E(i)
St
.
m
=
&
"CH-r)
j
-
>
Autocorr
.
&
covar
:
measure
dependence
between
R
that
are
periods
apart
-
·
autocouar
·
j(K)
=
E((r-M)
(H
-K
-
M)
-
>
&(4)
=
&TiP
(H
+
-
v)
(r
+
+
v)
·
autocar
.
p(K)
=
f(k)
/(20)
-
>
j(k)
=
j(k)
/Yo
=>
Box-Pierce
test
(join
sign.
of
1st
m
autocarr
.
)
:
Am
=
GL
K
*
+
X
&
(m)
but
M
+
2
caust
DE(s12
+
120
Volatility
Models
:
General
sel-up
#2
++
Stim
+
It
,
where
stWN
,
SEEPIIN)
#
=
M
+
+
Et
=
M
+
St
Cassume
ht
=
Mas
time-variation
in
is
moment
diff
.
to
capture)((st11tDa
·
Eft
white
noise
,
but
O
cond
,
mean
ELE
+
11t-)
=
0
an
start
2
②
time-var
.
Can
.
var
.
ECEF17
+
-1)
=
O
time-varying
G
·
0
ECH11
+
-1)
=
My
=
M
var
of
of
an
.
It
+:
My
②
UCr11
+
-1)
=
E((r-M
+>
11
+
-1)
=
of
M
-
M()
=
E
+
T
VIrel2t-17
=
Elli-Ecrt]
-
-
>
H-h
=
Et
=
+Et
,
Et
"20(
,
1)
St
.
of
captures
temporal
variation
of
Et
-z
(2)
M-MH)
=
2t2cE
+,
Et
=
std
.
W(0
,
1)
IN)
Of
12f1]
Of
Aim
:
develop
mic
model
to
estim
.
&
forecast
(need
to
specify
distr
of
Et)
*
Use
:
uncon-var.
of
Et
ELEI)
=
ElE(2f11
+
-))
=
E(o])
=
=2
Volatility
Clustering
def.
periods
of
large
I
small
movements
of
prices
alternate
·
volatility
def-
annualized
stder.
of
R
which
is
not
observable
Variance
Crisk)
o
=
E((H-1
>2)
,
where
M
=
Er
·
var
is
symmetric
:
weights
pos
,
der.
from
in
equally
to
neg
.
dev
.
-
>
downside
(tail)
risk
def
.
risk
associated
with
R
below
expec
.
R
measures
&
Value-at-Risk
(VaR)
②
Expected
Shartfall
(ES)
Value-at-Risk-quantile
-
prob
,
that
Ris
expec
,
to
be
exceeded
:
good
def
.
O
max
.
loss
(VaRt(q
,
h)
=
gth
quantile
of
distr
of
h-day
R
Fithih
=
Het
+...
+
With
②
min
R
P(rhin
<VaR
+
(1 -g
,
h))
=
g
(
=
)
Fithin
(VaR
+
(1-g
,
h))
=
9
pros
:
easy
to
calc
;
7
always
;
info
.
on
tail
risk
(left
&
right)
cans
:
uninfo
.
on
what
happens
once
VaR
has
been
breached
(gapI
break)
;
lack
of
sub-additivity
VaR t
(1-g
,
1
,
X
+
Y)
>
VaR
+
(1-g
,
1
,
X)
+
VaRt(1-g
,
1
,
Y)
St
.
discourage
diversification
-
>
estimation
Assume
all
it
has
same
distr
&
Historical
simulation
:
let
ra-
...
<
r()
St
.
VaRt(1-g
,
1)
=
regt)
1972
N
+
-
M
②
Parametric
model
for
uncon
·
density
-
>
Normal
=
T
=
M
+
0zt
-NC
,
2)
,
t
=
o
~
N(0
,
i)
St
.
VaRt(1-g
,
1)
=
M
+
zg0
-
>
VaRt4-aih)
=
Et
(thin)
+
Eg thin
~
assume
all
t
100
(1
,
02
&
def
.
sprt
rule
:
h-day
VaR
Square-root-of-time
Rule
let
Fithin
=
M
+
+
--
+
#th
&
#80(p
,
04
St
can
be
estim
,
as
o
time
0
ECrthin)
=
hh(
VaRt(g
,
h)
=
hi
+
EgoVaRt(-q
,
1)
I-day
VaR
estim
.
②
V
(rtthih)
=
Gez
o
11
&
of
constant
(i
=
2)
~
more
info
,
but
reg
,
finite
ist
moment
&
very
data-intensive
Expected
Shortfall - cand
.
expec
i
adressess
cans
of
VaR
2
def
.
expec
.
loss
given
VaRt(-g
,
h)
has
been
breached
(Fthh
VaRt(1-gin))
ESt(1-gih)
=
E
(Fthin
thin
-
VaRt(1-g
,
h))
=
E(rethin
I Crithin[ VaRt(1-gih))(
-
>
estimation
&
Hist
.
simulation
:
ESt(1-g
,
7)
=
(t)
,
qTEZ
②
Parametric
model
for
uncon
·
density
-
>
Normal
:
ESt(1-9
,
1)
=
4 - Efzlzq
Estimating
Oft
as
vola
.
#
observ
&
Historical
