Application of Bernstein copulas to the pricing of multi-asset derivatives Bertrand TAVIN ∗ Universit´e Paris 1 - Panth´eon Sorbonne [email protected] November 5, 2012

Abstract This paper deals with the application of Bernstein copulas to the pricing of derivatives written on several underlying assets. We review the main characteristics of this particular family of copulas. We then analyze their properties in a context of multi-asset derivatives pricing, with a focus on the approximation property. We finally give details about implementation steps and provide numerical evidences to illustrate the reviewed properties.

Keywords: Bernstein Copula, Multi-asset Derivative Pricing, Approximation, Multivariate Distribution. JEL Classification: C63, G13.

∗ This is a paper version of contributed chapter Tavin, B. (2013). Application of Bernstein Copulas to the Pricing of Multi-Asset Derivatives, In book: Copulae in Mathematical and Quantitative Finance, Proceedings of the Workshop Held in Cracow, 10-11 July 2012, Jaworski, Piotr; Durante, Fabrizio; H¨ ardle, Wolfgang Karl (Eds.), Springer-Verlag, Series Lecture Notes in Statistics, forthcoming 2013. Acknowledgments : I thank two anonymous referees and an editor for their comments on an earlier version. All remaining errors are mine. First version: July 25, 2012. Contact: Laboratoire PRISM - EA4101, 17 rue de la Sorbonne 75005 Paris France. Tel: +33140463170 Fax: +33140463011

1

1

Introduction

When facing a multivariate modeling problem, one needs a proper tool to model dependence. Copula functions are such a tool as they allow for the dependence to be modeled separately from the marginals, whenever these marginals are continuous. This paper focuses on a particular family of copulas, Bernstein copulas, and considers their application in finance. For further definitions, properties and references about copula functions, see [6] and the monograph [15]. [8] and [9] use Bernstein copulas in geology, for the modeling of dependence between petrophysical properties of oil reservoirs. In insurance, [5] uses Bernstein copulas to model the dependence between non-life insurance risks. In finance, [17] and [10] apply Bernstein copulas to the pricing of two-asset derivatives written on foreign exchange rates. Their approach focuses on the flexibility of the Bernstein copula that can be fitted to available market data, namely vanilla options on the cross exchange rate. In this paper we also work with Bernstein copulas for the pricing of multi-asset derivatives but our focus is different. It is on the approximation property of Bernstein copulas when a dependence model has already been chosen or fitted.

2

The Financial Framework

We consider a financial market with one period and n + 1 primary assets, t = 0 is the initial time and t = T < +∞ is the final time. The final prices of the primary assets are modeled as positive random
variables on (Ω, F , P) and are denoted by BT , ST1 , . . . , STn . The 0th asset, B, is a maturity T and risk-free zero-coupon bond. A European multi-asset derivative Z is a derivative that is written on up to n risky assets and that pays ZT at maturity. ZT is a positive random variable on (Ω, F , P) written ZT = z(ST1 , . . . , STn ) for a positive payoff function z on [0, +∞[n . In accordance with the First Fundamental Theorem of Asset Pricing, the time 0 price of Z can be obtained as the discounted expectation of its payoff under a risk-neutral probability measure Q. The time 0 price of Z is denoted by Z0 and writes Z0 = B0 EQ [ZT ]. A probability measure is said risk-neutral when it is equivalent to P and the discounted asset prices are Q-martingales. There are many ways to build such a measure for pricing purposes. An approach that is particularly suitable to our context is to construct the joint distribution of log-returns of the primary asset prices in two steps. In the first step their marginal distributions are computed from, or fitted to, the available vanilla option prices. And in the second step a dependence structure is applied to the marginals by means of a copula function. This two-step approach allows for a separated modeling of risk factors and for flexibility in the choice of the risk-neutral marginals. Background and details on this approach can be found, among others, in [2], [16], [20] and in the monograph [3]. Gaussian and Student copulas are derived from the associated multivariate distributions and both are popular choices. The former is parametrized with a correlation matrix R. The latter is parametrized with a correlation matrix R and a degree of freedom ν, it is symmetric and have tail dependence. See [4] for details. The skew t copula is the copula derived from the multivariate skew t distribution built in [1]. It works with a correlation matrix R, a degree of freedom ν and a skew vector α. This copula is able to describe asymmetric
dependence. See [1] and [12] for details. log-returns, Let Y01 , . . . , Y0n be the T -forward
prices of the n risky assets and define the associated
for k = 1, . . . , n, as XTk = ln STk /Y0k . F denotes the joint distribution of XT1 , . . . , XTn under Q, with F1 , . . . , Fn its marginals. ZT rewrites ZT = g(XT1 , . . . , XTn ) with g a positive function on ] − ∞, +∞[n . We denote by C (n) the set of n-dimensional copulas. Let C ∈ C (n) be the chosen risk-neutral copula to model the dependence structure of the asset price log-returns. We have Z
  g(x)dC(F1 (x1 ), . . . , Fn (xn )) = EQ [ZT ] = EQ g XT1 , . . . , XTn ]−∞,+∞[n Z = g(F1−1 (u1 ), . . . , Fn−1 (un ))dC(u) u∈[0,1]n

