Derivative pricing using multivariate affine generalized hyperbolic distributions

Journal of Banking & Finance 34 (2010) 1607–1617

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Journal of Banking & Finance journal homepage: www.elsevier.com/locate/jbf

Derivative pricing using multivariate affine generalized hyperbolic distributions q José Fajardo a,*, Aquiles Farias b a b

Brazilian School of Public and Business Administration, Getulio Vargas Foundation, Praia de Botafogo, 190, 5th Floor, Botafogo, Rio de Janeiro, Brazil Central Bank of Brazil

a r t i c l e

i n f o

Article history: Received 25 May 2009 Accepted 6 March 2010 Available online 10 March 2010 JEL classification: G13 C13

a b s t r a c t In this paper we use multivariate affine generalized hyperbolic (MAGH) distributions, introduced by Schmidt et al. (2006), to show how to price multidimensional derivatives when the underlying asset follows a MAGH distribution. We also illustrate the approach using market data from the BOVESPA (São Paulo Stock Exchange) and the exchange rate of the Brazilian Real vs. US Dollar to price some multidimensional derivatives. Ó 2010 Elsevier B.V. All rights reserved.

Keywords: Generalized hyperbolic distributions Multivariate distributions Derivative pricing Lévy processes

1. Introduction Since the introduction of generalized hyperbolic distributions (GH) by Barndorff-Nielsen (1977), the use of such distributions in many areas have increased enormously. In finance, it is well known that single assets are well fitted by the GH, as can be seen in Eberlein and Keller (1995) and Prause (1999), for German data and other international indexes and in Fajardo and Farias (2004) for Brazilian data. On the other hand, it is important to study the behavior of portfolio returns, for example to compute value at risk and also to price multidimensional derivatives. Then the extension to multidimensional generalized hyperbolic (MGH) distributions is natural. The first work to use MGH distributions to fit financial market data was done by Prause (1999). But, as Schmidt et al. (2006) pointed out, the algorithm used to estimate the distribution parameters needs intensive computational effort since all parame-

q We thank an anonymous referee for very helpful comments that help to improve the present work. The usual disclaimer applies. The first author gratefully acknowledges financial support from CNPq-Brazil. The views expressed are those of the authors and do not necessarily reflect those of the Central Bank of Brazil. * Corresponding author. Tel.: +55 21 4503 4162; fax: +55 21 4503 4168. E-mail addresses: [email protected] (J. Fajardo), [email protected] (A. Farias).

0378-4266/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2010.03.007

ters must be estimated simultaneously. Also, MGH distributions are not able to model tail-dependence. Schmidt et al. (2006) introduced multivariate affine generalized hyperbolic (MAGH) distributions. They showed that these distributions can circumvent in a simple way many of the shortcomings observed in the estimation procedure used by Prause (1999). This is possible because MAGH is defined as an affine transformation of a vector formed by unidimensional GHs. Very recently, Chen et al. (2010) and Broda and Paolella (2009) apply a very similar method to study portfolio risk management. They verify that this kind of approach is very fast and accurate to calculate value at risk and expected shortfall. In this paper we show that MAGH distributions can generate a multidimensional Lévy process and then we show how to price multidimensional derivatives, by obtaining the density resulting from the convolution of MAGH distributions. This result is not trivial since, as Schmidt et al. (2006) pointed out, MAGH distributions are not closed under convolution. To the best of our knowledge there is no other work to use these distributions to price derivatives. The paper is organized as follows: In Section 2 we present the MAGH distributions; in Section 3 we discuss the algorithm used to estimate the MAGH parameters; in Section 4 we study the derivative pricing problem; in Section 5 we present the pricing of exchange options; in Section 6 we price swaps; and in the last sections we present our conclusions and an appendix.

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Fig. 1. Density of a symmetric MAGH.

2. Multivariate affine generalized hyperbolic distributions We can write GHðx; x; d; lÞ1 as an affine transformation of a ~ kÞ. Then, we can define ~ ; b; canonical GH: GHðx; x; 1; 0Þ, where x :¼ ða MAGH as follows.