vola
.
using
rolling
windows
F
=**
CHT
-
r
,
where
F
=
&" r+ -
-
>
drawbacks
:
1)
estim
.
exhibit
ghost
features
;
of
suddenly
↑
Whenever
large
R
occurs
&
↓
Whenever
this
R
leaves
the
moving
window
due
to
equal
weighting
(*)
2)
Sensitive
to
7
;
the
smaller
T
,
the
more
prominent
impact
of
culliers
on
R
3)
assumes
R
over
past
T
days
have
same
var
②
Using
Exponentially
Weighted
Moving
Aug
.
F
=
(1
=
1)
[i]"(ri-F)"
=
Jo
+
(1-b)
(r-
1
-
F5
,
02x
-
>
Riskmetrics
+20
.
9h
Volatility
forecasts
Ethi
=
H
,
th
forecasts
made
at
t
for
future
periods
are
equal
to
th
-
>
Sart-of-time
rule
applies
when
120
.
97
:
h
=
Ch
Filt h
if
stationary
model
Volatility
Model
GARCH
(
model
of
=
w
+
def
+
Bot-110
,
Et
if
w
,
0
,
By
=
innovation
-
>
ARMACI
,
1
:
EF
=
w
+
(
+
B)
St
+
+
-
BV
+
-1
,
where
up
=
et-
of
&
Elvf11
++
)
=
0
M
covariance
stationary
if
<
+
B
<
1
(AR
coeff
.
(
-
>
EC)
2
B
EWMA
:
setting
w
=
o
,
1
-
x
=
B
=
J
=>
of
=
(1-5) F-1
+
JOF
-
>
IS
GARCHC
,
1)
good
model
for
of
estim
.?
-
S
=
of
*
law
of
prob
·
f(AIB)
=
f(A
,
B)
(f(B)
0
Non-normality
:
uncan
·
distr
of
of f(rt)
=
(f(r
+
S)ds
=
ff(rts)f(s)ds
=
mixture
of
Normal
#
~
N
*
yes
!
&
②
Autocarr
.
CauCF
,
+-j)
=
ECCF-M)
(r-j
-
>
)
=
E(z
+
o
+
E-jOj)
=
ElztE(t)
=
o
③
Autocar.
of
EF
:
If
X
small
&
X+B
close
to
1
:
g
,
=
X
small
,
but
decay
of
higher
oder
is
slow
-even
still
exponential
-
>
estim
.
GARCH
:
Max
.
Likelihood
function
Gy(t)
=
f(r
,
...,
Rit)indep
.
I
f(rt17
+
-
1
if)
st
.
EML
,
7
=
argmax(T(f)
=
argmax
22
,
et(e)
=
argm
.
2
In
(fCHIt
+
ie))
↓
·
forzN(0
,
1)
:
f(zt)
=
exph-f)
St
.
C
=
-EInCIT-IInCOF)-CF-M72
Of
-
>
evaluation
:
consider
properties
of
std
.
resid
.
Et
=
(H-)
/87
i
⑦
mean
o
&
var
.
I
③
K
=
3
if
normality
assumed
②
no
autocar
.
in
level
&
Sq
-
>
vola
.
forecast
2
1)
-Step
:
t
=
Elof11
+=
E(w
+*
zi
+
Bof11t)
=
w
+
af
+
Bo
=
o
2)
.
2
-
Step
:
H
=
E(OF217
+
)
=
w
+
(4
+
B)
of
as
ECEtilIt)
=
off
n).
n-
Step
:
…tthi
=
w
&
(h
+
B)
:
+
(a
+
B) h
+
C
ARCH(1
,
1)
model
of
=
w
+
EF
=
0
,
Et
if
w
,
do
if
do
;
it
homosk
as
of
w
-
-
>
AR(I)
:
SF
=
W
+
x[f1
+
N+
,
where
Ut
=
EF-
o
·
ARCH
Stationary
if
<1
:
shocks
have
transitory
effects
on
fij
-
>
volatility
clustering
:
if
1st-11
large
-
>
1841
expec
,
to
be
large
as
well
-
>
volatility
mean
reversion
:
0
(2
,
%
=
fraction
a
of
(EE
,
2)
*
pi
=
>
0
S
.
t
.
decline
exp.
towards
0
E
ARCHCI)
in
adequate
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PartI I Assel Returns sweighted sam ⑧ Simple R += (P+ -Pt-1) /Pt- 1 st. portfolio Rpt = &2 , WiRit , where wi= niPit- &niPi , t- ② Log # = In (P+ /P+ -1) St. multiple-period h = H + - - - + F- h + Stylized facts of asset returns & herizon sum ⑧ distr. of return is not normal Ke free , excess K: 3 : large & Small R occurs more often st. fat-tailed & peaked distr. neg. SK <0 : large neg. R occur more often than large pos. ones - which are more freq ② almost no sign. autocar in R ③ small , but very slowly declining autocorr in sq. & abs. R SK