2

When C is absolutely continuous, and c = EQ [ZT ] =

Z

∂nC ∂u1 ...∂un

u∈[0,1]n

3

is its density, the integral becomes

g(F1−1 (u1 ), . . . , Fn−1 (un ))c(u)du

(2.1)

Bernstein Copulas and Their Properties

The family of Bernstein copulas was introduced in [13] and [14]. This family of copulas is built with Bernstein polynomials as building blocks. Bivariate Bernstein copulas are studied in [7] and their multivariate extension is considered in [19] and [18]. [11] studies the asymptotic properties of the Bernstein copula estimator. Definition 3.1. Bernstein Polynomial m (Bi,m )i=0 are the m + 1 Bernstein polynomials of degree m ∈ N, defined for x ∈ [0, 1] as   m Bi,m (x) = xi (1 − x)m−i i Let Ln,m be a discretization of the n-dimensional unit hypercube [0, 1]n , with m ∈ N discretization steps in all dimensions, and written as nα o αn  1 Ln,m = ,..., αj ∈ N and 0 ≤ αj ≤ m for j = 1, . . . , n m m For ease of readability u = (u1 , . . . , un ) will denote an element of [0, 1]n and v = (v1 , . . . , vn ) will α denote an element of Ln,m , with
vj = mj for some αj ∈ N and 0 ≤ αj ≤ m (j = 1, . . . , n) and so that v αn α1 can also be written m , . . . , m .

Definition 3.2. Bernstein Copula m For ξ a given real-valued function on Ln,m , define CB : [0, 1]n −→ [0, 1] as m CB (u)

=

X

ξ(v)

v∈Ln,m

n Y

Bαi ,m (ui ) =

m X

···

α1 =0

i=1

m X

αn =0

ξ

n

α

αn  Y Bα ,m (ui ) ,..., m m i=1 i 1

!

(3.1)

m If ξ fulfills the two conditions stated below, then CB is a proper copula, named Bernstein Copula with parameter function ξ.

1. for 0 ≤ αj ≤ m − 1 (j = 1, . . . , n) and with δn = 0 or 1 whether n is even or odd, respectively. 1 X

l1 =0

···

1 X

(−1)(δn +

Pn

j=1 lj

ln =0

)ξ



α1 + l1 αn + ln ,..., m m



≥0

2. for v ∈ Ln,m 

max 

n X j=1



vj − n + 1, 0 ≤ ξ (v1 , . . . , vn ) ≤ min (vj ) j=1,...,n

m In the sequel and unless stated differently, we consider CB an order m Bernstein copula with parameter function ξ. Let ∆n ξ be the n-dimensional volume operator applied to the function ξ and defined, for 0 ≤ αj ≤ m − 1 (j = 1, . . . , n), as

