Khóa luận tiếng anh: Extreme Value Theory and Applications in Financial Market

Extreme Value Theory
Applications in Financial Markets
Extreme Value Theory and Applications in
Financial Market
Dao Minh Phuong
Thesis Advisor: Dr. Luu Hoang Duc
K53 Undergraduate Program in Mathematics
University of Science
Vietnam National University
8
th
December, 2012
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme 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
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Extreme Value Theory

What is Extreme Value Theory?
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Extreme Value Theory
What is Extreme Value Theory?
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Extreme Value Theory
What is Extreme Value Theory?
Extreme Value Theory studies extremal deviation from the
median of probability distribution.
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Extreme Value Theory
What is Extreme Value Theory?
Extreme Value Theory studies extremal deviation from the
median of probability distribution.
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market

Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Extreme 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.
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Extreme 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.
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method

Peaks- over- threshold method
Extreme 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.
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Extreme 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.
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method

Peaks- over- threshold method
Extreme 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
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Extreme 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
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market

Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Extreme 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
2
the Peaks- over - threshold method.
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Block Maxima Method
This method consists of dividing the series into non-overlapping
blocks of same length and then choosing the maximum from
every block.
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory

Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Block Maxima Method
This method consists of dividing the series into non-overlapping
blocks of same length and then choosing the maximum from
every block.
Limiting behavior of sample extrema
Let X
1
, X
2
, . . . , be iid random variables with distribution function
(df) F. Let M
n
= max(X
1
, . . . ,X
n
) be worst-case loss in a
sample of n losses.
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Block Maxima Method
This method consists of dividing the series into non-overlapping

blocks of same length and then choosing the maximum from
every block.
Limiting behavior of sample extrema
Let X
1
, X
2
, . . . , be iid random variables with distribution function
(df) F. Let M
n
= max(X
1
, . . . ,X
n
) be worst-case loss in a
sample of n losses.
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Block Maxima Method
This method consists of dividing the series into non-overlapping
blocks of same length and then choosing the maximum from
every block.
Limiting behavior of sample extrema
Let X
1
, X

2
, . . . , be iid random variables with distribution function
(df) F. Let M
n
= max(X
1
, . . . ,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
∈ R such
that (M
n
− b
n
)/a
n
converges in distribution, i.e:
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method

Block Maxima Method
This method consists of dividing the series into non-overlapping
blocks of same length and then choosing the maximum from
every block.
Limiting behavior of sample extrema
Let X
1
, X
2
, . . . , be iid random variables with distribution function
(df) F. Let M
n
= max(X
1
, . . . ,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
∈ R such
that (M
n
− b
n
)/a

n
converges in distribution, i.e:
P

M
n
− b
n
a
n
≤ x

= F
n
(a
n
x + b
n
)
n→∞
−→ H(x), (1)
for some non degenerate distribution function H(x).
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Generalized Extreme Value Distribution
The Generalized Extreme Value Distribution (GEV) is given as:

Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Generalized Extreme Value Distribution
The Generalized Extreme Value Distribution (GEV) is given as:
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Generalized Extreme Value Distribution
The Generalized Extreme Value Distribution (GEV) is given as:
H
ξ
(x) =

exp(−(1 + ξx)
−1/ξ
), ξ = 0.
exp(−e
−x
), ξ = 0.
(2)
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets

Introduction
Block Maxima Method
Peaks- over- threshold method
Generalized Extreme Value Distribution
The Generalized Extreme Value Distribution (GEV) is given as:
H
ξ
(x) =

exp(−(1 + ξx)
−1/ξ
), ξ = 0.
exp(−e
−x
), ξ = 0.
(2)
with x such that 1 + ξx > 0 and ξ is the shape parameter. This
parametrization is continuous in ξ where :
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Generalized Extreme Value Distribution
The Generalized Extreme Value Distribution (GEV) is given as:
H
ξ
(x) =


exp(−(1 + ξx)
−1/ξ
), ξ = 0.
exp(−e
−x
), ξ = 0.
(2)
with x such that 1 + ξx > 0 and ξ is the shape parameter. This
parametrization is continuous in ξ where :
ξ > 0: H
ξ
corresponds to Frechet family.
ξ = 0: H
ξ
corresponds to Gumbel family.
ξ < 0: H
ξ
corresponds to Weibull family.
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Generalized Extreme Value Distribution
The Generalized Extreme Value Distribution (GEV) is given as:
H
ξ
(x) =


exp(−(1 + ξx)
−1/ξ
), ξ = 0.
exp(−e
−x
), ξ = 0.
(2)
with x such that 1 + ξx > 0 and ξ is the shape parameter. This
parametrization is continuous in ξ where :
ξ > 0: H
ξ
corresponds to Frechet family.
ξ = 0: H
ξ
corresponds to Gumbel family.
ξ < 0: H
ξ
corresponds to Weibull family.
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market
Extreme Value Theory
Applications in Financial Markets
Introduction
Block Maxima Method
Peaks- over- threshold method
Fisher- 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:
Dao Minh Phuong Thesis Advisor: Dr. Luu Hoang Duc Extreme Value Theory and Applications in Financial Market