Khóa luận tiếng anh: Extreme Value Theory and Applications in Financial Market Extreme Value Theory Introduction Block Maxima Method Peaks over threshold method 2 Applications in Financial Markets Block Maxima Method Peaks over threshold method
Trang 1Extreme Value Theory Applications in Financial Markets
Extreme Value Theory and Applications in
Trang 2Extreme Value Theory Applications in Financial Markets
1 Extreme Value Theory
Introduction
Block Maxima Method
Peaks- over- threshold method
2 Applications in Financial Markets
Block Maxima Method
Peaks- over- threshold method
Trang 3Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method Peaks- over- threshold methodExtreme Value Theory
What is Extreme Value Theory?
Trang 4Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method Peaks- over- threshold methodExtreme Value Theory
What is Extreme Value Theory?
Trang 5Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method Peaks- over- threshold methodExtreme Value Theory
What is Extreme Value Theory?
Extreme Value Theory studies extremal deviation from the median of probability distribution.
Trang 6Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method Peaks- over- threshold methodExtreme Value Theory
What is Extreme Value Theory?
Extreme Value Theory studies extremal deviation from the median of probability distribution.
Trang 7Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method Peaks- over- threshold methodExtreme Value Theory
What is Extreme Value Theory?
Extreme Value Theory studies extremal deviation from the median of probability distribution.
It seeks for events that rarely happen but when happening, they have very important effects such as floods,
earthquakes, market crashes, etc.
Trang 8Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method Peaks- over- threshold methodExtreme Value Theory
What is Extreme Value Theory?
Extreme Value Theory studies extremal deviation from the median of probability distribution.
It seeks for events that rarely happen but when happening, they have very important effects such as floods,
earthquakes, market crashes, etc.
Trang 9Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method Peaks- over- threshold methodExtreme Value Theory
What is Extreme Value Theory?
Extreme Value Theory studies extremal deviation from the median of probability distribution.
It seeks for events that rarely happen but when happening, they have very important effects such as floods,
earthquakes, market crashes, etc.
Basically, there are two methods for identifying extremes in real data.
Trang 10Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method Peaks- over- threshold methodExtreme Value Theory
What is Extreme Value Theory?
Extreme Value Theory studies extremal deviation from the median of probability distribution.
It seeks for events that rarely happen but when happening, they have very important effects such as floods,
earthquakes, market crashes, etc.
Basically, there are two methods for identifying extremes in real data.
Trang 11Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method Peaks- over- threshold methodExtreme Value Theory
What is Extreme Value Theory?
Extreme Value Theory studies extremal deviation from the median of probability distribution.
It seeks for events that rarely happen but when happening, they have very important effects such as floods,
earthquakes, market crashes, etc.
Basically, there are two methods for identifying extremes in real data.
1 block maxima method
Trang 12Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method Peaks- over- threshold methodExtreme Value Theory
What is Extreme Value Theory?
Extreme Value Theory studies extremal deviation from the median of probability distribution.
It seeks for events that rarely happen but when happening, they have very important effects such as floods,
earthquakes, market crashes, etc.
Basically, there are two methods for identifying extremes in real data.
1 block maxima method
Trang 13Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method Peaks- over- threshold methodExtreme Value Theory
What is Extreme Value Theory?
Extreme Value Theory studies extremal deviation from the median of probability distribution.
It seeks for events that rarely happen but when happening, they have very important effects such as floods,
earthquakes, market crashes, etc.
Basically, there are two methods for identifying extremes in real data.
1 block maxima method
Trang 14Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold methodBlock Maxima Method
This method consists of dividing the series into non-overlappingblocks of same length and then choosing the maximum from
every block
Trang 15Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold methodBlock Maxima Method
This method consists of dividing the series into non-overlappingblocks of same length and then choosing the maximum from
every block
Limiting behavior of sample extrema
Let X1,X2, , be iid random variables with distribution function (df) F Let M n=max(X1, ,X n)be worst-case loss in a
sample of n losses.
Trang 16Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold methodBlock Maxima Method
This method consists of dividing the series into non-overlappingblocks of same length and then choosing the maximum from
every block
Limiting behavior of sample extrema
Let X1,X2, , be iid random variables with distribution function (df) F Let M n=max(X1, ,X n)be worst-case loss in a
sample of n losses.
Trang 17Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold methodBlock Maxima Method
This method consists of dividing the series into non-overlappingblocks of same length and then choosing the maximum from
every block
Limiting behavior of sample extrema
Let X1,X2, , be iid random variables with distribution function (df) F Let M n=max(X1, ,X n)be worst-case loss in a
sample of n losses.