####### : 3

  • 6 e
  • > Normality : SK = M3ln o

k = Myla(JB

= S + -3 Exice) , where R : HEL mj = E((r-E(i) St . m = &"CH-r) j

  • > Autocorr. & covar : measure
  • dependence between R that are periods apart · autocouar · j(K) = E((r-M) (H -K - M) - > &(4) = &TiP (H+ - v) (r+ + v) · autocar . p(K) = f(k) /(20) - > j(k) = j(k) /Yo => Box-Pierce test (join sign. of 1st m autocarr. ) : Am = GL K * + X & (m) but M+ 2 caust DE(s12+ 120
Volatility

####### Models : General sel-up #2++ Stim + It , where stWN , SEEPIIN)

= M + + Et = M + St Cassume ht = Mas time-variation in is moment diff. to capture)((st11tDa

·
   Eft white  noise, but  O  cond  ,  mean  ELE+ 11t-)  = 0 anstart 2
      ②  time-var. Can . var.  ECEF17  + -1)
         =O  time-varying
            G
· 0
   ECH11+ -1) = My = M var  of  of an.  It+:
      My

② UCr11+ -1) = E((r-M +> 11 + -1) = of M - M() = E+ T VIrel2t-

  • > - Elli-Ecrt] H-h = Et = +Et , Et "20(, 1) St . of captures temporal variation of Et -z (2) M-MH) = 2t2cE+,Et = std . W(0, 1) IN) Of 12f1] Of Aim : develop mic model to estim. & forecast (need to specify distr of Et)
    • Use : uncon-var. of Et ELEI) = ElE(2f11+-)) = E(o]) = = Volatility Clustering def. periods of large I small movements of prices alternate · volatility def- annualized stder. of R which is not observable Variance Crisk) o = E((H-1>2) , where M = Er · var is symmetric : weights pos, der. from in equally to neg. dev.
  • > downside (tail) risk def. risk associated with R below expec. R measures& Value-at-Risk (VaR) ② Expected Shartfall (ES) Value-at-Risk-quantile - prob, that Ris expec, to be exceeded : good def. O max. loss (VaRt(q , h) = gth quantile of distr of h-day R Fithih = Het +... + With ② min R P(rhin <VaR+ (1 -g , h)) = g ( =) Fithin (VaR+ (1-g , h)) = 9 pros : easy to calc ; 7 always ; info. on tail risk (left & right) cans : uninfo. on what happens once VaR has been breached (gapI break) ; lack of sub-additivity VaRt (1-g , 1 , X + Y) > VaR+ (1-g, 1 ,X) + VaRt(1-g , 1 , Y) St. discourage diversification
  • > estimation Assume all it has same distr & Historical simulation : let ra- ... < r() St. VaRt(1-g , 1) = regt) 1972 N+ - M ② Parametric model for uncon · density
  • > Normal = T = M + 0zt -NC, 2) , t = o ~ N(0, i) St. VaRt(1-g , 1) = M + zg0 - > VaRt4-aih) = Et (thin) + Eg thin ~ assume allt 100 (1 , 02 & def . sprt rule : h-day VaR Square-root-of-time Rule let Fithin = M+ + -- + #th & #80(p , 04 St can be estim , as o time 0 ECrthin) = hh( VaRt(g , h) = hi + EgoVaRt(-q , 1) I-day VaR estim. ② V (rtthih) = Gez o 11 & of constant (i = 2)

