copulas from fokker-planck equation

Copulas from Fokker-Planck equation

Hi Jun Choe

Dept of MathYonsei 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.

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


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


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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Sklar’s Theorem


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.

http://mathworld.wolfram.com/Copula.html

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Fokker-Planck equation for copulaP9

Copula function


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

indexP11

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)

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

indexP13

Σ = 12 ∫ |C(u,v)-uv| dudv

Φ = 90 ∫ |C(u,v) – uv|^2 dudv

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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.

http://en.wikipedia.org/wiki/Correlation

http://en.wikipedia.org/wiki/Normal_distribution

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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:

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Example of copula(Archimedian copula)


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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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.

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Example of copula(Bernstein copula)


Let

2-dimension case :

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Example of copula(Student-t copula)


Student-t copula :

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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 independentsituation (θ=0) to the situation of co-monotonicity (θ=1).

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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)

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Fokker-Planck equation for copula(cont’d)P22

Fokker-Planck equation and joint pdf


: joint pdf of at time t.

where

By integrating

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.

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.

,

where the distribution function is . .

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.

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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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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.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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.

If we define then,

Distribution function is and satisfies

with the boundary condition

and the initial condition

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Fokker-Planck equation and Copula


.

Change variable to new variablesand thus

The Copula satisfies the Fokker-Planck equation :

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Fokker-Planck equation and Copula


.

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 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.

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Fokker-Planck equation and theorem


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


.

Stochastic differential equations :

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Numerical Study (cont’d)P35

Quantile-Quantile


.

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Copular(independent SDE)


.

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Copular(dependent SDE)


.

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Thank you!!P38

Thank you !!


copulas from fokker-planck equation

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