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Copulas from Fokker-Planck equation
Hi Jun Choe
Dept of Math
Yonsei University
Seoul, Korea
Financial Crisis in 2007
Wired Magazine 02.23.09
Recipe for Disaster:The Formula That
Killed Wall Street by S. Salmon
Gaussian Copula by Davis X. Li
“On Default Correlation:A Copula Function Approach”,
The Journal of Fixed Income, 2000.
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Bond Market Investors needed clear probability
concept to manage risks.
Quants at Wall Street were excited by the convenience,
elegance and tractability of Gaussian copula and
adopted universally in risk management.
The amount of CDS(credit default swap) increased
from 920 billion dollar in 2001 to 62 trillion dollar by 2007.
The amount of CDO(collateral debt obligation) increased
from 275 billion dollar in 2000 to 4.7 trillion by 2006.
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Portfolio selection
Efficient portfolios are given by the mean variance optimization;
t
t
t
Sup a x with a ∑ a < c and a 1=1,
where x is expected return vector and ∑ is
covariance matrix .
The variance corresponds to the risk measure, but it implies
the world is Gaussian.
There arise two problems: Gaussian assumption and
joint distribution modeling.
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Danger of Uncertainty
Structure of Decision Makers: Quant-Trader-Sales
The correlation of financial quantities are notoriously
unstable and highly volatile.
The market is stable with 99% probability
although the 1% failure produces huge impact.
Thus everybody ignored the warning signal.
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Introduction
Black-Scholes formula is challenged in two aspects
1. Non-normality of asset return that appears as volatility smile and
structure form of Implied volatility(When there is smile effect, the
return shows non-normality and the linear correlation shows bias).
2. Market incompleteness.
Decides the asset value by a general stochastic differential
equation(SDE).
The complexity of financial market causes a significant difficulty in hedging
a large variety of different risks for a financial institute.
The derivative products are mutually connected and often exotic.
Copulas from Fokker-Planck equation
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Introduction(cont’d)
Chapman-Kolmogrov equation
One focuses on the marginal distributions of each product and
considers the correlation of them.
The Copulas are of great help to evaluation and hedging of
Derivative products.
A filtered probability space
process
generated by the stochastic
is Markov and the transition probability density
Function satisfies Chapman-Kolmogorov equation
Copulas from Fokker-Planck equation
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Introduction(cont’d)
Sklar’s Theorem
Let H be a two-dimensional distribution function with marginal distribution
functions F and G. Then there exists a copula C such that
Conversely, for any univariate distribution functions F and G and any
copula C, the function H is a two-dimensional distribution function with
marginals F and G. Furthermore, if F and G are continuous, then C is
unique.
Copulas from Fokker-Planck equation
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Fokker-Planck equation for copula
Copula function
is Copula if
for all
is continuous function satisfying
and
.
From condition (1), (2) and (3) we could prove that
and
for all
.
Copulas from Fokker-Planck equation
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Concordance
Definition: D is a measure of concordance for two random
variables X and Y whose copula is C if
1. -1 =K(X,-X)=< K(C) =< K(X,X)=1
2. K(X,Y)=K(Y,X)
3. If X and Y are independent, K(X,Y)=0
4. K(-X,Y)=K(X,-Y)=-K(X,Y)
5. If C1 < C2, then K(C1) < K(C2)
Example: Kendall’s tau, Spearman’s rho and Gini indices
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index
Τ= 4∫C(u,v) dC(u,v) -1
ρ= 12 ∫uv dC(u,v) -3
Г= 2 ∫|u+v-1| -|u-v| dC(u,v)
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Dependence
Definition: D is a measure of dependence for two randon variables
X and Y whose copula is C if
1. 0=D(uv) =< D(C)=<D(Min(u,v)) =1
2. D(X,Y)=D(Y,X)
3. D(uv)=D(X,Y)=0 if and only if X and Y are independent
4. D(X,Y)=D(Min(u,v))=1 if and onlly if each of X and Y
5. Is almost surely monotone increasing function of the other
6. D(h1(X),h2(X))=h(X,Y) for increasing functions h1 and h2
Example: Schweitzer and Wolff’s sigma and Hoeffding’s phi
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index
Σ = 12 ∫ |C(u,v)-uv| dudv
Φ = 90 ∫ |C(u,v) – uv|^2 dudv
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Introduction(cont’d)
Example of copula(Gaussian copula)
Gaussian copula function :
: the standard bivariate normal cumulative distribution function
with correlation ρ
: the standard normal cumulative distribution function
Differentiating C yields the copula density function:
is the density function for the standard bivariate Gaussian.
is the standard normal density.
Copulas from Fokker-Planck equation
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Introduction(cont’d)
Example of copula(Archimedian copula)
Unlike elliptical copulas (e.g. Gaussian), most of the Archimedean
copulas have closed-form solutions and are not derived from the
multivariate distribution functions using Sklar’s theorem.
One particularly simple form of a n-dimensional copula is
where
is known as a generator function.