∆n ξ

  1 1 P X α1 + l1 αn  X αn + ln δn + n lj ) ( j=1 = ··· (−1) ξ ,..., ,..., m m m m

α

1

l1 =0

ln =0

3

Definition 3.3. Bernstein Copula Density m The Bernstein copula CB is absolutely continuous and has density cm B defined as ! m−1 m−1 n α Y n m X X ∂ C α 1 n B cm ∆n ξ Bα ,m−1 (ui ) (u) = ··· ,..., B (u) = ∂u1 . . . ∂un m m i=1 i α =0 α =0

(3.2)

n

1

As it is proven in [19], a Bernstein copula can always be decomposed as a sum of the product copula and a perturbation term. This decomposition is written m CB (u) =

n Y

ui +

i=1

γ(v) = ξ(v) −

X

γ(v)

v∈Ln,m n Y

n Y

Bαi ,m (ui )

i=1

vi

i=1

m If C ∈ C (n) , then CB (C) defined, for u ∈ [0, 1]n , as

m CB (C)(u)

=

X

v∈Ln,m

C(v)

n Y

i=1

Bαi ,m (ui ) =

m X

···

α1 =0

m X

αn =0

C

! n αn  Y ,..., Bα ,m (ui ) m m i=1 i

α

1

(3.3)

is a proper copula named Bernstein copula approximation of C with order m. The associated parameter m function is written ξ(C)(v) = C (v1 , . . . , vn ), for v ∈ Ln,m . CB (C) uniformly converges to C as the order m grows, so that a given copula can be approximated to any precision level by a Bernstein copula. The decomposition result is useful to understand how the Bernstein copula approximation behaves. If the copula to be approximated is the product copula, then the perturbation term is zero as well as the approximation error. If a given copula to be approximated is different from the product copula, then the perturbation term departs from zero. In probabilistic terms, the product copula represents independence. We could expect that, for a fixed order m, the approximation quality worsens as the given copula represents a dependence structure that departs from independence. In order to investigate this behavior, we restrict ourselves to the bivariate case and we consider a measure of the approximation error that is defined, for C ∈ C (n) and m ∈ N, as m supu∈[0,1]2 |CB (C)(u) − C(u)|

(3.4)

This kind of sup-norm distance based measure of the approximation error is used in [7] in the same context. It takes only positive values and goes to zero as the order m grows. In the bivariate case, there are, at least, three ways for a copula to depart from independence, namely association, tail dependence and asymmetry. We now consider the approximation of different families of parametric copulas and we compute the approximation error (3.4) for different sets of parameters. In figures 1, 2 and 3 below, we plot the approximation error as a function of the Bernstein copula order m. Plotted values are obtained with Matlab minimization routine fmincon. In figure 1 the approximated copulas are Gaussian copulas with different correlation parameters, corresponding to different levels of association. The three curves of approximation error are ordered according to the levels of association. As expected, the lower the association level is, the lower the approximation error is. In figure 2 the approximated copulas are Student copulas with a fixed correlation parameter and different degrees of freedom, corresponding to different levels of tail dependence. Even though the three curves of approximation error are close to each other, they are ordered according to the levels of tail dependence. As expected, the approximation error worsens with the level of tail dependence. In figure 3 the approximated copulas are skew t copulas with fixed correlation and degree of freedom and different skew parameter vectors, corresponding to different cases of asymmetry. The three curves of approximation error are ordered according to the corresponding levels of asymmetry. As expected, the approximation error worsens with the level of asymmetry. Curves corresponding to asymmetric cases decrease at a slower rate than the curve corresponding to the symmetric case. 4

The behavior of the Bernstein copula approximation hence depends on the characteristics of the approximated copula. The initial intuition that this behavior usually depends on how the approximated copula departs from independence is thereby confirmed by these numerical investigations. 0.045 0.04 0.035

0.025

m

sup|CB (C)−C|

0.03

0.02 0.015 0.01 0.005 0

0

20

40

60 m ρ = 0.15 ρ = 0.40 ρ = 0.70

80

100

120

Figure 1: Error measure associated with Bernstein copula approximations of Gaussian copulas, as a function of the order m and for different correlation parameter values.

