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16th Annual Conference
Multinational Finance
Society in Crete (2009)
-
Statistical Distributions in
Finance
(invited presentation)
James B. McDonald
Brigham Young University
June 28- July 1, 2009
The research assistance of Brad Larsen and Patrick Turley is
gratefully acknowledged as are comments from Richard
Michelfelder and Panayiotis Theodossiou.
-
Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
3. Regression applications
4. Qualitative response models
5. Option pricing
6. VaR (value at risk)
7. Conclusion
-
Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
a. Families
3. Regression applications
4. Qualitative response models
5. Option pricing
6. VaR (value at risk)
7. Conclusion
-
Some families of statistical
distributions
a. Families f(y;θ), θ = vector of parameters
i. GB: GB1, GB2, GG (0
-
GB distribution tree
-
Probability Density Functions
( )( )( )( )
( ) ( )( )( )
11 1 1 /
; , , , , , 0 / 1
, 1 /
qaap
a a
p qaap
a y c y bGB y a b c p q y b c
b B p q c y b
−−
+
− −= −
+
( ) ( )( )( )( )
11 1 /
1 ; , , , ; , , 0, ,,
qaap
ap
a y y bGB y a b p q GB y a b c p q
b B p q
−− −
= = =
-
Probability Density Functions
( ) ( )( ) ( )( )
1
2 ; , , , ; , , 1, ,
, 1 /
ap
p qaap
a yGB y a b p q GB y a b c p q
b B p q y b
−
+= = =
+
( )( )
( )
/1
; , ,
ayap
ap
a y eGG y a p
p
−−
=
( )
0 / 1
,
a a
a controls peakedness
b is a scale parameter
c domain y b c
p q shape parameters
−
-
Probability Density FunctionsGB2 PDF evaluated at different parameter values:
-
Some families of statistical
distributions
a. Families
i. GB: GB1, GB2, GG
ii. EGB: EGB1, EGB2, EGG (Y is real valued)
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EGB distribution tree
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Probability Density Functions
( )( ) ( ) ( )( )
( ) ( )( )
1/ /
/
1 1; , , , ,
, 1
qp y m y m
p qy m
e c eEGB y m c p q
B p q ce
−− −
+−
− −=
+
- 1 - <
1
y mfor n
c
−
( )( ) ( )( )
( )
1/ /
11 ; , , ,
,
qp y m y m
e eEGB y m p q
B p q
−− −
−=
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Probability Density Functions
( )( )
( ) ( )( )
/
/2 ; , , ,
, 1
p y m
p qy m
eEGB y m p q
B p q e
−
+−
=+
( )( ) ( )
( )
//
; , ,
y mp y m ee eEGG y m p
p
−− −
=
,
m controls location
is a scale parameter
c defines the domain
p q are shape parameters
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Probability Density FunctionsEGB2 PDF evaluated at different parameter values:
-
Some families of statistical
distributions
a. Families
i. GB: GB1, GB2, GG
ii. EGB: EGB1, EGB2, EGG
iii. SGT (Skewed generalized t): SGED, GT, ST,
t, normal (Y is real valued)
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SGT distribution tree
SGT5 parameter
SGED GT
SLaplace SNormal t SCauchy
Laplace Uniform Normal Cauchy
4 parameter
3 parameter
2 parameter
λ=0 p=2q→∞
ST
GED
λ=0λ=0
λ=0 λ=0λ=0
p=2 p=2
p=2
p=1
p=1
q→∞ q→∞
q=1/2
q→∞ q=1/2
p→∞
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Probability Density Functions
( ); , , , ,SGT y m p q ( )
( )( )( )1/
1/
2 1/ , 11
p
p
p p
q p
p
y mq B p q
sign y m q
+
−+
+ −
=
( )( )( )( )
( )
/ 1
; , , ,2 1/
ppy m sign y m
peSGED y m p
p
− − + −
=
( )
= ( )
=
1 , -1 < < 1
2
, ,
m mode location parameter
scale
skewness area to left of m
p q shape parameters tail thickness moments of order pq df
− = =
= =
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Probability Density FunctionsSGT PDF evaluated at different parameter values:
-
Some families of statistical
distributions
a. Families
i. GB: GB1, GB2, GG
ii. EGB: EGB1, EGB2, EGG
iii. SGT (Skewed generalized t): SGED, GT, ST, t,
normal
iv. IHS
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Probability Density Functions
( )( )( ) sinh 0,1 /Y a b N k= + +
( )
( ) ( ) ( )
( )( )
22 22 2ln / / ln
2
22 2 2
; , , ,
2 /
ky y
keIHS y k
y
− − + + + − + − +
=
+ − +
( ) ( ) ( )2 2 2 2.5 .5.5 2 21/ , / , .5 , and .5 2 1k k k kw w w w we e e e e e
− − − −− + − += = = − = + + −
2
k
mean
variance
skewness parameter
tail thickness
=
=
=
=
( ); , lim ; , , , 0kN y IHS y k →= =
where
IHS
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Probability Density FunctionsIHS PDF evaluated at different parameter values:
-
Some families of statistical
distributions
a. Families
i. GB: GB1, GB2, GG
ii. EGB: EGB1, EGB2, EGG
iii. SGT (Skewed generalized t): SGED, GT, ST, t,
normal
iv. IHS
v. g-and-h distribution (Y is real valued)
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g-and-h distribution
Definition:
where Z ~ N[0,1]
( )2 / 2
,
1gZ hZg h
eY Z a b e
g
−= +
h>0 h
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g-and-h distribution
( ) 2 20,0 ~ ,Y Z a bZ N a b = + =
( ), 01gZ
g h
eY Z a b
g=
−= +
( )2 / 2
0,
gZ
g hY Z a bZe= = +
Is known as the g distribution
where the parameter g allows
for skewness.