Proposition 1. If X is a d-dimensional MAGH, i.e. X  MAGHðx; R; MÞ, then W ¼ BX is a set of d independent GHðx; d; lÞ distributions, where B is the inverse Cholesky factorization of the covariance matrix of X. Then, they use the following steps:

Definition 1. A d-dimensional random variable X is said to be distributed according to a MAGH distribution and we denote by X  MAGHðx; x; R; MÞ, with x :¼ ðx1 ; . . . ; xd Þ and xi :¼ ðki ; ai ; bi Þ0 ; i ¼ 1; . . . ; d, if:

(1) Get W ¼ BX, that is, a set of independent GHðxi ; d; lÞ. (2) Estimate the univariate GHs. (3) Translate the univariate parameters into the multivariate parameters.

(1) X can be expressed as an affine transformation of a vector Y: d

X ¼ A0 Y þ M;

ð1Þ

where A is an upper triangular matrix in Rdxd such that A0 A ¼ R is positive definite and the random vector Y consists of d mutually independent one-dimensional canonical GHðx; 1; 0Þ (for more details see Schmidt et al., 2006). The parameter M is a location parameter and R is a scaling factor. This definition is responsible for simplifying the estimation procedure. Fig. 1a and b shows the shape and contour graph of a bidimensional distribution. The MAGH distribution allows us to model more leptokurtic data. This distribution is extremely flexible, since k and a can be defined for each margin, improving the fit even if the margins have tails with different weights. Also, if R is a diagonal matrix, the margins would be independent and that fact is important in some situations. The simplification on the parameter estimation is due to a simple procedure that allows us to estimate d unidimensional distributions, instead of the simultaneous estimation of 5d þ ðd1ÞðdÞ 2 parameters of the d-dimensional distribution.

This procedure leads to less computational effort, since it consists of n univariate estimations, acquiring 5 parameters with each, parameters at instead of 1 multivariate estimation of 5d þ ðd1ÞðdÞ 2 once. For more details see Fajardo and Farias (2009). Another important issue in this procedure is the fact that unconstrained nonlinear optimization methods are used to estimate parameters, which allow us for robustness. Also, the optimization algorithm used belongs to the probabilistic, where first the objective function is evaluated at a random number of points being part of the search space and secondly, where the samples are transformed to become candidates for local optimization routines. Also, the above approach behave very well for higher dimensions, as Schmidt et al. (2006) pointed out and as Fajardo and Farias (2009) verified by estimating the distribution of a 6-dimensional portfolio. More details about the efficiency and robustness of the above procedure can be found in Section 6 in Schmidt et al. (2006). 4. Multidimensional derivative pricing Many important financial instruments are obtained by considering more than one asset, as for example spread options or swaps. In this section we show how to price these kinds of derivatives. Let us consider a market model with d assets and denote their prices by ðS1 ; S2 ; . . . ; Sd Þ. Now, assume that the price processes are given by:

3. Estimation algorithm

S1t ¼ S10 eX t ;

To estimate the parameter of the MAGH, we use the algorithm used by Fajardo and Farias (2009). First, they estimate the unidimensional distributions. Then, they use some propositions to simplify the procedure of the parameter estimation of the MAGH. The same approach was used by Schmidt et al. (2006) and Stützle and Hrycej (2005), and applied to several distributions. Basically, they use the following proposition.

where X ¼ ðX 1 ; X 2 ; . . . ; X d Þ is a d-dimensional Lévy process generated by a MAGH. This is possible because MAGH is infinitely divisible.2

1

See Appendix for the definition of GH distributions.

1

2

S2t ¼ S20 eX t ; . . . ;

d

Sdt ¼ Sd0 eX t ;

ð2Þ

4.1. Convolution of MAGHs Now, in order to price derivatives with maturity T, we need to obtain the following distribution: 2

See Appendix or Dwass and Teicher (1957).