0.025

0.015

m

sup|CB (C)−C|

0.02

0.01

0.005

0

0

20

40

60 m ρ = 0.40 ν = 12 ρ = 0.40 ν = 6 ρ = 0.40 ν = 3

80

100

120

Figure 2: Error measure associated with Bernstein copula approximations of Student copulas, as a function of the order m and for different degrees of freedom.

5

0.025

0.015

m

sup|CB (C)−C|

0.02

0.01

0.005

0

0

20

40

60 80 m ρ = 0.40 ν = 6 α = (0.0, 0.0) ρ = 0.40 ν = 6 α = (−0.02, −0.04) ρ = 0.40 ν = 6 α = (−0.05, −0.08)

100

120

Figure 3: Error measure associated with Bernstein copula approximations of skew t copulas, as a function of the order m and for different skew vectors.

4

The Pricing of Multi-asset Derivatives

Within the framework of Section 2 we consider the pricing of multi-asset derivatives by means of Bernstein copulas, the definition and properties of which were detailed in Section 3. Let C ∈ C (n) be the chosen m risk-neutral copula and CB (C) be its order m Bernstein copula approximation. An approximation of Z0 , the multi-asset derivative price, is obtained by replacing the chosen copula by its approximation. Bernstein copulas are absolutely continuous. The multiple integral to be computed then writes Z Q g(F1−1 (u1 ), . . . , Fn−1 (un ))cm (4.1) E [ZT ] ≈ B (C)(u)du u∈[0,1]n

This representation is particularly suitable for the use of numerical quadrature methods or quasi Monte Carlo integration methods. The choice between the two methods to solve this multiple integral usually depends on its dimensionality. In the sequel of this section we restrict ourselves to the two-asset case. We perform a numerical investigation for common multi-asset payoffs, namely calls on the equally weighted basket of S1 and S2 and puts on the maximum of the same pair of assets. We work with realistic market data conditions that could correspond to derivatives written on equity market indices such as the French and German euro-denominated market indices, CAC40 and DAX30. To compute the prices of basket options we use the Matlab quadrature routine quad2d. To compute the prices of puts on the maximum we use the onedimensional Matlab quadrature routine quadgk because the multiple integrals (2.1) and (4.1) simplify to one dimensional integrals, see [3] for details about this simplification. We consider different parametric copulas. G1 and G2 are Gaussian copulas with parameters ρ = 0.55 and ρ = 0.75, respectively. ST1 is a Student copula with parameters ρ = 0.55 and ν = 6. SKT1 and SKT2 are skew t copulas with respective parameters (ρ = 0.55, ν = 6, α1 = −0.02, α2 = −0.04) and (ρ = 0.55, ν = 6, α1 = −0.05, α2 = −0.08). We consider the risk-neutral marginal distributions of XT1 and XT2 to be Normal Inverse Gaussian (NIG) and Gaussian distributions. The NIG distributions for the three months log-returns have parameters (a1 = 22.8, b1 = −16.0, m1 = 0.10, d1 = 0.11) and (a2 = 26.9, b2 = −18.3, m2 = 0.10, d2 = 0.11) and the Gaussian distributions for the same log-returns are parametrized with the corresponding ATM volatilities, respectively σ1 = 0.208 and σ2 = 0.188. The NIG distributions for the six months log-returns have parameters (a1 = 16.4, b1 = −12.5, m1 = 0.15, d1 = 0.14) and (a2 = 17.8, b2 = −13.4, m2 = 0.15, d2 = 0.15). The other market parameters are as follows. Three and six months forward prices of S1 and S2 6

are both set equal to 100. The three and six months zero-coupon bond prices are 0.9975 and 0.9950, respectively. In tables 1, 2 and 3 below we have gathered the exact and approximated prices, as well as the differences between both, for basket options struck below, above and at the underlying forward value. Their strikes are denoted by K. In table 4 we have gathered the exact and approximated prices, as well as the differences between both, for puts on the maximum struck below, above and close to the underlying forward value. Their strikes are also denoted by K. For basket options, the considered Bernstein copula order is m = 50 and for puts on the maximum, this order is m = 120. These values are chosen for an illustrative purpose and correspond, at the same time, to an acceptable precision for the approximated price and to a reasonable computational load to handle. The considered order is larger for puts on the maximum than for basket options because the price of the former is more sensitive to the dependence structure. All computations are done with Matlab routines on a personal laptop and the required computation times are indicated in the last column of the tables. Copula