Is known as the h distribution
• Symmetric
• Allows for thick tails
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Probability Density Functionsg-and-h PDF evaluated at different parameter values with h>0:
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Probability Density Functionsg-and-h PDF evaluated at different parameter values with h
-
Some families of statistical
distributions
a. Families f(y;θ)
i. GB: GB1, GB2, GG
ii. EGB: EGB1, EGB2, EGG
iii. SGT (Skewed generalized t): SGED, GT, ST, t,
normal
iv. IHS
v. g-and-h distribution
vi. Other distributions: extreme value, Pearson
family, …
-
Some families of statistical
distributions
a. Families f(y;θ)
i. GB: GB1, GB2, GG
ii. EGB: EGB1, EGB2, EGG
iii. SGT (Skewed generalized t): SGED, GT, ST, t, normal
iv. IHS
v. g- and h-distribution
vi. Other distributions: extreme value, Pearson family, …
vii. Extensions: ( )( )1. x = , 2. Multivariate
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Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
a. Families
b. Properties
3. Regression applications
4. Qualitative response models
5. Option pricing
6. VaR (value at risk)
7. Conclusion
-
Some families of statistical
distributions
b. Properties
i. Moments
1. GB family
( )( )( ) 2 1
/ , / ; / , F
/ ;,
h
h
GB
p h a h a cb B p h a qE Y
p q h aB p q
++ =
+ +
for h < aq with c=1
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Some families of statistical
distributions
b. Properties
i. Moments
1. GB family
a. GB1
( )( )( )1
/ ,
,
h
h
GB
b B p h a qE Y
B p q
+=
-
Some families of statistical
distributions
b. Properties
i. Moments
1. GB family
a. GB1
b. GB2
( )( )
( )2/ , /
- / ,
h
h
GB
b B p h a q h aE Y p h a q
B p q
+ −=
-
Some families of statistical
distributions
b. Properties
i. Moments
1. GB family
a. GB1
b. GB2
c. GG
( )( )( )
/ /
h
h
GG
p h aE Y for h a p
p
−=
-
Some families of statistical
distributions
b. Properties
i. Moments
1. GB family
2. EGB family
( ) ( )( )( ) 2 1
, ; c,
p+q+t,
t
ty
EGB
p t te B p t qM t E e F
B p q
++ = =
/ σ with 1for t q c =
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EGB moments
( )p + ( ) ( )p p q + − + ( ) ( )p q + −
( )2 ' p ( ) ( )2 ' 'p p q − + ( ) ( )
2 ' 'p q +
( )3 '' p ( ) ( )3 '' ''p p q − + ( ) ( )
3 '' ''p q −
( )4 ''' p ( ) ( )4 ''' '''p p q − + ( ) ( )
4 ''' '''p q +
EGG EGB1 EGB2
Mean
Variance
Skewness
Excess kurtosis
( )( )d n s
sds
=
-
EGB2 moment space
-
Some families of statistical
distributions
b. Properties
i. Moments
1. GB family
2. EGB family
3. SGT family
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SGT family
( ) ( ) ( ) ( )( )/
1 1
1,
1 1 12 1
,
h p
hh h h h
SGT
h hq B q
p pE y m
B qp
+ +
+−
− = + + − −
( ) ( ) ( ) ( )( )1 11
1 1 12 1
hh h h h
SGED
h
pE y m
p
+ +
+
− = + + − −
for h < pq=d.f.
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SGT moment space
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SGT family moment space
-
Some families of statistical
distributions
a. Families
b. Properties
i. Moments
1. GB family
2. EGB family
3. SGT family
4. IHS
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IHS moment space
-
Some families of statistical
distributions
a. Families
b. Properties
i. Moments
1. GB family
2. EGB family
3. SGT family
4. IHS
5. g-and-h family
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g- and h-family
( )( )
( )
( )
2
2 1
0
,
1
1
i j gi ihj
njn n i i
g h ii
ie
n jE X a b
i g ih
− −
=−
=
−
= −
Moments exist up to order 1/h (0
-
g-and-h moment space (h>0)(visually equivalent to the IHS)
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Moment space for g-and-h (h>0)
and g-and-h (h real)
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Moment space of SGT, EGB2,
IHS, and g-and-h
-
Some families of statistical
distributions
b. Properties
i. Moments
ii. Cumulative distribution functions (see
appendix)
• Involve the incomplete gamma and beta
functions
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Some families of statistical
distributions
b. Properties
i. Moments
ii. Cumulative distribution functions (see appendix)
• Involve the incomplete gamma and beta functions
iii. Gini coefficients (G)
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Gini Coefficients (G)
Definition:
( ) ( )0 0
1: :
2G x y f x f y dxdy
= −
( )( )
( )( )
2
0
0
11
1
F y dy
F y dy
−−
−
( )G =
( )G = (Dorfman, 1979, RESTAT)
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Gini Coefficients
Interpretation:
G = 2A
-
Gini Coefficients
Application: Stochastic Dominance
-
Some families of statistical
distributions
b. Properties
i. Moments
ii. Cumulative distribution functions (see appendix)
iii. Gini coefficients (G)
iv. Incomplete moments
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Incomplete moments
Definition: ( )( )
( );
y
h
h
s f s ds
y hE Y
−=
Applications:
Option pricing formulas
Lorenz Curves
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Incomplete moments
Convenient theoretical results:
( );y h
( )2 2; ,LN y h +
( ); , , /GG y a p h a +
( )2 ; , , / , /GB y a b p h a q h a+ −
Distribution
LN
GG
GB2
-
Some families of statistical
distributions
b. Properties
i. Moments
ii. Cumulative distribution functions (see appendix)
iii. Gini coefficients (G)
iv. Incomplete moments
v. Mixture models
-
Mixture Models
Let denote a structural or conditional
density of the random variable Y where
and denote vectors of distributional
parameters. Let the density of be given by
the mixing distribution . The observed
or mixed distribution can be written as
( ); ,f y
( );g
( ) ( ) ( ); , ; , ;h y f y g d =
-
Mixture Models
Observed model Structural
model
Mixing
distribution
( ); , , , ,SGT y m p q
( ); , ,GT y p q
( )2 ; , , ,EGB y p q
( )2 ; , , ,GB y a b p q
( ); , ,LT y q
( ); ,t y q
( ); , , ,SGED y m s p
( ); ,GED y s p
( )( ); , ln ,EGG y s p
( ); , ,GG y a s p
( ); ,LN y s
( ); ,N y s
( )1/; , ,pIGG s p q q
( )1/; , ,pIGG s p q q
1; , ,IGG s e q
( ); , ,IGG s a b q
( )1/ 2; 1,IGG s a q =
( )1/ 2;IGA s q
-
Some families of statistical
distributions
b. Properties
i. Moments
ii. Cumulative distribution functions (see appendix)
iii. Gini coefficients (G)
iv. Incomplete moments
v. Mixture models
vi. Hazard functions (Duration dependence)
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Hazard functions
Definition:
Let denote the pdf of a spell (S) or duration of an
event.
is the probability that that S>s.
The corresponding hazard function is defined by
which can be thought of as representing the rate or
likelihood that a spell will be completed after surviving
s periods.
( )f s
( )1 F s−
( )( )
( )1
f sh s
F s=
−
-
Hazard functions
Applications:
⚫ Does the probability of ending a strike, unemployment spell, expansion, or stock run depend on the length of the strike, unemployment spell, or of the run?
⚫ With unemployment,⚫ A job seeker might lower their reservation wage and become more likely to find a
job Increasing hazard function
⚫ However, if being out of work is a signal of damaged goods, the longer they are out of work might decrease employment opportunities Decreasing hazard function.