We say that F ∈the maximum domain of attraction of H
(MDA(H)) , if there exists real numbers a n>0 and b n∈ Rsuch that(Mn−bn)/an converges in distribution, i.e:
Trang 18Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold methodBlock Maxima Method
This method consists of dividing the series into non-overlappingblocks of same length and then choosing the maximum from
every block
Limiting behavior of sample extrema
Let X1,X2, , be iid random variables with distribution function (df) F Let M n=max(X1, ,X n)be worst-case loss in a
sample of n losses.
We say that F ∈the maximum domain of attraction of H
(MDA(H)) , if there exists real numbers a n>0 and b n∈ Rsuch that(Mn−bn)/an converges in distribution, i.e:
Trang 19Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold methodGeneralized Extreme Value Distribution
The Generalized Extreme Value Distribution (GEV) is given as:
Trang 20Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold methodGeneralized Extreme Value Distribution
The Generalized Extreme Value Distribution (GEV) is given as:
Trang 21Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold methodGeneralized Extreme Value Distribution
The Generalized Extreme Value Distribution (GEV) is given as:
Hξ(x) =
exp(−(1+ ξx)− 1 /ξ), ξ 6=0
Trang 22Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold methodGeneralized Extreme Value Distribution
The Generalized Extreme Value Distribution (GEV) is given as:
Trang 23Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold methodGeneralized Extreme Value Distribution
The Generalized Extreme Value Distribution (GEV) is given as:
ξ >0: Hξcorresponds to Frechet family.
ξ =0: Hξ corresponds to Gumbel family.
ξ <0: Hξcorresponds to Weibull family.
Trang 24Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold methodGeneralized Extreme Value Distribution
The Generalized Extreme Value Distribution (GEV) is given as:
ξ >0: Hξcorresponds to Frechet family.
ξ =0: Hξ corresponds to Gumbel family.
ξ <0: Hξcorresponds to Weibull family.
Trang 25Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold methodFisher- Tippett Theorem
Trang 26Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold methodFisher- Tippett Theorem
Theorem
If appropriately normalized maxima converge in distribution to a non-degenerate limit, then the limit distribution must be an
extreme value distribution, that is:
If F ∈MDA (H) then H is of type Hξfor someξ.
where Hξ isGeneralized Extreme Value Distribution.
Trang 27Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold methodFisher- Tippett Theorem
Theorem
If appropriately normalized maxima converge in distribution to a non-degenerate limit, then the limit distribution must be an
extreme value distribution, that is:
If F ∈MDA (H) then H is of type Hξfor someξ.
where Hξ isGeneralized Extreme Value Distribution.
Trang 28Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Example
Trang 29Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Example
1 Fr´echet case (ξ >0):
Heavy- tailed distributions are belong to Fr´echet family A
typical one is the Pareto distribution,
F(x) =1−
K
K +x
α
, α, K >0, x ≥0,
is in MDA(H1/α) if we take a n= Kn1/α/α, b n= Kn1/α− K
Trang 30Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Example
1 Fr´echet case (ξ >0):
Heavy- tailed distributions are belong to Fr´echet family A
typical one is the Pareto distribution,
F(x) =1−
K
K +x
α
, α, K >0, x ≥0,
is in MDA(H1/α) if we take a n= Kn1/α/α, b n= Kn1/α− K
2 Gumbel case F ∈MDA (H0) :
The exponential distribution: F(x) =1−e−λx, λ >0,x ≥0
We take a n=1/λ,b n= (log n)/λ,ξ =0
Trang 31Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Example
1 Fr´echet case (ξ >0):
Heavy- tailed distributions are belong to Fr´echet family A
typical one is the Pareto distribution,
F(x) =1−
K
K +x
α
, α, K >0, x ≥0,
is in MDA(H1/α) if we take a n= Kn1/α/α, b n= Kn1/α− K
2 Gumbel case F ∈MDA (H0) :
The exponential distribution: F(x) =1−e−λx, λ >0,x ≥0
We take a n=1/λ,b n= (log n)/λ,ξ =0
3 Weilbull case (ξ <0) :
Trang 32Extreme Value Theory
Applications in Financial Markets
Introduction Block Maxima Method
Peaks- over- threshold method
Peaks- over- threshold method
Set a threshold and then collect the exceedances over a
threshold
Trang 33Extreme Value Theory
Applications in Financial Markets
Introduction Block Maxima Method
Peaks- over- threshold method
Peaks- over- threshold method
Set a threshold and then collect the exceedances over a
threshold
Trang 34Extreme Value Theory
Applications in Financial Markets
Introduction Block Maxima Method
Peaks- over- threshold method
Peaks- over- threshold method
Set a threshold and then collect the exceedances over a
Trang 35Extreme Value Theory
Applications in Financial Markets
Introduction Block Maxima Method
Peaks- over- threshold method
Peaks- over- threshold method
Set a threshold and then collect the exceedances over a
Trang 36Extreme Value Theory
Applications in Financial Markets
Introduction Block Maxima Method
Peaks- over- threshold method
Peaks- over- threshold method
Set a threshold and then collect the exceedances over a
threshold
Model data using the Generalized Pareto distribution whichcalculate the probability of recording extreme events
exceed the threshold
Generalized Pareto distribution (GPD)
The GPD is a two parameter distribution with distribution
Trang 37Extreme Value Theory
Applications in Financial Markets
Introduction Block Maxima Method
Peaks- over- threshold method
POT method
The excess distribution
Consider an unknown distribution function F of a random variable.