K = 95

K = 98

K = 100

K = 102

K = 105

Comp. time

Gaussian G1 Gaussian G2 Student ST1 Skew t SKT1 Skew t SKT2

6.747 6.943 6.728 6.762 6.797

4.658 4.865 4.627 4.675 4.724

3.462 3.667 3.428 3.486 3.547

2.451 2.643 2.420 2.488 2.561

1.309 1.465 1.290 1.373 1.464

5.7 sec. 4.0 sec. 6.9 sec. 197 sec. 196 sec.

Bernstein ξ(G1 ) Bernstein ξ(G2 ) Bernstein ξ(ST1 ) Bernstein ξ(SKT1 ) Bernstein ξ(SKT2 )

6.725 6.915 6.708 6.747 6.788

4.637 4.839 4.608 4.661 4.718

3.442 3.642 3.410 3.474 3.542

2.433 2.620 2.403 2.477 2.559

1.292 1.444 1.275 1.365 1.466

9.1 sec. 5.0 sec. 11.2 sec. 10.2 sec. 10.2 sec.

price diff. (G1 ) price diff. (G2 ) price diff. (ST1 ) price diff. (SKT1 ) price diff. (SKT2 )

0.022 0.028 0.020 0.016 0.01

0.021 0.026 0.019 0.013 0.006

0.02 0.025 0.018 0.012 0.004

0.019 0.023 0.017 0.010 0.002

0.017 0.020 0.015 0.008 0.002

Table 1: Prices and approximation errors for call options on the equally weighted basket of S1 and S2 . Maturity is T = 3 months, marginals are NIG and the Bernstein copulas order is m = 50.

7

Copula

K = 95

K = 98

K = 100

K = 102

K = 105

Comp. time

Gaussian G1 Gaussian G2 Student ST1 Skew t SKT1 Skew t SKT2

6.444 6.623 6.423 6.450 6.478

4.523 4.732 4.490 4.531 4.575

3.469 3.685 3.433 3.485 3.540

2.596 2.809 2.563 2.626 2.694

1.604 1.795 1.583 1.660 1.746

0.1 sec. 0.1 sec. 0.2 sec. 173.1 sec. 173.8 sec.

Bernstein ξ(G1 ) Bernstein ξ(G2 ) Bernstein ξ(ST1 ) Bernstein ξ(SKT1 ) Bernstein ξ(SKT2 )

6.425 6.599 6.406 6.437 6.469

4.503 4.706 4.472 4.517 4.567

3.448 3.659 3.414 3.471 3.533

2.575 2.783 2.544 2.612 2.687

1.583 1.769 1.564 1.647 1.740

price diff. (G1 ) price diff. (G2 ) price diff. (ST1 ) price diff. (SKT1 ) price diff. (SKT2 )

0.019 0.024 0.017 0.014 0.009

0.021 0.025 0.018 0.014 0.008

0.021 0.026 0.019 0.014 0.008

0.021 0.026 0.019 0.014 0.007

0.021 0.026 0.019 0.014 0.006

7.7 5.6 7.6 7.6 7.6

sec. sec. sec. sec. sec.

Table 2: Prices and approximation errors for call options on the equally weighted basket of S1 and S2 . Maturity is T = 3 months, marginals are Gaussian and the Bernstein copulas order is m = 50.

Copula

K = 95

K = 98

K = 100

K = 102

K = 105

Comp. time

Gaussian G1 Gaussian G2 Student ST1 Skew t SKT1 Skew t SKT2

8.339 8.652 8.300 8.362 8.426

6.382 6.698 6.334 6.410 6.490

5.214 5.525 5.163 5.249 5.341

4.166 4.464 4.117 4.213 4.317

2.836 3.104 2.796 2.908 3.030

7.1 sec. 4.7 sec. 8.9 sec. 253.3 sec. 244.4 sec.