⚫ An alternative example might deal with attempts to model the time between stock trades. ⚫ Engle and Russell (1998) Autoregressive conditional duration: a new model for
irregularly spaced transaction data. Econometrica 66: 1127-1162
⚫ Hazard function of time between trades is decreasing as t increases or the longer the time between trades the less likely the next trade will occur.
-
Hazard functions
Applications:
⚫ Bubbles⚫ McQueen and Thorley (1994) Bubbles, stock returns, and duration dependence.
Journal of Financial and Quantitative Analysis, 29:379-401
⚫ Efficient markets hypothesis, stock runs should not exhibit duration dependence (constant hazard function)
⚫ McQueen and Thorley argue that asset prices may contain “bubbles” which grow each period until they “burst” causing the stock market to crash. Hence, bubbles cause runs of positive stock returns to exhibit duration dependence—the longer the run the less likely it will end (decreasing hazard function), but runs of negative stock returns exhibit no duration dependence
⚫ Grimshaw, McDonald, McQueen, and Thorley. 2005, Communications in Statistics—Simulation and Computation, 34: 451-463.
⚫ What model should we use to characterize duration dependence?⚫ Exponential—constant
⚫ Gamma—the hazard function can increase, decrease, or be constant
⚫ Weibull—the hazard function can increase, decrease, or be constant
⚫ Generalized Gamma: the hazard function can be increasing, decreasing, constant, -shaped, or -shaped
-
Hazard functions
Possible shapes for the GG hazard functions
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Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
a. Families
b. Properties
c. Model selection
3. Regression applications
4. Qualitative response models
5. Option pricing
6. VaR (value at risk)
7. Conclusion
-
Some families of statistical
distributionsc. Model selection
i. Goodness of fit statistics
• Log-likelihood values
o for individual data
o for grouped data
Partition the data into g groups,
Empirical frequency:
Theoretical frequency:
( ) ( )( )1
:n
i
i
n f y =
=
( ) ( ) ( )( ) ( ) 1
!g
i i i
i
n n n n p n n =
= + −
)1, , 1,2,...,i i iI Y Y i g−= =
1
/ , g
i i i
i
p n n n n=
= =
( ) ( );i
i
I
p f y dy =
-
Model Selection
i. Goodness of fit statistics
• Log-likelihood values
• Possible Measures
( )1
g
i i
i
SAE p p =
= −
( )( )2
1
g
i i
i
SSE p p =
= −
( ) ( ) ( )2
2 2
1
/ ~ # 1g
ii i
i
nn p p g parameters
n
=
= − − −
-
Model Selection
i. Goodness of fit statistics
• Log-likelihood values
• Possible Measures
• Akaike Information Criterion (AIC)
• A tool for model selection
• Attaches a penalty to over-fitting a model
( )( )2AIC k= −
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Model Selection
i. Goodness of fit statistics
ii. Testing nested models
Examples:
1.
2.
( ): 0OH g =
: : 0O OH SGT GT H = =
: : 2, 0, O OH SGT Normal H p and q= = = →
-
Testing nested models
Likelihood ratio tests
where r denotes the number
of independent restrictions
Wald test
( ) ( )22 * ~aLR r= −
( ) ( )21 2 * ~ 1a
SGT GTLR = −
( ) ( )22 2 * ~ 3a
SGT NormalLR = −
( )( ) ( )( )( ) ( ) ( )1
2' var ~aMLE MLE MLEW g g g r −
=
( ) ( )( ) ( ) ( )1
2
1ˆ ˆ ˆ0 0 ~ 1aW Var
−
= − −
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Statistical Distributions in
Finance1. Introduction
2. Some families of statistical distributions
a. Families
b. Properties
c. Model selection
d. An example: the distribution of stock returns
3. Regression applications
4. Qualitative response models
5. Option pricing
6. VaR (value at risk)
7. Conclusion
-
An example: the distribution of
stock returns
( ) 1 11/ ~ 1t t t
t t t
t t
P P Py n P P
P P
+ +−
−= = −
Daily, weekly, and monthly excess returns (1/2/2002 –
12/29/2006) from CRSP database (NYSE, AMEX, and
NASDAQ)— 4547 companies
H0: skewness = 0
H0: excess kurtosis = 0
H0: returns ~ N(μ, σ2)
JB =
( ).95 2 6/ , 2 6/CI n n= −( ).95 2 24/ , 2 24/CI n n= −
( )( )
222
.05
~ 2 5.99
6 24
excess kurtosisskewn
+ =
( ).95 0 5.99CI JB=
-
An example: the distribution of
stock returns (continued)
% of stocks for which excess returns statistics are in 95% C.I.
HO: Skewness=0 HO:Excess kurtosis=0 HO: Normal
Daily 16.38% 0.04% 0.09%
Weekly 30.61% 4.88% 4.75%
Monthly 66.79% 56.65% 53.77%
-
An example: the distribution of
stock returns (continued)
Daily excess returns plotted with admissible moment space of flexible distributions
-4 -3 -2 -1 0 1 2 3 40
10
20
30
40
50
60
Skewness
Kurt
osis
CRSP daily stocks--excess returns
CRSP stock
EGB2
SGT
IHS
bound
-
An example: the distribution of
stock returns (continued)
Weekly excess returns plotted with admissible moment space of flexible distributions
-4 -3 -2 -1 0 1 2 3 40
10
20
30
40
50
60
Skewness
Kurt
osis
CRSP weekly stocks--excess returns
CRSP stock
EGB2
SGT
IHS
bound
-
An example: the distribution of
stock returns (continued)
Monthly excess returns plotted with admissible moment space of flexible distributions
-4 -3 -2 -1 0 1 2 3 40
10
20
30
40
50
60
Skewness
Kurt
osis
CRSP monthly stocks--excess returns
CRSP stock
EGB2
SGT
IHS
bound
-
An example: the distribution of
stock returns (continued)
Fraction of stocks in the admissible skewness-kurtosis
space
daily weekly monthly
EGB2 15.48% 43.81% 50.80%
IHS 83.92% 84.39% 61.97%
SGT 87.62% 89.00% 95.10%
g-and-h 100.00% 99.98% 98.99%
-
An example: the distribution of
stock returns (continued)
Fitting a PDF to normal excess returns
Company Name Skew Kurtosis Jb Stat
US Steel 0.06 3.308 5.62
Estimated PDF logL SSE SAE Chi^2
Normal 2753.52 0.001 0.12 27.81
EGB2 2756.83 0.001 0.11 23.38
IHS 2756.76 0.001 0.11 23.46
SGT 2758.78 0.001 0.12 28.19
-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.20
2
4
6
8
10
12
14
16
18
20
Excess returns