~ more info , but reg, finite ist moment & very data-intensive

Expected Shortfall - cand. expec i adressess cans of VaR

2

def. expec. loss given VaRt(-g , h) has been breached (Fthh VaRt(1-gin))

ESt(1-gih)

= E

(Fthin thin - VaRt(1-g , h))

= E(rethin

I Crithin[ VaRt(1-gih))(

- > estimation

& Hist . simulation : ESt(1-g , 7) = (t) , qTEZ

② Parametric model for uncon · density - > Normal : ESt(1-9 , 1) = 4 - Efzlzq

Estimating Oft as vola. # observ

& Historical vola.

using rolling windows F

=**CHT - r, where F = &" r+ -

- > drawbacks :

1) estim . exhibit ghost features ; of suddenly ↑ Whenever large R occurs & ↓ Whenever this R leaves

the moving window due to equal

weighting (*)

2) Sensitive to 7 ; the smaller T, the more prominent impact of culliers on R

3) assumes R over past T days have same var

② Using Exponentially Weighted Moving Aug. F = (1 = 1)[i]"(ri-F)" = Jo + (1-b) (r- 1 - F5 , 02x

- > Riskmetrics +20 . 9h

Volatility forecasts Ethi

=

H , th

forecasts made at t for future periods are equal to th

  • >

Sart-of-time rule applies when 120. 97

:

h

= Ch

Filt h

if stationary model

Volatility

Model

GARCH ( model of = w + def + Bot-110 , Et if w , 0 , By

=

innovation

- > ARMACI

, 1 : EF = w + ( + B) St + + - BV+-1 , where up = et- of & Elvf11 ++ ) = 0

M

covariance

stationary if

< + B < 1 (AR coeff. ( - >

EC) 2 B

EWMA : setting w = o , 1 - x = B = J => of = (1-5) F-1 + JOF

- > IS GARCHC

, 1) good model for of estim .? - S = of * law of prob· f(AIB) = f(A, B) (f(B)

0 Non-normality : uncan · distr of of f(rt) = (f(r+ S)ds = * ff(rts)f(s)ds = mixture of Normal # ~ N

yes!

&

② Autocarr. CauCF , +-j) = ECCF-M) (r-j - >) = E(z+ o+ E-jOj) = ElztE(t) = o

③ Autocar. of EF : If X small & X+B close to 1 : g, = X small , but decay of higher oder is slow

-even still exponential

- > estim

. GARCH : Max. Likelihood function Gy(t) = f(r, ..., Rit)indep

.