Bernstein ξ(G1 ) Bernstein ξ(G2 ) Bernstein ξ(ST1 ) Bernstein ξ(SKT1 ) Bernstein ξ(SKT2 )

8.306 8.611 8.270 8.341 8.415

6.351 6.659 6.306 6.391 6.483

5.183 5.487 5.137 5.233 5.337

4.137 4.428 4.091 4.198 4.315

2.810 3.071 2.772 2.896 3.032

12.5 sec. 6.9 sec. 12.6 sec. 12.8 sec. 12.4 sec.

price diff. (G1 ) price diff. (G2 ) price diff. (ST1 ) price diff. (SKT1 ) price diff. (SKT2 )

0.033 0.041 0.030 0.021 0.010

0.031 0.039 0.028 0.018 0.006

0.03 0.038 0.027 0.016 0.003

0.029 0.036 0.026 0.015 0.001

0.027 0.033 0.024 0.012 0.003

Table 3: Prices and approximation errors for call options on the equally weighted basket of S1 and S2 . Maturity is T = 6 months, marginals are NIG and the Bernstein copulas order is m = 50.

8

Copula

K = 105

K = 108

K = 110

K = 112

K = 115

Comp. time

Gaussian G1 Gaussian G2 Student ST1 Skew t SKT1 Skew t SKT2

3.799 4.516 3.888 3.960 4.037

5.599 6.416 5.695 5.782 5.877

7.062 7.927 7.164 7.261 7.366

8.702 9.600 8.812 8.915 9.028

11.402 12.327 11.522 11.631 11.752

0.7 sec. 0.6 sec. 0.6 sec. 12.6 sec. 12.6 sec.

Bernstein ξ(G1 ) Bernstein ξ(G2 ) Bernstein ξ(ST1 ) Bernstein ξ(SKT1 ) Bernstein ξ(SKT2 )

3.763 4.445 3.836 3.906 3.981

5.560 6.339 5.639 5.724 5.816

7.020 7.846 7.106 7.199 7.301

8.658 9.516 8.751 8.851 8.961

11.356 12.239 11.458 11.563 11.680

2.2 2.2 2.3 2.3 2.4

price diff. (G1 ) price diff. (G2 ) price diff. (ST1 ) price diff. (SKT1 ) price diff. (SKT2 )

0.036 0.071 0.052 0.054 0.055

0.040 0.077 0.056 0.059 0.061

0.042 0.081 0.059 0.062 0.064

0.044 0.084 0.061 0.065 0.067

0.046 0.089 0.064 0.068 0.072

sec. sec. sec. sec. sec.

Table 4: Prices and approximation errors for put options written on the maximum of S1 and S2 . Maturity is T = 3 months and marginals are NIG distributions. The Bernstein copulas order is m = 120. Tables 1 and 4 show that the use of Bernstein copulas can lead to acceptable levels of approximation while keeping computation time reasonable. It is particularly true for basket options, for which the pricing error magnitude is around two cents of the underlying forward value. Tables 2 and 3 confirm this feature as the magnitude of pricing errors is little affected by a change in marginal distributions or in time to maturity. Across our numerical results, the main driving factors of the approximation behavior are the choice of the approximated copula, particularly its correlation parameter and the derivative payoff. This is in accordance with the remarks made in Section 2. From tables 1, 2 and 3, pricing with Bernstein copulas appears particularly suitable when the chosen copula is slow to compute because it has a complex expression, like the skew t copula. In such a case, the approach with Bernstein copulas offers a clear reduction of the computation time required to value a portfolio of derivatives. The reduction of computational time, however smaller, is confirmed for rainbow options by table 4. This smaller edge is explained by the possibility to compute the exact prices faster with one dimensional integrals and by the higher order of a Bernstein copula required to have an acceptable level of precision.