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XT ¼

T X

X t  X t1 :

ð3Þ

Table 1 Sample statistics (%).

t¼1

Observe that we have:

/X T ðuÞ ¼

T Y

/X1 ðuÞ ¼ /TX 1 ðuÞ:

ð4Þ

t¼1

Since MAGH are not closed under convolutions, due to the fact that GH are not closed under convolutions, i.e. the convolutions of GHs are not necessarily a GH, only in the particular case of NIG do we observe this property. Then, to simplify the computation of the convolution of MAGHs, it is advisable to choose characteristic functions without imaginary component, see Press et al. (1992). This can be done by considering symmetric and centered distributions. Proposition 2. Let X  MAGHðx; x; R; MÞ, then the density of d

XT ¼

T X

d

X t  X t1 ¼

t¼1

T X t¼1

with X 0 ¼ 0, denoted by MAGHT ðx; x; R; MÞ can be expressed as: d Y

ebi yi

i¼1

MT0i ðbi Þ

T

ghi ðyi ; ai ; 0; ki ; 1; 0Þ;

ð5Þ

where MT0i represents the moment generating function of the GH symmetric distribution of the i margin powered by T and y ¼ ðA0 Þ1 ðx  TMÞ. Proof. See Appendix.

Tnlp4

Mean S.D. Asymmetry Kurtosis Minimum Maximum

0.152 3.498 0.422 7.109 19.583 24.512

0.060 3.060 1.074 18.661 15.563 38.977

Correlation Matrix

1.000 0.618

0.618 1.000

Then, we can use this risk-neutral measure to price any European derivative f:

Price ¼ erT EQ f ðST Þ:

X1;

MAGHT ðx; x; R; MÞ ¼ jAj1

Tnlp3

ð8Þ

In the next sections we use the above results to price a spread option of Telemar ON (TNLP3) vs. Telemar PN (TNLP4) and a Dollar swap vs. Ibovespa (Bovespa index). The former is not traded in the market, but is a very interesting instrument that can capture the voting premium,5 while the latter is traded on the OTC market of the BM&F (Brazilian Mercantile and Futures Exchange). Moreover, the assets present different dependence structures. In one case we have a positive correlation while in the other we have a negative one.

h

In this way, the distribution of MAGH convoluted T times is reduced to a function of the convoluted unidimensional symmetric and centered GHs. This convolution was already obtained (see Prause, 1999; Fajardo and Farias, 2004) by using IFFT.3

5. Pricing of spread options using MAGH

Remark 1. In the particular case of ki ¼ 0:5; 8i ¼ 1; . . . ; d, i.e. NIG distributions, Eq. (5) transforms into:

PO ¼ maxðS3T  S2T  K; 0Þ;

MAGHT ðx; x; R; MÞ ¼ jAj1

d Y

ghi ðyi ; ai ; bi ; 1=2; T; 0Þ;

ð6Þ

i¼1

In the market introduced in Section 4, let us consider a European spread option, which has a payoff, denoted by PO and given by:

ð9Þ

where K represents the strike price and T is the option maturity. It is very well known that the price c of a European derivative is given by:

due to the fact that NIG are closed under convolution.

c ¼ erT EQ maxðS3T  S2T  K; 0Þ;

4.2. Esscher transforms for MAGH

where EQ denotes the expectation under the risk- neutral measure Q and r is the risk-free rate. Using the dynamics of S3 and S2 , we have:

To obtain a risk-neutral measure Q, we use the multidimensional version of the Esscher transform. In our setting, we denote that transform by MAGHT;h , where h denotes the Esscher parameter.4 It is given by the following proposition. ~ kÞ; R; MÞ is given by: ~ ; b; Proposition 3. The density of MAGHT;h ðx; ða

~ kÞ; R; MÞ ~ ; b; MAGHT;h ðx; ða 0

¼

eh x M TMAGH ðhÞ

jAj1

d Y

ebi yi

i¼1

M T0i ðbi Þ

T

ð7Þ

where MT0i represents the moment generating function of the GH symmetric distribution of the i margin powered by T; y ¼ ðA0 Þ1 ðx  TMÞ and MMAGH denotes the moment generating function of the MAGH, given by Eq. (19) in Appendix. Proof. See Appendix.

3

h i 3 2 c ¼ erT EQ maxðS30 eX T  S20 eXT  K; 0Þ : Then, we obtain:

c ¼ erT

Z

1

Z



1

h

The same technique can be applied for unidimensional exotic options for more details see Fusai and Meucci (2008). 4 To see how to obtain the Esscher parameter h, see Eq. (21) in Appendix.