I f(rt17 +- 1 if)

st. EML , 7 = argmax(T(f) = argmax 22 , et(e) = argm. 2 In (fCHIt + ie))

· ↓

forzN(0 , 1)

:

f(zt)

= exph-f) St. C = -EInCIT-IInCOF)-CF-M72Of

- > evaluation : consider

properties of std . resid. Et

= (H-) /87 i

⑦ mean o & var. I ③ K = 3 if

normality assumed

② no autocar. in level & Sq

- > vola .

forecast 2

1) -Step : t = Elof11 += E(w +* zi + Bof11t) = w + af + Bo = o

2). 2 - Step : H = E(OF217+ ) = w + (4+ B) of as ECEtilIt) = off

n). n-Step : Ötthi = w & (h + B) : + (a+ B) h + C

ARCH(1 , 1) model of = w + EF= 0 , Et if w, do if do ; it homosk as of w

- > AR(I) : SF = W + x[f1 +

N+ , where Ut = EF- o

· ARCH

Stationary if <

: shocks have

transitory effects on fij

  • >

volatility clustering

: if 1st-

large-> 1841 expec , to be large as well

- >

volatility mean reversion

: 0 (2,% =

fraction a of (EE , 2)

*

pi

= > 0 S .t. decline

exp. towards 0 E ARCHCI) in adequate

Part 2 4

Portfolio Management diversification helps to reduce risk

risk : measure of uncertainty in asset R - > diversification is crucial in order to obtain higher mean R than risk-free rate fully invested portfolios : weights sum up to assume ↓ <? Variance of Portfolio Returns OPIN) = 0 F/1-pr mmmm, 02) reduction due to Op + 022 - 2 p 1 2002 diversification - > no gains from divers, if : 17. O, or = 0 2) 0 = 22pi = 1 3) Pi = 210 = 1 Sit. WT = 0

  • the more assets are combined in a portfolio , the smaller the var of portfolio R = O(WE , N) 1 Op(WY , M) , ANIM but this occurs at a decreasing rate & the var can't be reduced to 0 st. - limits to diversification -> some systematic (undiversifiable) risk remains INX1] Minimum Variance Portfolios min . op c) = V st. = 1 => = + Proof Lagrangean -J. (E-1) [NXN] [NX1] iy
  • E · FOC : VE - X, = 0 => w = J , U
  • E · Soc : W = 1 = d = 1/(u + E)

Other portfolio constraints &

reg - input Om & Covar, matrix v O reg , of a target R : M = Mp ② no short-sales constraints : wiso ③ upper bounds on exposure to indiv . assets : Wit Ui Estimating Covar. Matrix * V contains #(ENCN + 1) unighe elem .

Historical U : use data over past 7 days to estim . / forecast covar- at t

  1. equal weights Git = 2 (ht-1-) (bit-c - 5j) , i = &Fit & reg. TIN + 1 = pos , definite Covar. matrix v
  2. EWMA Ojt = (1-b) CEd" Crit-c -ri) (bjit -1 - Fj) = X0jt = + (1 - b) (ht- 1 - mi) (Dit- 1 - Fj) , 0 <
  • E Multivariate GARCH Models

Factor Models + reduce

dimensionality : set KN idea : cross-section of NAR Rit can be desor. int terms of K common factors fit Rit = Xi + Birfit + --- + Bikfk++ Eit => R + = x + Bft + Et [NXK] 4 B : factor loadings 4 idiosyncratic component

· assumptions 1), factor realizations of are stationary with uncon · movements : common factor E(t) = Mf , V (ft) = o [4] 2) dit are uncarr· with each of fit : Cor (fit , Eis) = 0 , i,jis 3) sit are serial uncarr· Cou(Eit , Ejs) =

/Gifij

ts

  • > V = BrB' + 0 , where D = diag (0,, ..., Or St- U contains NK + EK(k+ 1) + N unique elem- · Precision Matrix U + = 0 + - 0 "B(M+ + B'O + B) "B'O- · Cor (Rit , Rit) = BirBj + Ca (Eit , Ejt) common factors lead to caar. betw . AR ; if idiosyncratic components are uncorr. => covar , is completely due to their common dependence on risk factors It · Dimensionality reduction using factor structure of :
  1. level - > aug-

####### 2) slope + 0 (3 months, 120 months ( I capture 99. 5 % of variation

  1. curvature - > 2x24M-ROM-3M yields Altern. motivation FMs : diversification can't elim . systematic risk due to risk factors to which individual assets are exposed s. t. R = function of risk factors + idiosynoratic component