Estimated PDFs for US Steel daily excess returns
Returns
Normal
EGB2
IHS
SGT
-
An example: the distribution of
stock returns (continued)
Company Name Skew Kurtosis Jb Stat
iShares -29.06 965.09 48733899.02
Fitting a PDF to leptokurtic excess returns
Estimated PDF logL SSE SAE Chi^2
Normal 2516.86 0.099 0.93 1433.33
EGB2 3713.99 0.002 0.13 43.47
IHS 3795.21 0.001 0.12 33.43
SGT 3810.07 0.003 0.21 79.35
-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.080
5
10
15
20
25
30
35
40
45
50
Excess returns
Estimated PDFs for iShares daily excess returns
Returns
Normal
EGB2
IHS
SGT
-
Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
3. Regression applications
4. Qualitative response models
5. Option pricing
6. VaR (value at risk)
7. Conclusion
-
Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
3. Regression applicationsa. Background
4. Qualitative response models
5. Option pricing
6. VaR (value at risk)
7. Conclusion
-
Regression applications--
background
Model:
1xK vector of observations on the explanatory
variables
Kx1 vector of unknown coefficients
independently and identically distributed random
disturbances with pdf
t t tY X = +
tX
t( );f
-
Regression applications--
background
⚫ If the errors are normally distributed
⚫ OLS will be unbiased and minimum variance
⚫ However, if the errors are not normally distributed
⚫ OLS will still be BLUE
⚫ There may be more efficient nonlinear estimators
-
Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
3. Regression applicationsa. Background
b. Alternative estimators
4. Qualitative response models
5. Option pricing
6. VaR (value at risk)
7. Conclusion
-
Alternative Estimators
i. Estimation
OLS
LAD
Lp
( )2
1
arg minn
OLS t t
t
Y X =
= = −
1
arg minn
LAD t t
t
Y X =
= = −
1
arg minp
pn
L t t
t
Y X =
= = −
-
Alternative Estimators
(continued)
i. Estimation (continued)
M-estimators:
⚫ Includes OLS, LAD, and Lp as special cases
⚫ Includes MLE (QMLE or partially adaptive estimators) as a special case where
▪ SGT
▪ SGED
▪ EGB2
▪ IHS
( ) ( ); ;n f = −
( ),1
arg min ;n
MLE t t
t
Y X =
= = −
( )1
arg minn
M t t
t
Y X =
= = −
-
Alternative Estimators
(continued)
i. Estimation
ii. Influence functions: ( ) '( ) =
OLS LADRedescending
influence function
-
Alternative Estimators
(continued)
i. Estimation
ii. Influence functions
iii. Asymptotic distribution of extremum
estimators
where
( )min H
1 1ˆ ~ ;a sandwichN A BA − − =
( )2 and
' '
d H dH dHA E B E
d d d d
= =
-
Alternative Estimators
(continued)
i. Estimation
ii. Influence functions
iii. Asymptotic distribution of extremum estimators
iv. Other estimators
⚫ Semiparametric (Kernel estimator, Adaptive MLE)
where
denotes a kernel, and h is the window width
( )1
arg min n
SP K t t
t
n f Y X =
= − = −
( )1
1 n iK
i
ef K
nh h
=
− =
i i i OLSe Y X = −
( )K
-
Regression applications
(continued)
iv. Other estimators (continued)
⚫ Generalized Method of Moments (GMM)
where
Z denotes a vector of instruments (can be X)
Q is a positive definite matrix
( ) ( )arg min 'GMM g Qg =
( ) ( )1
n
i i i i
i
g Z h Y X =
= = −
( )1( )Q Var g −=
-
Statistical Distributions in
Finance1. Introduction
2. Some families of statistical distributions
3. Regression applicationsa. Background
b. Alternative estimators
c. A Monte Carlo comparison of alternative estimators
4. Qualitative response models
5. Option pricing
6. VaR (value at risk)
7. Conclusion
-
A Monte Carlo comparison of
alternative estimators
c. A Monte Carlo comparison of alternative estimators
⚫ Model:
⚫ Error distributions: (zero mean and unitary variance)
Normal:
Mixture:
Skewness =0
Kurtosis =24.3
Skewed:
Skewness=6.18
Kurtosis=113.9
0;1N
.9* 0,1/ 9 .1* 0,9N N+
( )( ) ( ).50,1 / 1LN e e e− −
1t t ty X = − + +
-
A Monte Carlo comparison of
alternative estimators
Skewness
Kurtosis
Skewed
Mixture
Normal
-
A Monte Carlo comparison of
alternative estimators
Estimators Normal Mixture-thick tails Skewed
OLS .275 .287 .280
LAD .332 .122 .159
SGED .335 .128 .060
ST .293 .112 .054
GT .314 .133 .135
SGT .335 .125 .073
EGB2 .287 .125 .049
IHS .285 .119 .054
SP = AML .285 .114 .128
GMM .319 .115 .088
Sample size = 50, T=1000 replications
RMSE for slope estimators
-
Statistical Distributions in
Finance1. Introduction
2. Some families of statistical distributions
3. Regression applicationsa. Background
b. Alternative estimators
c. A Monte Carlo comparison of alternative estimators
d. An application: CAPM i. Error distribution effects
ii. ARCH effects
4. Qualitative response models
5. Option pricing
6. VaR (value at risk)
7. Conclusion
-
An application: CAPM
i. CAPM and the error distribution
Daily, weekly, and monthly excess returns
(1/2/2002 – 12/29/2006) from CRSP database
(NYSE, AMEX, and NASDAQ)— 4547
companies
HO: Skewness=0 HO:Excess kurtosis=0 HO: Normal (JB)
Daily 14.14% 0.02% 0%
Weekly 28.13% 3.91% 3.43%
Monthly 67.56% 57.14% 54.76%
Percent of stocks for which OLS residual statistics are in 95% C.I.
-
An application: CAPM with and
without ARCH effects (ST)
i. CAPM and the error distribution
Daily, weekly, and monthly excess returns
(1/2/2002 – 12/29/2006) from CRSP database
(NYSE, AMEX, and NASDAQ)— 4547
companies
HO: Skewness=0 HO:Excess kurtosis=0 HO: Normal (JB)
Daily 14.05% 0.02% 0%
Weekly 28.82% 3.83% 3.39%
Monthly 64.04% 54.72% 51.48%
Percent of stocks for which ST residual statistics are in 95% C.I.
-
An application: CAPM with and
without ARCH effects (IHS)
i. CAPM and the error distribution
Daily, weekly, and monthly excess returns
(1/2/2002 – 12/29/2006) from CRSP database
(NYSE, AMEX, and NASDAQ)— 4547
companies
HO: Skewness=0 HO:Excess kurtosis=0 HO: Normal (JB)
Daily 13.99% 0.02% 0%
Weekly 27.89% 3.83% 3.36%
Monthly 65.54% 55.71% 52.32%
Percent of stocks for which IHS residual statistics are in 95% C.I.