S30 ex3  S20 ex2  K



fS30 ex3 >ðS20 ex2 þK Þg   ~ k; R; M dx3 dx2 : ~ ; b;  MAGHT;h ðx2 ; x3 Þ; a 1

ghi ðyi ; ai ; 0; ki ; 1; 0Þ;

ð10Þ

We can rewrite the integration limits:

c ¼ erT

Z

1 1

Z

1

ln



S2 ex2 þK 0 S3 0





S30 ex3  S20 ex2  K



  ~ k; R; M dx3 dx2 : ~ ; b;  MAGHT;h ðx2 ; x3 Þ; a

5 TNLP4 and TNLP3 are the voting and non-voting stocks, respectively. It is documented that shares with superior voting rights usually trade at a premium relative to shares with inferior voting rights. This premium is often called the ‘‘voting premium”.

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Table 2 Estimated parameters for MAGH and its subclasses. Sample Tnlp3 vs. Tnlp4

GH Nig Hyp

a~

~ b

k

M

R

0.0895 0.4301 0.3401 0.9023 0.0904 0.6309

0.7979 0.1774 0.1869 0.0552 0.0949 0.0433

1.3207 1.8788 0.5 0.5 1 1

0.1239 0.0951 0.1518 0.0959 0.2233 0.1361

17.0810 14.4360 7.4062 6.4008 0.9509 1.1533

Finally, we obtain:

c ¼ erT S30

Z

 erT S20  erT K

1

Z

1



1

ln

Z

Z

1

1

1

Z



1



S2 ex2 þK 0 S3 0

ln

1

Z

S2 ex2 þK 0 S3 0

1

ln



S2 ex2 þK 0 S3 0

  ~ k; R; M dx3 dx2 ~ ; b; ex3 MAGHT;h ðx2 ; x3 Þ; a





ex2 MAGH

T;h



Loglikehood 14.4360 17.9440 6.4008 7.9559 1.1533 1.4334

1991.9 2538.4 2013.4 2541.8 2072.3 2547.7

Table 3 TNLP3 and TNLP4 correlation and pseudo correlation.

 ~ k; R; M dx3 dx2 ~ ; b; ðx2 ; x3 Þ; a

Distribution

q and q

MAGH MANig MAHyp Sample

0.8246 0.8339 0.9879 0.6178

  ~ k; R; M dx3 dx2 : ~ ; b; MAGHT;h ðx2 ; x3 Þ; a ð11Þ

Table 4 TNLP3 and TNLP4 Esscher parameters. h

T;h

~ k; R; MÞ is obtained using ~ ; b; Remember that MAGH ððx2 ; x3 Þ; a Proposition 2, all the terms are known and the integral can be computed numerically. If we set6 K ¼ 0, we have the price of an exchange option.

10%pa

15%pa

20%pa

25%pa

30%pa

GH

1.1885 3.1714

1.2172 3.1078

1.2406 3.0434

1.259 2.9779

1.2724 2.9111

h1 þ h2

1.9829

1.8906

1.8028

1.7189

1.6387

5.1. Telemar ON (TNLP3) vs. Telemar PN (TNLP4) exchange option

Nig

1.7865 4.3434

1.7895 4.2539

1.7913 4.1672

1.7938 4.085

1.7953 4.0051

To price this derivative, we use a sample of daily returns from 09/22/1998 to 05/30/2006. In Table 1 we present some sample statistics. To make the statistics uniform and have confidence in the correlations, we interpolated data when one of the assets was not traded in the market on a particular day. Now, using the approach presented in Section 3, we estimate the parameter. The results are presented in Table 2, where Nig and Hyp are the GH subclasses obtained when k ¼ 1=2 and k ¼ 1, respectively. Now let us introduce a measure that will help us to interpret the results.

h1 þ h2

2.5569

2.4644

2.3759

2.2912

2.2098

Hyp

1.6908 2.5216

1.6944 2.5191

1.6978 2.5166

1.7011 2.5142

1.7042 2.512

h1 þ h2

0.8308

0.8247

0.8188

0.8131

0.8078

Definition 2 (Pseudo Correlation of MAGH). Let a matrix Rdd be scale parameter of a MAGH defined by:

2

3

r21 r12    r1d 7 r22    r2d 7 7

6 6 r21 R¼6 6 . 6 .. 4

.. .