9

References [1] Azzalini, A., Capitanio, A.: Distribution generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J. R. Stat. Soc. Ser. B Stat. Methodol. 65, 367-389 (2003) [2] Cherubini, U., Luciano, E.: Bivariate option pricing with copulas. Appl. Math. Finance 9, 69-82 (2002) [3] Cherubini, U., Luciano, E., Vecchiato, W.: Copula Methods in Finance, Wiley Finance. John Wiley and Sons, Chichester (2004). [4] Demarta, S., McNeil, A.: The t Copula and Related Copulas. Int. Stat. Rev. 73, 111-129 (2005) [5] Diers, D., Eling, M., Marek, S.: Dependence modeling in non-life insurance using the Bernstein copula. Insur. Math. Econ. 50(3), 430-436 (2012) [6] Durante, F., Sempi, C.: Copula Theory: An Introduction. In: Jaworski, P., Durante, F., H¨ ardle, W., Rychlik, T.: (eds.) Copula Theory and Its Applications, Proceedings of the Workshop Held in Warsaw 25-26 September 2009, Lecture Notes in Statistics – Proceedings, vol. 198. Springer, Dordrecht (2010). [7] Durrleman, V., Nikeghbali, A., Roncalli, T.: Copulas approximations and new families. Working Paper, Groupe de Recherche Op´erationnelle du Cr´edit Lyonnais (2000) [8] Erdely, A., D´ıaz-Viera, M.: Nonparametric and Semiparametric Bivariate Modeling of Petrophysical Porosity-Permeability Dependence from Well Log Data. In: Jaworski, P., Durante, F., H¨ ardle, W., Rychlik, T.: (eds.) Copula Theory and Its Applications, Proceedings of the Workshop Held in Warsaw 25-26 September 2009, Lecture Notes in Statistics – Proceedings, vol. 198. Springer, Dordrecht (2010). [9] Hern´andez-Maldonado, V., D´ıaz-Viera, M., Erdely, A.: A joint stochastic simulation method using the Bernstein copula as a flexible tool for modeling nonlinear dependence structures between petrophysical properties. J. Petrophys. Sci. Eng. 90-91, 112-123 (2012) [10] Hurd, M., Salmon, M., Schleicher, C.: Using copulas to construct bivariate foreign exchange distributions with an application to the sterling exchange rate index. Bank of England, Working paper 334 (2007) [11] Janssen, P., Swanepoel, J., Veraverbeke, N.: Large sample behavior of the Bernstein copula estimator. J. Stat. Plan. Inference 142(5), 1189-1197 (2012) [12] Kollo, T., Pettere, G.: Parameter Estimation and Application of the Multivariate Skew t -Copula. In: Jaworski, P., Durante, F., H¨ ardle, W., Rychlik, T.: (eds.) Copula Theory and Its Applications, Proceedings of the Workshop Held in Warsaw 25-26 September 2009, Lecture Notes in Statistics – Proceedings, vol. 198. Springer, Dordrecht (2010). [13] Li, X., Mikusi´ nski, P., Sherwood, H., Taylor, M.: On approximations of copulas. In: Beneˇs, V., ˇ ep´an, J.: (eds) Distributions with given marginals and moment problems, Kluwer, Dordrecht (1997). Stˇ [14] Li, X., Mikusi´ nski, P., Taylor, M.: Strong approximation of copulas. J. Math. Anal. Appl. 225(2), 608-623 (1998) [15] Nelsen, R.B.: An introduction to copulas, Lecture Notes in Statistics, vol. 139. Springer-Verlag, New York (1999). [16] Rosenberg, J.: Nonparametric Pricing of Multivariate Contingent Claims. J. Deriv. 10(3), 9-26 (2003) [17] Salmon, M., Schleicher, C.: Pricing Multivariate Currency Options with Copulas. In: Rank, J. (ed.): Copulas: from theory to application in Finance. RISK Books, London (2006).

10

[18] Sancetta, A.: Nonparametric estimation of distributions with given marginals via BernsteinKantorovitch Polynomilas: L1 and pointwise convergence theory. J. Multivar. Anal. 98(7), 1376-1390 (2007) [19] Sancetta, A., Satchell, S.: The Bernstein copula and its application to modeling and approximations of multivariate distributions. Econom. Theory 20(3), 535-562 (2004) [20] van der Goorbergh, R., Genest, C., Werker, B.: Bivariate option pricing using dynamic copula models. Insur. Math. Econ. 37(1), 101-114 (2005)

11