-
An application: CAPM with
alternative error distributions
Company Name Skewness Kurtosis JB stat
UNITED NATURAL FOODS INC -0.074 2.8004 0.1543
99 CENTS ONLY STORES 1.7541 7.6594 85.0456
Statistics of OLS residuals
Company Name OLS T GT SGED EGB2 IHS ST SGT
UNITED NATURAL FOODS INC 0.313 0.313 0.335 0.334 0.303 0.302 0.314 0.335
99 CENTS ONLY STORES 0.184 0.125 0.125 0.110 0.109 0.106 0.110 0.110
Estimated Betas
-
An application: CAPM with and
without ARCH effects
i. CAPM and the error distribution
ii. CAPM: how about ARCH effects?
⚫ Review:
⚫ If errors are normal and no ARCH effects, OLS is MLE
⚫ If errors are not normal and no ARCH effects OLS is
BLUE, but not MLE nor efficient
⚫ If errors are normal and have ARCH effects OLS is
BLUE, but not efficient
⚫ If errors are not normal and have ARCH effects OLS
is BLUE,but not efficient
-
An application: CAPM with and
without ARCH effects
ii. CAPM: ARCH effects (continued)
⚫ Model:
Percent of stocks exhibiting ARCH(1) effects (OLS)
(% rejecting )1: 0OH =
0.10 level 0.05 level
Daily 63.2% 60.0%
Weekly 29.2% 24.1%
Monthly 18.7% 13.7%
t t tY X = +
.52
0 1 1t t tu − = +
-
An application: CAPM with and
without ARCH effects
Percent of stocks exhibiting ARCH(1) effects (ST)
(% rejecting )
Percent of stocks exhibiting ARCH(1) effects (IHS)
(% rejecting )
0.10 level 0.05 level
Daily 63.2% 59.9%
Weekly 29.1% 23.9%
Monthly 16.9% 12.3%
1: 0OH =
0.10 level 0.05 level
Daily 63.3% 60.0%
Weekly 29.3% 24.1%
Monthly 18.9% 13.9%
1: 0OH =
-
An application: CAPM with and
without ARCH effects
ii. CAPM: ARCH effects (continued)
⚫ ARCH Simulations▪ , t= 1, …, 60
▪ X monthly excess market returns, 1/2002 to 12/31/2006
▪ Error distributions
( ) 0 .9 t t ty X excess market return = + = +
2~ 0,t N
( ) ( ) 1
.52
0 11 : where ~ 0,1tN t t tARCH u u N −= +
( ) ( )1
.52
0 11 : where ~ (5)tt t t tARCH u u t −= +
-
An application: CAPM with and
without ARCH effects
ARCH Simulations (continued)
Errors
Est imat ion N on- A R C H A R C H N on- A R C H A R C H N on- A R C H A R C H
OLS/Normal 0.352 0.356 0.347 0.291 0.353 0.300
LAD 0.444 0.446 0.397 0.369 0.315 0.297
T 0.358 0.363 0.338 0.293 0.283 0.265
GED 0.381 0.389 0.357 0.318 0.306 0.285
GT 0.387 0.396 0.362 0.322 0.306 0.286
SGED 0.406 0.417 0.374 0.341 0.318 0.297
EGB2 0.371 0.376 0.352 0.312 0.300 0.281
IHS 0.368 0.377 0.348 0.319 0.291 0.275
ST 0.375 0.382 0.350 0.310 0.293 0.277
SGT 0.409 0.420 0.376 0.344 0.316 0.297
Root Mean Square Error (RMSE) for 10,000 replications
N ( 0 ,σ^2 ) N ( 0 ,1) , A rch( 1) t ( 5) , A rch( 1)
-
Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
3. Regression applications
4. Qualitative response models
5. Option pricing
6. VaR (value at risk)
7. Conclusion
-
Qualitative Response Models
-
Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
3. Regression applications
4. Qualitative response models
a. Basic framework
5. Option pricing
6. VaR (value at risk)
7. Conclusion
-
Qualitative Response—
Basic Framework
⚫ Model:
if and 0 otherwise
⚫ Log-likelihood function:
*
i i iy X = −
1 iy =* 0iy
( ) ( )*Pr 1 Pri i i i i iy X y X X = = = −
( ) ( ) ( )Pr ; ;iX
i i iX f s ds F X
−
= = =
( ) ( )( ) ( ) ( )( ) 1
, ; 1 1 ;n
i i i i
i
y n F X y n F X =
= + − −
-
Qualitative Response—
Basic Framework (continued)
⚫ MLE of will be consistent and asymptotically distributed as
if the model is correctly specified.
⚫ Probit and logit estimators will be inconsistent if⚫ The error distribution is incorrectly specified
⚫ heteroskedasticity exists, e.g. unmeasured heterogeneity is present
⚫ relevant variables have been omitted
⚫ The index appears in a nonlinear form
⚫ Similar results are associated with Censored & Truncated regression models
12
ˆ ~ ;'
a dN Ed d
− = −
̂
-
Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
3. Regression applications
4. Qualitative response models
a. Basic framework
b. An application: fraud detection
5. Option pricing
6. VaR (value at risk)
7. Conclusion
-
⚫ Prediction of corporate fraud (Y=1 fraud)
⚫ Compare financial ratios of companies with averages
of five largest companies (“virtual” firm)
⚫ 228 companies (114 fraud and 114 non-fraud)
⚫ Variables: accruals to assets, asset quality, asset
turnover, days sales in receivables, deferred charges
to assets, depreciation, gross margin, increase in
intangibles, inventory growth, leverage, operating
performance margin, percent uncollectables,
receivables growth, sales growth, working capital
turnover.
⚫ SGT, EGB2, & IHS formulations improve predictions
Qualitative response—
An application: fraud detection
-
Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
3. Regression applications
4. Qualitative response models
a. Basic framework
b. An application
c. Some related issues
5. Option pricing
6. VaR (value at risk)
7. Conclusion
-
Qualitative response—
Some related issues
⚫ Cost of misclassification
⚫ Choice-based sampling
⚫ Heterogeneity
⚫ Semi-parametric estimation procedures
-
Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
3. Regression applications
4. Qualitative response models
5. Option pricing: European call option
6. VaR (value at risk)
7. Conclusion
-
Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
3. Regression applications
4. Qualitative response models
5. Option pricing: European call option
a. The Black-Scholes option pricing formula
6. VaR (value at risk)
7. Conclusion
-
Option pricing—
Black-Scholes
a. The Black Scholes option pricing formula
The equilibrium price of a European call option is equal to the present value of its expected return at expiration:
where involve “normalized
incomplete” moments
( ) ( )( ) ( ) ( )0, , ,0 ,
;1 ;0
rT rt
f T T
X
rT
T
T t
C S T X e E C S e S X f S S T dS
X XS e X
S S
− −
−
= = −
= −
( )( )
( )
( )
( ); 1
yhh
y
h h
s f s dss f s ds
y hE y E y
−= − =
( ) ( )( ); 1 ;y h y h = −
.