..

.

: .. 7 . 7 5

rd1 rd2    r2d

A matrix q , called a pseudo correlation matrix of MAGH, is defined by:

ri;j ; qi;j ¼ pffiffiffiffiffiffiffiffiffiffiffi rii rjj

ð12Þ

where qi;j represents the element in line i and column j of q and ri;j the element in line i and column jof R. Notice that q would be the correlation matrix if R were the covariance matrix. The pseudo correlation and sample correlation between Tnlp3 and Tnlp4, for GH and its subclasses, are presented in Table 3. The pseudo correlation is a reasonable measure for the linear dependence. 6 This choice was made to facilitate calculations. The case where K – 0 only needs the computation of one more integral. Later in Fig. 7 we present the result for several K. In a very recent paper, Hurd and Zhou (2009) also address the case of K – 0 by using the Fourier transform of the spread option payoff.

In what follows we calculate h, the Esscher parameter. As we mention before, we need to solve numerically the Eq. (21). The estimation results, for several values of the risk-free interest rate, are in Table 4. There are many works that study the impact of the skewness of the risk-neutral distribution on option pricing. It is closely related to a notion of symmetry in the market and the value of the Esscher parameter (see Fajardo and Mordecki (2006b) and Fajardo and Mordecki (2007) for more details). In Fig. 2a and b, we show the densities of the estimated MAGH and the risk-neutral distribution obtained with the Esscher parameter and with possible values of the Brazilian interest rate.7 Then, in Fig. 3a and b, we show the level curves, and finally, in Fig. 4a and b, we have the respective graphs for the difference between the riskneutral and estimated densities. We can see in the last graph (Fig. 4b), looking at level curves, that the effect of the Esscher transform is to give greater weight to events considered ‘‘worst”, and lower weights to events considered ‘‘best”. The numerical implementation for the calculation of the call price must be done carefully. Due to the fact that the algorithm used for this calculation, the FFT, needs variables to be equally spaced, particularly in the multivariate case, attention is called for since the inverse transform must be applied to variable y transformed with Proposition 2. The software used in the numerical implementation was MatlabÓ, using the same precision levels used in the numerical optimizations. The integral was computed with the trapezoidal method

7

At this writing, this rate is 13.5%, but in the past it was almost 26%.

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Fig. 2. Density Tnlp3 vs. Tnlp4.

Fig. 3. Level curves Tnlp3 vs. Tnlp4.

Fig. 4. Risk-neutral density–estimated density.

with a grid of 1000  1000 points. The FFT algorithm was used with 212 iterations. Fig. 5 shows the prices of exchange options ðK ¼ 0Þ with maturity in 10 days, with interest rate 10% per annum, for several initial prices of Tnlp3 and Tnlp4. For comparison, Fig. 6 shows the graph of the difference between the prices of the exchange option computed with MAGH and the prices computed with the formula obtained by Margrabe, when returns follow a bivariate normal distribution.8

Table 5 Sample statistics (%).

Mean S.D. Asymmetry Kurtosis Minimum Maximum Correlation matrix

8

See Margrabe (1978). Here, we used historical volatilities.

Dollar

Ibov

0,008 1.005 0.651 13.543 9.360 4.926

0.079 1.850 0.143 4.021 9.634 8.400

1.000 0.269

0.269 1.000

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Fig. 5. Price of a Tnlp4 vs. Tnlp3 exchange option with maturity 10 days.

Fig. 6. MAGH price–Margrabe price.

Another important sensitivity analysis is with respect to strikes and maturities. We use the following parameters to get some intuition: S20 ¼ S30 ¼ 50 and interest rate of 10% per annum. Fig. 7 shows the results.