Any generator function which satisfies the properties below is
the basis for a valid copula:
Copulas from Fokker-Planck equation
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Introduction(cont’d)
Example of copula(Archimedian copula)
Gumbel copula :
Frank copula :
Periodic copula :
In 2005 Aurélien Alfonsi and Damiano Brigo introduced new families of
copulas based on periodic functions. They noticed that if ƒ is a 1-periodic
non-negative function that integrates to 1 over [0, 1] and F is a double
primitive of ƒ, then both
are copula functions, the second one not necessarily exchangeable.
This may be a tool to introduce asymmetric dependence, which is absent
in most known copula functions.
Copulas from Fokker-Planck equation
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Introduction(cont’d)
Example of copula(Empirical copulas)
Empirical copulas :
When analysing data with an unknown underlying distribution, one can
transform the empirical data distribution into an "empirical copula" by
warping such that the marginal distributions become uniform.
Mathematically the empirical copula frequency function is calculated by
where x(i) represents the ith order statistic of x.
Less formally, simply replace the data along each dimension with
the data ranks divided by n.
Copulas from Fokker-Planck equation
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Introduction(cont’d)
Example of copula(Bernstein copula)
Let
2-dimension case :
Copulas from Fokker-Planck equation
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Introduction(cont’d)
Example of copula(Student-t copula)
Student-t copula :
Copulas from Fokker-Planck equation
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Introduction(cont’d)
Example of copula(Marshall-Olkin copula)
Marshall-Olkin copula:
The Marshall-Olkin copula is a function
With an appropriate extension of its domain to
, the copula is a joint
distribution function with marginals uniform on [0,1].
This copula depends on a parameter θ∈[0,1](we consider the case
in which the variables are exchangeable) that reflexes the dependent
structure existing between the marginals, from the stochastic independent
situation (θ=0) to the situation of co-monotonicity (θ=1).
Copulas from Fokker-Planck equation
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Introduction(cont’d)
Maximum and Minimum copulas
Maximum copula: M(u,v) = Min (u,v)
Minimum copula:
W(u,v) = Max (u+v-1,0)
W (u,v) =< C(u,v) =<
M(u,v)
Copulas from Fokker-Planck equation
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Fokker-Planck equation for copula(cont’d)
Fokker-Planck equation and joint pdf
: joint pdf of
at time t.
where
By integrating
Copulas from Fokker-Planck equation
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Fokker-Planck equation for copula(cont’d)
Fokker-Planck equation and joint pdf
.
Copulas from Fokker-Planck equation
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Fokker-Planck equation for copula(cont’d)
Fokker-Planck equation and joint pdf
.
,
where the distribution function
is
Copulas from Fokker-Planck equation
..
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Fokker-Planck equation for copula(cont’d)
Fokker-Planck equation and joint pdf
Hence
.
Copulas from Fokker-Planck equation
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Inference Function of Margin
In market, we have to deal with hundreds or thounds
financial data which are correlated.
Finding the joint probability density function is very difficult.
Further, if time is a main parameter, it is almost impossible
to find their joint pdf.
Therefore, we only Consider each data separately, namely,
find the marginal distribution of each data.
The correlation is obtained using the marginal distributions.
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Fokker-Planck equation for copula(cont’d)
Fokker-Planck equation and joint pdf
Relation between copula and marginal distribution function
satisfies the Fokker-Planck equation.
.
From inference function of margin, we consider separable structure SDE
Under Markov property, the joint pdf satisfies Fokker-Planck equation
Copulas from Fokker-Planck equation
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Fokker-Planck equation for copula(cont’d)
Fokker-Planck equation and joint pdf
We find that the marginal distribution functions satisfy
where
.
and from the separable structure of SDE, the marginal distribution
functions
and
can be solved independently.
Copulas from Fokker-Planck equation
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Fokker-Planck equation for copula(cont’d)
Fokker-Planck equation and joint pdf
,
If we define
Distribution function
is
.
then
and satisfies
with the boundary condition
and the initial condition
Copulas from Fokker-Planck equation
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Fokker-Planck equation for copula(cont’d)
Fokker-Planck equation and Copula
Change variable to new variables
and thus
The Copula
Fokker-Planck equation :
satisfies the
.
Copulas from Fokker-Planck equation
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Fokker-Planck equation for copula(cont’d)
Fokker-Planck equation and Copula
Considering the equation for marginal distributions
In
For all
.
with the boundary condition
and
and the initial condition
Copulas from Fokker-Planck equation
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Fokker-Planck equation for copula(cont’d)
Fokker-Planck equation and copula
Conversely, if C is a solution to
with the copula boundary condition, then C is copula.
The maximum principle for the derivatives of C is key
ingredient for proof.
Copulas from Fokker-Planck equation
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Fokker-Planck equation for copula(cont’d)
Fokker-Planck equation and theorem
Theorem.
We consider the solution
to
for a large k. Then we find that
are independent standard
Brownian processes.
Copulas from Fokker-Planck equation
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Numerical Study
Marginal distribution function
Stochastic differential equations :
.
Copulas from Fokker-Planck equation
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Numerical Study (cont’d)
Quantile-Quantile
.
Copulas from Fokker-Planck equation
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Numerical Study (cont’d)
Copular(independent SDE)
.
Copulas from Fokker-Planck equation
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Numerical Study (cont’d)
Copular(dependent SDE)
.
Copulas from Fokker-Planck equation
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Thank you!!
Thank you !!
Copulas from Fokker-Planck equation