-
Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
3. Regression applications
4. Qualitative response models
5. Option pricing: European call option
a. The Black-Scholes option pricing formula
b. Some background and alternative formulations
6. VaR (value at risk)
7. Conclusion
-
Option pricing– Some background
and alternative formulations
⚫ The Black Scholes (1973) option pricing formula corresponds to being the lognormal
⚫ , the cdf for the lognormal
⚫ The Black Scholes formula (Bookstaber and McDonald, 1991) corresponding to the Generalized Gamma is obtained from
⚫ , the cdf for the GG
⚫ The Black Scholes formula ( Bookstaber and McDonald, 1991) corresponding to the GB2 is obtained from
⚫ , the cdf for the GB2
⚫ Rebonato (1999) applied to the Deutschemark
( )f s
( ) ( )2 2; ; ,LN y h LN y h = +
( ); ; , ,GGh
y h GG y a pa
= +
( )2 ; 2 ; , , ,GBh h
y h GB y a b p qa a
= + −
( )2 , ,GB TC S T X
-
Option pricing– Some background
and alternative formulations
⚫ Sherrick, Garcia, and Tirupattur (1996) used to price soybean futures.
⚫ Theodosiou (2000) developed the
⚫ Savickas (2001) explored the use of
⚫ Dutta and Babbel (2005) explore the g- and h- family (4-parameter) of option pricing formulas, , based on Tukey’s nonlinear transformation of a standard normal.
⚫ Applied the g-and-h to pricing 1-month and 3-month London Inter Bank Offer Rates (LIBOR)
⚫ g- and- h distribution and GB2 perform much better (errors fairly highly correlated) than the Lognormal, Burr 3, and Weibull distributions
( )& , ,g h TC S T X− −
( ), ,SGED TC S T X
( ), ,Weibull TC S T X
( )3 , ,Burr TC S T X−
-
Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
3. Regression applications
4. Qualitative response models
5. Option pricing: European call option
a. The Black-Scholes option pricing formula
b. Some background and alternative formulations
c. A comparison of pricing behavior
6. VaR (value at risk)
7. Conclusion
-
A comparison of pricing
behaviorc. A comparison of pricing behavior (Dutta and Babbel, Journal of
Business, 2005) ⚫ Calculates the difference between the market price and predicted price
for the g-and-h, GB2, lognormal, Burr3, and Weibull distributions
-
Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
3. Regression applications
4. Qualitative response models
5. Option pricing: European call option
6. VaR (value at risk)
7. Conclusion
-
Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
3. Regression applications
4. Qualitative response models
5. Option pricing: European call option
6. VaR (value at risk)
a. Background and definitions
7. Conclusion
-
VaR—Background and
definitions
i. Value at risk (VaR) is the maximum expected loss on a portfolio of assets over a certain time period for a given probability level.
⚫ R is the return on the asset
⚫ θ denotes the distributional parameters
⚫ α is the predetermined confidence level or coverage probability
⚫ is the corresponding maximum expected loss or conditional threshold
( )( )
;R
f R dR
−
=
( ) ( )1 :RR F
−=
( )R
-
VaR—Background and
definitions
R z = +
( )( )
( )( )( ) ( )1 :; ,
R
Z Z
Z
Ff z dz Z
−=
−
−==
( ) ( )1 :ZR F
−= +
ii. Standardized returns
-
VaR—Background and
definitions
iii. Unconditional VaR formulation
Estimate f(R;θ)
-
VaR—Background and
definitions
iv. Conditional VaR formulation (AR(1) ABS-
GARCH(1,1))
0 1 1t t t t t t tR R Z z −= + + = +
0 1 1 1 2 1t t t tz − − −= + +
( )
( )1 :t t t ZR F
−= +
-
Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
3. Regression applications
4. Qualitative response models
5. Option pricing: European call option
6. VaR (value at risk)
a. Background and definitions
b. Models and applications
7. Conclusion
-
VaR—
Models and applications
i. Unconditional VaR formulation
⚫ Exponential: (Hogg, R. V. and S. A. Klugman (1983))
⚫ Gamma: (Cummins, et al. 1990)
⚫ Log-gamma: (Ramlau-Hansen (1988)), (Hogg, R. V. and
S. A. Klugman (1983))
⚫ Lognormal: (Ramlau-Hansen (1988))
⚫ Stable: (Paulson and Faris (1985)
⚫ Pareto: (Hogg, R. V. and S. A. Klugman (1983))
⚫ Log-t: (Hogg, R. V. and S. A. Klugman (1983))
⚫ Weibull: (Cummins et al. (1990))
-
VaR—
Models and applications
i. Unconditional VaR formulation (continued)
⚫ Burr: (Hogg, R. V. and S. A. Klugman (1983))
⚫ Generalized Pareto: (Hogg, R. V. and S. A.
Klugman (1983))
⚫ GB2: (Cummins (1990, 1999, 2007)
⚫ Pearson family: Aiuppa (1988)
⚫ Extreme value distribution: Bali (2003), Bali and
Theodossiou (2008)
⚫ IHS: Bali and Theodossiou (2008)
-
VaR—
Models and applications
ii. Conditional VaR formulations
(Bali and Theodossiou, JRI, 2008)
⚫ Data:
▪ S&P500 composite index, 1/4/50 – 12/29/2000 (n=12,832)
▪ Daily percentage log-returns: (Sample mean = .0341,
maximum=8.71, minimum=-22.90
⚫ standard deviation = .874
⚫ skewness =1.622
⚫ kurtosis=45.52
-
VaR—
Models and applications
ii. Conditional distributions (Bali and Theodossiou, JRI, 2008) (continued)
⚫ Models
⚫ Generalized extreme value
⚫ EGB2
⚫ SGT
⚫ IHS
⚫ Findings
⚫ Out of sample VaR estimates are rejected for most unconditionalspecifications
⚫ Thresholds exhibit time varying behavior
⚫ Out of sample VaR estimates for the conditional specifications corresponding to the SGT, IHS, and EGB2 perform better than the
extreme value distributions
-
Selected references for option pricing and VaR
⚫ Aiuppa, T. A. 1988. “Evaluation of Pearson curves as an approximation of the maximum probable annual aggregate loss.” Journal of Risk and Insurance 55, 425-441
⚫ Bali, T. G., 2003. “An Extreme Value Approach to Estimating Volatility and Value at Risk,” Journal of Business, 76:83-108
⚫ Bali, T. G. and P. Theodossiou, 2007. “A Conditional-SGT-VaR Approach with Alternative GARCH Models,” Annals of Operations Research, 151: 241-267.