6. Pricing of swaps using MAGH The pricing of swaps is very similar to the pricing of exchange options. For that reason we omit some steps. Now let us consider

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Subclass

a~

~ b

k

M

R

Dollar vs. Ibvsp

GH

0.2396 4.0283 0.5402 4.0749 0.3132 3.9730

0.0923 0.0950 0.0448 0.0901 0.0229 0.0792

1.3910 1.0038 0.5 0.5 1 1

0.0638 0.4115 0.0723 0.4143 0.0729 0.4227

1.3125 2.2166 0.7670 2.0054 0.2415 1.3986

Nig Hyp

2.2166 15.1710 2.0054 13.7250 1.3986 9.5725

2355.6 2593.1 2360.3 2593.2 2377.7 2593.4

a swap with the payoff ðS3T  S2T Þ, where T is the maturity. Then, the price of the swap, denoted by s, is given by:

Table 7 Dollar and Ibvsp correlation and pseudo correlation. Distribution

q and q

MAGH MANig MAHyp Sample

0.4967 0.6181 0.9198 0.269

s ¼ erT EQ

h i S3T  S2T ;

ð13Þ

where EQ denotes the expectation under the risk-neutral measure Q and r is the risk-free rate. Using the dynamics of S3 and S2 , we have:

s ¼ erT

Z

1 1

Table 8 Dollar and Ibvsp Esscher parameters.

Z

1

1

  ~ k; R; M dx3 dx2 : ~ ; b; ðS30 ex3  S20 ex2 ÞMAGHT;h ðx2 ; x3 Þ; a

Then,

h 10%pa

LL

Z

Z

  ~ k; R; M dx3 dx2 ~ ; b; ex3 MAGHT;h ðx2 ; x3 Þ; a 1 Z 1 1 Z 1   ~ k; R; M dx3 dx2 : ~ ; b;  ert S20 ex2 MAGHT;h ðx2 ; x3 Þ; a 1

1

15%pa

20%pa

25%pa

30%pa

8.6926 1.7292

13.133 3.5647

17.329 5.3179

21.339 6.9987

25.15 8.6094

h1 þ h2

10.4218

16.6977

22.6469

28.3377

33.7594

Nig

16.403 4.579

25.539 7.6106

34.281 10.515

42.653 13.3

50.675 15.971

h1 þ h2

20.9820

33.1496

44.7960

55.9530

66.6460

Hyp

3.245 0.93696

3.2488 0.92967

3.2524 0.92268

3.2558 0.91597

3.259 0.9095

6.1. Dollar  Ibovespa swap

h1 þ h2

4.1820

4.1785

4.1751

4.1718

4.1685

Now we consider an important contract traded in the BM&F OTC market: the Dollar  Ibovespa swap. To price this contract,

GH

s ¼ ert S30

1

1

ð14Þ In this way, after obtaining the risk-neutral density and computing the resulting integral, we obtain the price of the swap.

Fig. 7. Price of spread option Tnlp4 vs. Tnlp3 for several strikes.

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Fig. 8. Density exchange rate R$/US$ vs. Ibovespa.

Fig. 9. Level curves exchange rate R$/US$ vs. Ibovespa.

Fig. 10. Risk-neutral–sample densities.

we use the following sample9 from 02/01/1999 to 05/30/2006. Table 5 presents some sample statistics: 9 From 1994 to 02/01/1999, the Central Bank of Brazil followed a system of administering the exchange rate between the Brazilian Real and Dollar.

It is important to notice that in our sample we use the exchange rate quotation at the close of the market, given by the Central Bank of Brazil, called PTAX. We consider this quotation because the BM&F uses it to make settlements and cash flow adjustments. Also, we need to interpolate some data due to the fact that on Brazilian

J. Fajardo, A. Farias / Journal of Banking & Finance 34 (2010) 1607–1617

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Fig. 11. Price of Dollar vs. Ibovespa swap with 10-day maturity.

Fig. 12. Prices of Dollar vs. Ibovespa swap for several rates and maturities.

holidays the Dollar is quoted but there is no trading on the Bovespa. The estimated parameters using MAGH and its subclasses are presented in Table 6. Table 7 shows pseudo correlation for MAGH and its subclasses and the sample correlation.

The Esscher parameters computed, using several risk-free interest rates, are in Table 8. Observe that, in contrast with the Esscher parameters presented in Table 4, here the Esscher parameters have the same sign. This comes from the fact that Tnlp3  Tnlp4 shows a positive correlation, while in the Dollar  Ibovespa case the correlation is negative.

