copulas from fokker-planck equation hi jun choe dept of math yonsei university seoul, korea
TRANSCRIPT
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.
Bond Market Investors needed clear probability concept to manage risks.
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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.
Quants at Wall Street were excited by the convenience, elegance and tractability of Gaussian copula and
adopted universally in risk management.
Portfolio selectionP4
Efficient portfolios are given by the mean variance optimization;
Sup a x with a ∑ a < c and a 1=1,t t t
where x is expected return vector and ∑ is covariance matrix .
The variance corresponds to the risk measure, but it impliesthe world is Gaussian.
There arise two problems: Gaussian assumption and joint distribution modeling.
Danger of UncertaintyP5
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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IntroductionP6
Black-Scholes formula is challenged in two aspects
Copulas from Fokker-Planck equation
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.
The complexity of financial market causes a significant difficulty in hedginga large variety of different risks for a financial institute.
The derivative products are mutually connected and often exotic.
Decides the asset value by a general stochastic differentialequation(SDE).
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Introduction(cont’d)P7
Chapman-Kolmogrov equation
Copulas from Fokker-Planck equation
One focuses on the marginal distributions of each product andconsiders the correlation of them.
The Copulas are of great help to evaluation and hedging of Derivative products.
A filtered probability space generated by the stochastic process
is Markov and the transition probability density Function satisfies Chapman-Kolmogorov equation
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Introduction(cont’d)P8
Sklar’s Theorem
Copulas from Fokker-Planck equation
Let H be a two-dimensional distribution function with marginal distributionfunctions 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.
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Fokker-Planck equation for copulaP9
Copula function
Copulas from Fokker-Planck equation
is Copula if is continuous function satisfying
for all and From condition (1), (2) and (3) we could prove that
.
and
for all .
ConcordanceP10
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)=04. 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
DependenceP12
Definition: D is a measure of dependence for two randon variablesX 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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Introduction(cont’d)P14
Example of copula(Gaussian copula)
Copulas from Fokker-Planck equation
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.
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Introduction(cont’d)P15
Example of copula(Archimedian copula)
Copulas from Fokker-Planck equation
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:
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Introduction(cont’d)P16
Example of copula(Archimedian copula)
Copulas from Fokker-Planck equation
Gumbel copula :
Frank copula :
Periodic copula :
In 2005 Aurélien Alfonsi and Damiano Brigo introduced new families ofcopulas 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 absentin most known copula functions.
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Introduction(cont’d)P17
Example of copula(Empirical copulas)
Copulas from Fokker-Planck equation
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.
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Introduction(cont’d)P18
Example of copula(Bernstein copula)
Copulas from Fokker-Planck equation
Let
2-dimension case :
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Introduction(cont’d)P19
Example of copula(Student-t copula)
Copulas from Fokker-Planck equation
Student-t copula :
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Introduction(cont’d)P20
Example of copula(Marshall-Olkin copula)
Copulas from Fokker-Planck equation
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 independentsituation (θ=0) to the situation of co-monotonicity (θ=1).
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Introduction(cont’d)P21
Maximum and Minimum copulas
Copulas from Fokker-Planck equation
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)
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Fokker-Planck equation for copula(cont’d)P22
Fokker-Planck equation and joint pdf
Copulas from Fokker-Planck equation
: joint pdf of at time t.
where
By integrating
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Fokker-Planck equation for copula(cont’d)P23
Fokker-Planck equation and joint pdf
Copulas from Fokker-Planck equation
.
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Fokker-Planck equation for copula(cont’d)P24
Fokker-Planck equation and joint pdf
Copulas from Fokker-Planck equation
.
,
where the distribution function is . .
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Fokker-Planck equation for copula(cont’d)P25
Fokker-Planck equation and joint pdf
Copulas from Fokker-Planck equation
.
Hence
Inference Function of MarginP26
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)P27
Fokker-Planck equation and joint pdf
Copulas from Fokker-Planck equation
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
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Fokker-Planck equation for copula(cont’d)P28
Fokker-Planck equation and joint pdf
Copulas from Fokker-Planck equation
.where
We find that the marginal distribution functions satisfy
and from the separable structure of SDE, the marginal distributionfunctions and can be solved independently.
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Fokker-Planck equation for copula(cont’d)P29
Fokker-Planck equation and joint pdf
Copulas from Fokker-Planck equation
.
If we define then,
Distribution function is and satisfies
with the boundary condition
and the initial condition
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Fokker-Planck equation for copula(cont’d)P30
Fokker-Planck equation and Copula
Copulas from Fokker-Planck equation
.
Change variable to new variables
and thus
The Copula satisfies the Fokker-Planck equation :
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Fokker-Planck equation for copula(cont’d)P31
Fokker-Planck equation and Copula
Copulas from Fokker-Planck equation
.
Considering the equation for marginal distributions
In with the boundary condition
For all and and the initial condition
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Fokker-Planck equation for copula(cont’d)P32
Fokker-Planck equation and copula
Copulas from Fokker-Planck equation
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.
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Fokker-Planck equation for copula(cont’d)P33
Fokker-Planck equation and theorem
Copulas from Fokker-Planck equation
Theorem.
We consider the solution to
for a large k. Then we find that
are independent standard Brownian processes.
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Numerical StudyP34
Marginal distribution function
Copulas from Fokker-Planck equation
.
Stochastic differential equations :