⚫ Bali, T. G. and P. Theodossiou, 2008. “Risk Measurement Performance of Alternaitve Distribution Functions,” Journal of Risk and Insurance, 75: 411-437.
⚫ Black, F (1976). The Pricing of Commodity Contracts. Journal of Financial Economics 3:169-179.
⚫ Cummins, J. D., G. Dionne, J. B. McDonald, and B. M. Pritchett 1990. “Applications of the GB2 family of distributions in modeling insurance loss processes.” Insurance: Mathematics and Economics 9, 257-272.
⚫ Cummins, J. D., C. Merrill, and J. B. McDonald, 2007. “Risky Loss Distributions and Modeling the Loss Reserve Pay-out Tail,” Review of Applied Economics 3.
⚫ Cummins, J. D., R. D. Phillips, and S. D. Smith 2001. “Pricing Excess of Loss Reinsurance Contracts against catastrophic loss.” In Kenneth Froot, ed., The Financing of Catastrophe Risk (Chicago: University of Chicago Press)
⚫ Dutta, K. K. and D. F. Babbel 2005. “Extracting Probabilistic Information from the Prices of Interest Rate Options: Tests ofDistributional Assumptions.” Journal of Business 78:841-870
⚫ Hogg, R. V. and S. A. Klugman, 1983. “On the Estimation of Long Tailed Skewed Distributions with Actuarial Applications.” Journal of Econometrics 23, 91-102.
⚫ McDonald, J. B. and R. M. Bookstaber (1991). “Option Pricing for Generalized Distributions.” Communications in Statistics: Theory and Methods, 20(12), 4053-4068.
⚫ Rebonato, R. (1999). Volatility and correlations in the pricing of equity. FX and interest-rate options. New York: John Wiley.
⚫ Paulson, A. S. and N. J. Faris (1985). “A Practical Approach to Measuring the Distribuiton of Total Annual Claims.” In J. D. Cummins, ed., Strategic Planning and Modeling in Property-Liability Insurance. Norwell, MA: Kluwer Academic Publishers.
⚫ Ramlau-Hansen, H. (1988). “A Solvency Study in Non-life Insurance. Part 1. Analysis of Fire, Windstorm, and Glass Claims.” Scandinavian Actuarial Journal, pp. 3-34.
⚫ Rebonato, R. 1999. Volatility and correlations in the pricing of equity, FX and interest-rate options. New York: John Wiley.
⚫ Reid, D. H. (1978). “Claim Reserves in General Insurance,” Journal of the Institute of Actuaries 105: 211-296
⚫ Savickas, R. (2001). A Simple option-pricing formula. Working paper, Department of Finance, George Washington University, Washington, DC.
⚫ Sherrick, B. J., P. Garcia, and V. Tirupattur (1996). Recovering probabilistic information for options markets: Tests of distributional assumptions. Journal of Futures Markets 16:545-560.
⚫ Theodossiou, Panayiotis, “Skewed Generalized Error Distribution of Financial Assets and Option Pricing,”
-
Statistical Distributions in
Finance
1. Introduction
2. Some families of statistical distributions
3. Regression applications
4. Qualitative response models
5. Option pricing: European call option
6. VaR (value at risk)
7. Conclusion
-
Conclusion
-
END OF PRESENTATION
-
Appendices
⚫ Cumulative distribution functions
1. GB, GB1, GB2, GG
2. EGB2
3. SGT
4. SGED
5. IHS
6. g-and-h distribution
⚫ Option pricing basics
⚫ VaR—Models and applications discussion
-
Appendices—
Cumulative distribution functions
1. GB, GB1, GB2, and GG
where and
denotes the incomplete beta function
( )
( )
( )
2 1 ,1 ; 1;1 ; , , ,
,
,
p
z
z F p q p zGB y a b p q
pB p q
B p q
− +=
=
( )/a
z y b=
( )( )
( )
11
0
1
,,
zqp
z
s s ds
B p qB p q
−− −
=
-
Appendices—
Cumulative distribution functions
1. GB, GB1, GB2, and GG (continued)
where
( )
( )
( )
2 1 ,1 ; 1;2 ; , , ,
,
,
p
z
z F p q p zGB y a b p q
pB p q
B p q
− +=
=
( )
( )
/
1 /
a
a
y bz
y b=
+
-
Appendices—
Cumulative distribution functions
1. GB, GB1, GB2, and GG (continued)
where
and
denotes the incomplete gamma function
Abramowitz and Stegun (1970, p. 932), McDonald (1984), and Rainville (1960,p. 60 and 125)
( )( ) ( )
( )( )
/
1 1
/; , , 1; 1; /
1
a apyae y
GG y a b p F p yp
−
= + +
( )z p=
( )/a
z y =
( )( )
1
0
z
p s
z
s e ds
pp
− −
=
-
Appendices—
Cumulative distribution functions
2. EGB2
where
3. SGT
where
( ) ( )2 ; , , , ,zEGB y m p q B p q =
( )
( )
/
/1
y m
y m
ez
e
−
−=
+
( )( )( )
( ) ( )11
; , , , , 1/ ,2 2
z
sign y mSGT y m p q sign y m B p q
+ −−= + −
( )( )1
p
pp p
y mz
y m q sign y m
−=
− + + −
-
Appendix—
Cumulative distribution functions
4. SGED
where
( )( )
( ) ( )11
; , , , 1/2 2
z
sign y mSGED y m p sign y m p
+ − −= + −
( )( )1
p
pp
y mz
sign y m
−=
+ −
-
Appendices—
Cumulative distribution functions
5. IHS
where
( ) ( ) ( ); , , , Pr PrIHS y k Y y Z z = =
( ) ( )2; 0, 1 PrN z Z z = = = 2
1 1
1 1 3 ; ;
2 2 2 22
z zF
−= +
( )( )( )2 / 2
1 1
2 2 2z
sign z = +
2
1y a y a
z k n kb b
− − = + + −
/ wb = = ( ) ( )2 2 2.5 .52 2/ .5 2 1k k ke e e − − −+ − + + + −
( )( )2.5.5 kwa b b e e e
−−= − = − −
and with
-
Appendices—
Cumulative distribution functions
6. g- and h-distribution
⚫ Numeric procedures, based on the use of order statistics as outlined in Exploring Data Tables, Trends, and Shapes by Hoaglin,, Mosteller, and Tukey (1985), Wiley.