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In Fig. 8a and b, we present the densities of the estimated MAGH and the risk-neutral density given by the Esscher transform. The level curves of the respective densities are presented in Fig. 9a and b. And, as in the previous section, in Fig. 10a and b, we show the graph and level curves of the difference between the risk-neutral distribution and the sample distribution. We observe the same behavior of the risk-neutral distribution as in the last section. Fig. 11 shows the price of swaps, with maturity T ¼ 10 days and interest rate of 10% per annum, for several initial values of the exchange rate and Ibovespa (in relatives terms). In some cases the price is negative, such as when the expectation of Dollar variation is greater than the expectation of the Ibovespa variation. In that case, to exchange the Dollar return for the Ibovespa return, investors will demand to receive something (negative price). We also examine the sensitivity with respect to interest rate and maturity. The parameters used are, S20 ¼ S30 ¼ 1. The intuition is presented in Fig. 12.

where l is a location parameter, d is a scale factor, comparable to Gaussian r; a and b determine the distribution shape and k defines the tail fatness,10 and thus are subclasses of GH, and k

ða2  b2 Þ2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; aðk; a; b; dÞ ¼ pffiffiffiffiffiffiffi 1 2paðk2Þ dk K k d a2  b2 is a norming factor to make the curve area equal to 1 and

1 2

K k ðxÞ ¼

Z

1

0

   1  yk1 exp  x y þ y1 dy; 2

is the modified Bessel function of third kind with index k. The domains of the parameters are:

l; k 2 R a 0: Also is interesting to use the following parameter transformation:

6.2. Remark

b

In very recent works Chen et al. (2010) and Broda and Paolella (2009) use independent component analysis and GH distributions to model multivariate distributions. The former assumes that drift of returns is zero. While the latter not, which coincide with our model. Both models use the FastICA algorithm of Hyvärinen (1999) to obtain the transformation matrix that maximize the negentropy while our estimation uses cross entropy, which is less efficient than FastICA as pointed out by Broda and Paolella (2009). Also, to obtain convolutions of the GHs, Chen et al. (2010) use the FFT. While Broda and Paolella (2009) use saddle point approximations (SPA). The SPA can be seen as an approximation of inversion of the characteristic function of the convoluted distribution without integration, it allows to a fast estimation. Then, Quantiles and Expected shortfalls must be calculated. Our approach is based on FFT and then only numerical integration is need to price derivatives. It seems that the estimation procedure than combines FastICA and SPA can be more efficient than the procedure suggested by Schmidt et al. (2006).

a~ ¼ ad and b~ ¼ : a

ð15Þ

A.2. MAGH geometric motion Proposition 4. MAGH is infinitely divisible. Proof 1. By definition, if X  MAGHða; b; k; R; MÞ, d-dimensional, d we have X ¼ A0 Y þ M, where Y i  GHðai ; bi ; ki ; 1; 0Þ. We know that GH is infinitely divisible. Then, for each Y i , there 1=d are fY ij gj2f1;...;dg r.v. independents such that: d

Yi ¼

d X

Y 1=d ij :

ð16Þ

j¼1 1=d

Now denote by Y k d

X ¼ A0

d X

!

1=d

1=d

the vector in which line i is given by Y ik . Then,

d 1 d X 1 1=d 1=d A0 Y k þ d  M ¼ A0 Y k þ M : d d k¼1 k¼1

d

d X

þ

1 M, d

þM¼

Yk

k¼1

7. Conclusion

ð17Þ

In this paper we priced some multidimensional derivatives written in terms of assets that follow a Lévy process generated by a MAGH. In particular, we priced the Tnlp3 vs. Tnlp4 exchange option, which can capture the voting premium, and also an exchange rate vs. Ibovespa swap, available for trading in the BM&F OTC market. Of course other examples of multidimensional derivatives can be priced with the approach used in this paper. It is noteworthy that the dependence structure obtained is a simplification and a more realistic nonlinear dependence structure can be obtained using more complex models. It is very relevant for measuring the risk exposure in our contracts, otherwise we will need to estimate risk measures for our contracts as is done by Sorwar and Dowd (forthcoming) for unidimensional options, it will be a topic for future research.