CAPM E(Rit-rf) = BiE(Rmt-rf) , where Rm+= R on market portfolio 5 · V(Rit) = BiV(Rmt) + V(Eit) E) Total Risk = systematic + idiosyncratic risk Factor Models Expected- Return Rit = Xi + Bif + Eit = E(Rit) = ai + BiE(f) di : unexplained part = abnormal part DifFM correctly specified + i = 0 , i · BijE(fit) = reward/compensation for exposure to Crisk) factors FMs and portfolios Rpt = CWiRit = ERt & Rit = Xi + Bif + dit & portfolio's factor loadings Bpj = wiBj ② portfolio's total risk into syncr. & idio : BprBp = #BrB & Ulept) =* Ow ③ restr. on exposure to risk factors : Bpj = Bj Ufj

  • Macroeconomic FM factors observ , loadings unknown
  • > How many & which factors fit to use? · depends on : 17. market portfolio 2) interest rates 3). exchange rates · Fama-French factors Rit = di + Bim (M - + -rf) + BiHMLHML+ + BiSMBSMB+ + Eit CAPM fails in capturing system. risk excess on market portfolio
  • > Estimation : OLS using historical data
  1. expanding window EW ; all T (ai , bil) = argmin [T (Rit-xi-Brifit) => blEN) = [ (fit-fi) (Rit-Ri) = Cor(fit , Rit) & ( fit - Fil < U ( fit)
  2. moving MW ; only most recent t b, MW) same as EW , but with all summations running from t = T-m+ 1 to 7
  3. EWMA ; all 7 , but more weights on recent - > weight dit at + for OCJ

bilENMA) = 21 ji-+ ( fit-Fi) (Rit = Ri) or ols in Rit = dit + Bintfit + E 2) ji - + (fit - F, > & Fundamental FMs loadings observ , factors unknown

  • > idea : size & value are relevant system. risk factors for asset , but only specific observable factors represent these + choose Bij to force models to ind . these factors
  • > Nelson - Siegel Yield Curve Model y+ (Ti) = f(t + (
  • e -

####### 37)

f2t + 1-e-57-

####### e-bi)

fat + dit force latent fit , in 1. 2 , 3 to ↓ Ti ( ↓ Ti represent level , slope & currature

  • > estimation : cross-sectional regression model Rit = Bilfit + --- + Bikfit
  • Eit assume ai = o T cross-sectional regn. suffer from heterosk . as ECEF) = of varies across observ. St. OLS unbiased & consistent but NOT efficient + need WLS Homosk- : Std . Observ . by Ost. regn, with homosk. E = Cito EC : 1 , i O apply OLS to estim factor realizations fit in cross-sectional regr. & obtain estim . of op: t ③ use of to form weighted regn. St. OLS delivers efficient estim , of fit, ..., filt

Statistical FMs both factors & loadings unknown - > Exploring Data Analysis

  • > idea : deser. (Co-) variation in WAR with limited #of K factors fit, - .., fit with KN St. am- to use It that can best deser. Principal Components of Cavar. matrix of Rit best f+& B : min . SSE = min &" (R+ -BfI)'(R+ -Bft) not unique ( max. var (It)
  • > Normalization needed to identify PCs : 1). B'B = In 2). & diagonal 3 ). V( f+ ) Y 4 columns of loading matrix are arthogonal
  • > estimation : OLS in cross-sectional regr. At = (BIB) "B'RNBR+

Part 3 7 Derivatives def. fin , instruments whose value depends on value of other asset can be used for speculation & hedging (elim . / neutralize risk · traded OTC M 2 on exchanges ↓ Forward Contract def. agreement to buy or sell an asset at a specific future time for a specific price in contrast to spot contract (buy or sell today). · trading = OTC

  • > long position : agree to buy the underlying asset
  • > short position : agree to sell the underlying asset ! At the time forward agreem, is made , forward P is set s. t. no paym. need to be made now => current value of forward contract = o
  • > Value : if Spot rate = forward contract worth & (viceversa) payoff from long position in forw. can . = St-K , where St = spot P at expiration date 7 Profit, a profit K = agreed upon forw. P

payoff long. > ST ST payoff short Future contract def. agreem · betw. 2 parties to buy or sell an asset at a specific + in future for a specific P · traded on organized exchange : futures with std . features

