Procedure to Generate Uniform Random Variates from Each ...

[Pages:3]Procedure to Generate Uniform Random Variates from Each Copula

The Gaussian Copula The Gaussian copula may be generated by first obtaining a set of correlated normally distributed variates v1 and v2 using Choleski's decomposition, and then transforming these to uniform variables u1 = (v1 ) and u2 = (v2 ) , where is the cumulative standard normal. Then, the pair ( u1 , u2 ) represents draws from the Gaussian copula.

The FGM Copula

Generating draws from the FGM copula is easiest done using the conditional distribution

approach (see Nelsen, 2006; pg 41). Note that the conditional distribution function of U2 , given

U1 = u1 , may be written as:

C2|1 (u1 , u2 )

=

Pr[U 2

u2

|U1

=

u1 ] =

Lim C

u1 0

(u1

+ u1 , u2 ) - C u1

(u1 , u2 )

=

C (u1 , u2 ) u1

(1)

Thus, a general algorithm to draw from a copula C (u1 , u2 ) would be (see Johnson, 1987, Ch.

3):

(1) Draw two independent uniform random variates (u1, v2 ) .

(2)

Set

u2

=

C -1 2|1

(u1

,

v2

).

,

where

C -1 2|1

denotes

the

pseudo-inverse

of

C 2|1 .

The vector (u1, u2 ) is generated from the copula C. For the FGM copula, the above algorithm is:

(1) Draw two independent uniform random variates (u1, v2 ) .

(2) Set u2 = 2v2 /(B + A), A = 1 + (1 - 2u1 ), B = A2 - 4( A - 1)v2 .1

(2)

1 Note that C2|1 (u1, u2 ) = u2 A + [-( A - 1)]u2 2 , where A is as just defined. Setting this equal to v2 (0 < v2 < 1) and

A- solving the quadratic for u2 in the unit interval provides one root solution: u2 =

A2 - 4( A - 1)v2 . Multiplying 2( A - 1)

and dividing by A + A2 - 4( A - 1)v2 and simplifying, we get the desired result.

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The Clayton Copula

For the Clayton copula, we draw variates using the conditional distribution approach discussed

earlier.

(1) Draw two independent uniform random variates (u1, v2 ) .

(2)

Set

u 2

=

[u1 -

(v

- 2

/(1+

)

-1) + 1]-1/

(3)

The Gumbel Copula

For the Gumbel copula, the conditional distribution is not directly invertible (see Venter, 2001),

and so we use another way to generate variates using the following general algorithm (see

Nelsen, 2006, Genest and Rivest, 1993):

(1) Generate two independent uniform variates (v1, v2 ) .

(2)

Set

w

=

K

- 1 C

(

v

2

),

K

C

(t

)

=

t - (t) ' (t)

(4)

(3) Set u1 = -1[v1 (w)] and u2 = -1[(1 - v1 ) (w)] .

The desired pair is then (u1 , u2 ) . In the above algorithm, the function KC (t) is the distribution function of the random variable C (U1 ,U 2 ) , where U1 and U2 are uniform random variables with an Archimedean copula C generated by . For the Gumbel copula, the above algorithm is:

(1) Generate two independent uniform variates (v1, v2 ) .

(2)

Set

KC (w) =

w1 -

ln(w)

=

v2 , and

solve

numerically for

0<

w < 1.

(5)

(3) Set u1 = exp[v11/ ln(w)] and u2 = exp[(1 - v1 )1/ ln(w)] .

The Frank Copula

Using the conditional distribution approach discussed earlier, one can draw variates from the

Frank copula as follows:

(1) Draw two independent uniform random variates (u1, v2 ) .

(2)

Set

u2

=

-1

ln1 +

v2 (1 - e - ) v2 (e -u1 - 1) - e -u1

(6)

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The Joe Copula

Random variates from the Joe copula can be obtained using the general algorithm for

Archimedean copulas, discussed under Gumbel's copula:

(1) Generate two independent uniform variates (v1, v2 ) .

[ ][ ] (2)

Set

KC

(w)

=

w

-

1

ln(1-(1- w) ) (1-(1- w) (1- w)-1

= v2, and solve numerically for 0 < w < 1. (7)

[ [ ] ] [ [ ] ] (3)

Set

u1

= 1 - 1 - 1 - (1 - w)

v1

1/

and

u2

= 1- 1- 1- (1- w)

1- v1

1/

.

References:

Genest, C., and L.-P. Rivest (1993) Statistical Inference Procedures for Bivariate Archimedean Copulas. Journal of the American Statistical Association, 88(423), 1034-1043.

Johnson, H. (1987) Options on the Maximum or the Minimum of Several Assets. The Journal of Financial and Quantitative Analysis, 22(3), 277-283.

Nelsen, R. B. (2006) An Introduction to Copulas (2nd ed.), Springer-Verlag, New York.

Venter, G. G. (2001) Tails of Copulas. Presented at ASTIN Colloquium, International Actuarial Association, Washington D.C.

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