Let u be the high threshold The distribution function F uis called the conditional excess distribution function and is defined as:
Trang 38Extreme Value Theory
Applications in Financial Markets
Introduction Block Maxima Method
Peaks- over- threshold method
POT method
The excess distribution
Consider an unknown distribution function F of a random variable.
Let u be the high threshold The distribution function F uis called the conditional excess distribution function and is defined as:
F u(x) =P(X −u≤x|X >u) =F(x+u) −F(u)
1 −F(u) (4)
for 0 ≤x ≤x F −u where x F = sup {x ∈ R : F(x) < 1 } ≤ ∞ is the
right end point of F
Trang 39Extreme Value Theory
Applications in Financial Markets
Introduction Block Maxima Method
Peaks- over- threshold method
POT method
The excess distribution
Consider an unknown distribution function F of a random variable.
Let u be the high threshold The distribution function F uis called the conditional excess distribution function and is defined as:
F u(x) =P(X −u≤x|X >u) =F(x+u) −F(u)
1 −F(u) (4)
for 0 ≤x ≤x F −u where x F = sup {x ∈ R : F(x) < 1 } ≤ ∞ is the
right end point of F
Example
Trang 40Extreme Value Theory
Applications in Financial Markets
Introduction Block Maxima Method
Peaks- over- threshold method
POT method
The excess distribution
Consider an unknown distribution function F of a random variable.
Let u be the high threshold The distribution function F uis called the conditional excess distribution function and is defined as:
F u(x) =P(X −u≤x|X >u) =F(x+u) −F(u)
1 −F(u) (4)
for 0 ≤x ≤x F −u where x F = sup {x ∈ R : F(x) < 1 } ≤ ∞ is the
right end point of F
Example
1 F(x) = 1 −eλx , λ > 0 ,x ≥ 0 then F u(x) =F(x),x ≥ 0
Trang 41Extreme Value Theory
Applications in Financial Markets
Introduction Block Maxima Method
Peaks- over- threshold method
POT method
The excess distribution
Consider an unknown distribution function F of a random variable.
Let u be the high threshold The distribution function F uis called the conditional excess distribution function and is defined as:
F u(x) =P(X −u≤x|X >u) =F(x+u) −F(u)
1 −F(u) (4)
for 0 ≤x ≤x F −u where x F = sup {x ∈ R : F(x) < 1 } ≤ ∞ is the
right end point of F
Example
1 F(x) = 1 −eλx , λ > 0 ,x ≥ 0 then F u(x) =F(x),x ≥ 0
Trang 42Extreme Value Theory
Applications in Financial Markets
Introduction Block Maxima Method
Peaks- over- threshold method
POT method
The excess distribution
Consider an unknown distribution function F of a random variable.
Let u be the high threshold The distribution function F uis called the conditional excess distribution function and is defined as:
F u(x) =P(X −u≤x|X >u) =F(x+u) −F(u)
1 −F(u) (4)
for 0 ≤x ≤x F −u where x F = sup {x ∈ R : F(x) < 1 } ≤ ∞ is the
right end point of F
Example
1 F(x) = 1 −eλx , λ > 0 ,x ≥ 0 then F u(x) =F(x),x ≥ 0
2 F(x) =Gξ,β (x)then F u(x) =Gξ,β+ξu (x)