  1. asset : which & quality 3) delivery location : where
  2. contract size : how much 4) delivery time : when - using margin account P& L on futures are settled at end of every trading day + Marking to market
  • > the right futures P : at time new future contr. are issued , P is set s. no paym. needed => current value = 0 Options

& with forw. 2 futures : option gives the right to buy/sell , not the obligation O Call def , gives buyer right to buy underlying asset by a specific date & P s .t. max (St-K , 0) ② Put def. gives buyer right to sell underlying asset by a specific date & P s . t. max(K-ST , 0) trade on UTC & exchanges * lang Eh

  • > terminology : 1) . delivery P ; exercise /strike price K St. Short--long
  1. delivery date ; expiration datel maturity 7
  • > types : 1). European

####### !

right can be exercised only on 7 2) American right can be exercised at any time up to T 3). Bermudan options ; exercised on a limited #of specific dates before T

  • > Moneyness a lang call a long put position & In-the-money (ITM) : pos , payoff long : expec. ↑ ② At-the-money (ATM) = 0 payoff. 1st i 1 Sy Short : expect ③ Cut-of-the-money (OTM) = neg - payoff
  • > trading strategies

####### X A Short call

a Short put option ① protective put : long put & long position K K Se lang · : invester 757 same ② bull spread : buy call at K2 sell at higher 12 Short : Writer

for S S . beton Pr of

underlying with limited P & L putal bear spread : sell put at K . 2 buy at higher he abull a bear a butterfly 5 .to bet on PV with limited PRL & butterfly spread : buy call at K , and K3 & ↓ hist in iist in inst sell at e st. beton vola . being low

a straddle a strangle ③ straddle : buy call & put at K & st. bet on vola . being high i 15 is ins strangle : buy put at K , & buy call at higher te Forward Prices E strategy that generates riskless profit

  • > P for which the forw. PK + current Spot P of asset , grossed up to maturity with of st. 7 arbitrage opportunity
  • > assumptions : 1). transaction costs
  1. all trading profits are taxed at same rate
  2. Of is same for borrowing & lending 4). market participants will take advan of arbitrage appor. as they occur
  • > K = Spert , where = of if K < Soert : arbitragers can buy for So , borrow same #2 sell forward (CF = 0 => at 7 : pay back lan for Soert and sell fork St. riskless profit = K-Soert > o Factors
affecting Option

Prices T ↑ expec. doesn't matter K + ECSy) forw. 2 future Prices are obtained fairly using no-arbitrage arguments = Soertst . deviation but option Prices are trickier ; as options gives the right (not obligation) is arbitrage oppor .

  • > factors : 1) . So 3 ). T 5). r = if
  1. I 1) vola. O 6) dividends o c, 2 PP
  • > notation <p = value European option So t - C, P : value American option K - + T t ?,+ Upperbond So & piPIK o t t
  • > EU put : pLKericK , T= o & ro r t - howerbound D - t
  • > EU call : max (So-Ker , 0) 1 0 - So 3 nonlinear
  • > EU put : max(KerT-So, 0) < PLA Put-call Parity def. both are worth the same at to 2 t = T st. c + Keri = p + So Binomial Market 3 components bond P : Bo Boerst ② Underlying stock with prices Po , Su claim f(u) ② riskless band that determines time-value- of money So ⑤ asset that derives its value from stock e. g. option t= 0 1-p. > Sd f(d) Price of stock only 2 time points : t = 0 & At += xt spot value S : can only attain 2 values at Ut ; Sucup) & Sa (down) & prob, moving up = po s. prob, moving down = 1-po
  • > bond : band can be bought Isold for Bo st. will be worth Boerst at t = At
  • > use flu) 2f(d) : derivative's value when stock P goes upI down s .t. payoff at = et is known but price at + = o is unknown - > idea : Set up replicating portfolio 4 but at t = o is unknown slock & band portfolio d : units of stock & 4 : units of band · t = 0 + cost = So + yBo += - -> worth : 17. OSutyBoerot , if stock p
  1. 4Sa + y Boerst, if stock PV Replicating Portfolio
  • > solving at + = 0 : &SutyBoert = f(u) < = fiul-f(d) &H = Bierot(f(n) - PSu pSd + 4 Boertt = f(d) Su-Sd