⚫ For h > 0, the transformation
is one-to-one, (Martinez, J. and B. Iglewicz . 1984. “Some Properties of Tukey g and h family of distributions,” Communications in Statistics—Theory and Methods 13, 353-369). Even without an explicit functional form for the inverse, numerical “MLE” estimates” can be obtained.
( )2 / 2
,
1gZ hZg h
eY Z a b e
g
−= +
-
Appendices
⚫ Cumulative distribution functions
⚫ Option pricing basics
1. European call option
2. Put option
3. Definitions of terms
4. Assumptions
5. Volatility
6. The Greeks
⚫ VaR—Models and applications discussion
-
Appendices—
Option pricing basics
1. European call option
2. Put option
( ) ( )( ) ( ) ( )
( ) ( )( )
0
T 1 2
, , , ,0, , ,
;1 ;0 BS: S d
rT rt
f T T
X
rT rT
T
T t
C S T X r e E C S X r e S X f S S T dS
X XS e X e X d
S S
− −
− −
= = −
= − −
( ) ( )2 1 : -rT
TBS Put formula e X d S d− − −
-
Appendices—
Option pricing basics
3. Definitions of terms:⚫ T = time to expiration
⚫ ST = Current market price
⚫ r = interest rate (risk free rate)
⚫ X = strike price (or exercise price) ▪ call options: price at which the instrument can be purchased
up to expiration
profit per share gained upon exercising or selling the option
>0 in the money
-
Appendices—
Option pricing basics
4. Assumptions:⚫ Can short sell the underlying instrument
⚫ No arbitrage opportunities
⚫ Continuous trading in the instrument
⚫ No taxes or transaction costs
⚫ Securities are perfectly divisible
⚫ Can borrow or lend at a constant risk free rate
⚫ The instrument does not pay a dividend
5. Volatility (in the BS option pricing formula—based on the LN)
-
Appendices—
Option pricing basics
6. The Greeks:
⚫ (delta) measures the change in value of the instrument to a change in the current market price
⚫ (kappa or vega) measures the responsiveness of the value of the instrument in response to a change in volatility
⚫ (theta) responsiveness of the value of the instrument to T (time to expiration)
⚫ (rho) responsiveness to changes in the risk free rate
( )( ), , ,;1
f T
T T
C S T X r X
S S
= =
( )( ), , ,( )
f TC S T X r
volatility
=
( )( ), , ,f TC S T X rT
= −
( )( ), , ,f TC S T X rr
=
-
Appendices
⚫ Cumulative distribution functions
⚫ Option pricing basics
⚫ VaR—Models and applications
discussion
-
Appendices—VaR: Models and
applications discussion
⚫ Paulson and Faris (1985) used the stable family and Aiuppa (1988) used the Pearson family to model insurance losses
⚫ Ramlau-Hansen (1988) modeled fire, windstorm, and glass claims using the log-gamma and lognormal
⚫ Cummins, et al. (1990) modeled fire losses using the GB2
⚫ Cummins, Lewis, and Phillips (1999) used the LN, Burr 12, and GB2 to model hurricane and earthquake losses.
⚫ Hogg, R. V. and S. A. Klugman, 1983. “On the Estimation of Long Tailed Skewed Distributions with Actuarial Applications.” Journal of Econometrics 23, 91-102
⚫ Models loss distributions (a. Hurricaines (1949-1980), b. malpractice claims paid for insured hospitals in 1975)
⚫ Considers exponential, pareto (mixture of an exponential and inverse gamma), generalized pareto (mixture of gamma and inverse gamma), Burr distribution (mixture of a Weibull and inverse gamma), log-t (mixture of a lognormal and inverse gamma) and a log-gamma.
⚫ Consider alternative estimation procedures: maximum likelihood and minimum distance estimators
⚫ Many loss distributions are characterized by skewness and long tails such as associated with the flexible distributions coming from mixtures.
-
Appendices—VaR: Models and
applications discussion
⚫ Cummins, J. D., G. Dionne, J. B. McDonald, and B. M. Pritchett, 1990. “Applications of the GB2 family of distributions in modeling insurance loss processes.” Insurance: Mathematics and Economics 9, 257-272.
⚫ Models fire losses
⚫ Considers the GB2 and special cases GG, BR3, BR12, LN, W, and GA to model the fire loss data. MLE estimates of distributional parameters and Maximum Probably Yearly Aggregate Loss (MPY) were obtained at the .01 level.
⚫ Important to use distributions which permit thick tails
⚫ Bali, T. G., 2003. “An Extreme Value Approach to Estimating Volatility and Value at Risk,” Journal of Business, 76:83-108
-
Appendices—VaR: Models and
applications discussion
⚫ Cummins, J. D., C. Merrill, and J. B. McDonald, 2007. “Risky Loss Distributions and Modeling the Loss Reserve Pay-out Tail,” Review of Applied Economics 3.
⚫ Estimate aggregate loss distribution associated with claims incurred in a given year, but settled in different years
▪ Data: U.S. products liability insurance paid claims (Insurance Services Office (ISO))
▪ Mixture model:
▪ Consider different GB2 distributions for each cell (year)
▪ Multinomial distribution for fraction of claims settled at different lags
▪ Single aggregate GB2 distribution for each year GB2 provides a significantly better fit to severity data than the LN, gamma, Weibull, Burr12, or generalized gamma
▪ The Aggregate GB2 distribution has a thicker tail than does the mixture distribution
-
Appendices—VaR: Models and
applications discussion
⚫ Bali, T. G. and P. Theodossiou, 2008. “Risk Measurement Performance of Alternative Distribution Functions,” Journal of Risk and Insurance, 75: 411-437.
⚫ Models: Unconditional formulations▪ Generalized Pareto
▪ Generalized extreme value
▪ Box-Cox extreme value
▪ SGED
▪ SGT
▪ EGB2
▪ IHS
⚫ Models: Conditional formulations (model time-varying VaR thresholds)
0 1 1t t t t t t tR R z z −= + + = +
0 1 1 1 2 1t t t tz − − −= + +tL
•
-
Appendices—VaR: Models and
applications discussion
⚫ Bali, T. G. and P. Theodossiou, 2008. “Risk Measurement Performance of Alternative Distribution Functions,” Journal of Risk and Insurance, 75: 411-437. (continued)
⚫ Data
▪ S&P500 composite index (1/4/1950 to 12/29/2000)
▪ Daily percentage log-returns: (n=12,832
▪ maximum=8.71
▪ minimum=-22.90
▪ skewness =1.622
▪ kurtosis=45.52
⚫ Findings
▪ Out of sample VaR estimates are rejected for most unconditional specifications
▪ Thresholds exhibit time varying behavior
▪ Out of sample VaR estimates for the conditional specifications corresponding to the SGT, IHS, and EGB2 perform better than the
extreme value distributions
-
END OF APPENDICES
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