Now let d

X¼

d X

1=d Xk

0

¼A

1=d Yk

then:

1=d

Xk : 

ð18Þ

k¼1

More details on sufficient and necessary conditions for a random vector of the form AY to be infinitely divisible can be found in Dwass and Teicher (1957). Proposition 5. The moment generating function of a d-dimensional d MAGH X ¼ A0 Y þ M is given by: 0

MX ðzÞ ¼ ez M

d Y

MY i ð~zi Þ;

ð19Þ

i¼1

where M Y i ð~zi Þ represents the moment generating function of Y i and ~z ¼ Az.

Appendix A d

Proof of Proposition 2. From X ¼ A0 Y þ M, we obtain for T ¼ 2: A.1. Generalized hyperbolic distributions

  2 F 2 ðA0 Þ1 ðx  2MÞ : X ðxÞ ¼ F Y

The probability density function of the one-dimensional GH is defined by:

 ð Þ ghðx; a; b; k; d; lÞ ¼ aðk; a; b; dÞ d2 þ ðx  lÞ2 K k1 2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2  a d þ ðx  lÞ exp ðbðx  lÞÞ; k1 2 2

Then, the densities are related by:

  fX2 ðxÞ ¼ fY2 ðA0 Þ1 ðx  2MÞ jAj1 : 10 Of course there are many other processes with fat tails as for example the tempered stable process considered by Kim et al. (2008) and Kim et al. (forthcoming).

1617

J. Fajardo, A. Farias / Journal of Banking & Finance 34 (2010) 1607–1617

By definition of Y, we have:

fX2 ðxÞ ¼

d Y

fY2i ðyi ÞjAj1 ¼ jAj1

i¼1

¼ jAj

Using Proposition 2: d Y

  ~ kÞ; R; M ¼ ~ ; b; MAGHT;h x; ða

fY2i ðyi Þ

i¼1

d Y 1

2

ghi ðyi ; ai ; bi ; ki ; 1; 0Þ;



ð20Þ

i¼1

where i represents the margins and y ¼ ðA Þ ðx  2MÞ. Using the equation of a centered GH (Prause, 1999): d Y

ghi ðyi ; ai ; 0; ki ; 1; 0Þ:

As we said, to obtain an equivalent martingale measure we use the one given by the Esscher transform (see Gerber and Shiu, 0 1994), generalized to the multivariate case: let Mðh; tÞ ¼ Eðeh X t Þ be the moment generating function, then define the following function:

Mðz þ h; tÞ : Mðh; tÞ

ð21Þ

Now, we look for a vector h such that, the density  0

dQht eðh Þ Xt ; ¼  0 dP Eðeðh Þ Xt Þ

ð22Þ

is an equivalent martingale measure. To this end, as in Gerber and Shiu (1994), take X 1t ¼ rt. Then, it is enough to find h such that:

Sj0 ¼ E ðert Sjt Þ 8j;

8t; 

where E is the expectation under probability Qh . Now let 0 1 1j ¼ @0; . . . ;

1 ; . . . ; 0A, then the solution of the following |{z}

jcoordinate

equation give us the vector h :

r ¼ ln½Mð1j ; 1; h Þ ¼ ln

ebi yi

i¼1

M T0i ðbi Þ

T

ghi ðyi ; ai ; 0; 1; 0; ki Þ:

2

2 i¼1 M 0i ðbi Þ

A.3. Equivalent martingale measures



d Y

References

e bi y i

The case T is straightforward. h

Mðz; t; hÞ ¼

jAj1

We finish the proof. h

0 1

fX2 ðxÞ ¼ jAj1

0

eh x M TMAGH ðhÞ



Mð1j þ h ; 1Þ : Mðh ; 1Þ

For other applications of Esscher transform to multidimensional derivative pricing see Fajardo and Mordecki (2006a). Proof Proposition 3. Using the Esscher transform definition, we have

  ~ kÞ; R; M ¼ ~ ; b; MAGHT;h x; ða

0

eh x MTMAGH ðhÞ

  ~ kÞ; R; M : ~ ; b; MAGHT x; ða

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