hyperbolic distributions in finance

8/8/2019 Hyperbolic Distributions in Finance

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International Statistical Institute (ISI) and Bernoulli Society for Mathematical Statistics

and Probability

Hyperbolic Distributions in FinanceAuthor(s): Ernst Eberlein and Ulrich KellerSource: Bernoulli, Vol. 1, No. 3 (Sep., 1995), pp. 281-299Published by: International Statistical Institute (ISI) and Bernoulli Society for Mathematical Statistics andProbabilityStable URL: http://www.jstor.org/stable/3318481

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Bernoulli 1(3), 1995, 281-299

Hyperbolic distributions in f i n a n c eERNST EBERLEIN and ULRICH KELLERInstitutfiir MathematischeStochastik, Universitit Freiburg,Hebelstrasse 27, 79104 Freiburg,GermanyDistributional ssumptionsor the returnson the underlying ssetsplaya keyrolein valuation heories orderivative ecurities.Basedon a data set consistingof daily pricesof the 30 DAX sharesovera three-yeaperiod,we investigatehe distributionalorm of compound eturns.Afterperforming numberof statisticatests, t becomes lear hatsomeof the standard ssumptionsannotbejustified.nstead,weintroduceheclassof hyperbolic istributions hichcan be fitted o the empiricaleturnswithhigh accuracy.Two modelsbasedon hyperbolicLevymotion are discussed.By studying he Esscher ransform f theprocesswithhyperbolireturns,we derivea valuation ormula or derivative ecurities.The resultsuggestsa correction f standardBlack-Scholespricing, speciallyoroptionscloseto expiration.Keywords:bsolutecontinuous hangeof measure; yperbolic istributions;yperbolicLevymotion;optionpricing; tatistical nalysisof stockpricedata

1. IntroductionIn valuation theories for derivative securities as well as in other questions in finance thedistributional form of the returns on the underlying assets plays a key role. In this paper, afterinvestigating classical assumptions, in particular the normality hypothesis, we introduce a modelwhich fits the data with high accuracy and draw some conclusions concerning option pricing.

Let (Pt,),o denote the price process of a security, in particular of a stock. In order to allowcomparison of investments in differentsecurities,we shall investigate the rates of returndefinedbyXt = log Pt - log Pt-1. (1)

Like most authors, we prefer these rates, which correspond to continuous compounding, to thealternativeYt = (Pt - Pt- 1) Pt-1. (2)The reason for this is that the return over n periods, for example n days, is then just the sum

Xt + ...--+ Xt+n-1 = log Pt+n-1 - log Pt-1. (3)This does not hold for Yr.Another aspect we have in mind is that the underlying price process is acontinuous-time process from which discrete time series are drawn at equidistant time-points. Butfor continuous-time processes returns with continuous compounding are the natural choice. Thefact that the underlyingprocess is a continuous-time process led us to use t both as a continuous andas a discrete parameter.What is actually meant should be clear from the context. Numerically thedifferencebetween Xt and Ytis negligible since Y - Xt - 1X2 + X3 +... and X, is typically of theorder 10-21350-7265 c 1995 Chapman & Hall


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Hyperbolicdistributions nfinance 283

L-

Lo

0.0 0.2 0.4 0.6 0.8 1.0Time scale

Figure 1. Simulation of geometric Brownian motion

DaimlerBenzoo

C-

crash

0 200 400 600tradingdays

Figure 2. Daily stock prices from October 1989 to September 1992


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284 E. Eberlein and U. Keller

10.5 11 0 11.5 12.0 12 5 13.0 13.5Decimal time scale

Figure3. Intraday alueof Siemens hares,2 March 1992distributions can be fitted to the empirical distributions with high accuracy. The data we areexploring aredaily KASSA pricesof 10 of the 30 stocks which compose the DAX, the Germanstockindex, duringthe three-year period from 2 October 1989 to 30 September1992. This gives time seriesof 745 data-points each for the returns.The data are corrected for dividend payouts, that is to say,the returns on ex-dividend days are defined by

X = log(Pt + d) - log P1, (6)where d, is the amount paid in dividend on day t. Note that dividends are paid only once a year forGerman stocks. Two unusual price changes occurred in this period: the crash on 16 October 1989and the drop as a consequence of the coup in Moscow on 19August 1991. The lattercan be seen as adeep notch in Fig. 2. The 10 stocks considered here were chosen on account of their large tradingvolume and also because of the specific activity of the company in order to get a reasonablerepresentation of the market. This choice has no influence on the conclusions.

2. Testing for classical assumptionsIt is well known that the normal distribution is a poor model for stock returns. In this section weshow explicitly how large the discrepancy actually is. A qualitative yet very powerful method fortesting the goodness of fit are quantile-quantile plots. Figure 4 shows normal QQplots for thereturnsof BASF and Deutsche Bank. The deviation from a straight line and thus from normality isobvious. Also shown are the correspondingempiricaldensities and the normal density. It is evident

0- --0.4 -

0-000:

t-

1051.C151.01D 301.0eia0im clFiue3CnrdyvleoDSeessae,2Mrh19


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Hyperbolicdistributions nfinance 285U)0 u 0

-oa 9

((0

, I oQuantiles of Standard Normal Quantiles of Standard Normal

BASF Deutsche Bank

0o 0- 0

-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3Quantiles f StandardNormal Quantiles f StandardNormaluantiles of Standard Normal Quantiles of Standard Normal

Figure4. NormalQQplotsanddensityplotsthat there is considerably more mass around the origin and in the tails than the standard normaldistribution can provide.The order of magnitude of the deviation can be seen from Table 3 below. Counting the numberofreturnsin the interval (-ku, ku), for k < 5, we compare the relative frequencieswhich are given inthe first line for each stock with the expected normal frequencieswhich are given in the first line ofthe table. The values for the 10companies considered are quite uniform and differ from the normalat the origin, that is to say, in the interval (-a, a) by 0.1. In particular, note that the empiricaldistributions have roughly 5000 times the mass of the normal distribution in the tails startingat 5a.We continue with X2tests for normality. To avoid any problems arisingfrom partition sensitivity,three differentestimation procedureswere considered. Let *2denote the test statisticcomputed withcells of equal probability (1/k), while ^2is used for cells of equal width. The second cell structurewas modified by collapsing outer cells, such that the expected value of observations becomes greaterthan 5. The thirdprocedureis very much the same as the second, but startingwith k = 40 instead ofk = 22. For all stocks the null hypothesis is rejectedat the level a = 0.01. As an example we cite thecorresponding values for the two stocks considered above. Full-length tables are available inEberlein and Keller (1994). 2X-1;0.99enotes the 0.99 quantile of the X2 distribution with k - 1degrees of freedom.


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286 E. Eberlein and U. Keller-2 2 _2 2 -2 2XI Xk-1;0.99 X2 Xk-1;0.99 X3 Xk-1;0.99

BASF 104.02 38.93 62.54 18.48 93.74 27.69DeutscheBank 88.02 38.93 55.88 18.48 87.92 30.58

Another standard method of testing for normality is to compute certain functions of the momentsof the sample data and to compare them with the expected values for a normal population. We usetwo such tests which, moreover, have the favourable feature of scale and location invariance, so weare able to test the composite hypothesis by means of these tests. If we denote bymk= n-1 1 i- V)k the sample moment of order k, then the test statistics are given byK = m4/m2 - 3 and S = m3/m/2, measuring kurtosis and skewness of the sample, which shouldboth be zero under the assumption of normality. Again for brevity'ssake we mention the results forBASF and Deutsche Bank. For the former the skewness is 0.52 and the kurtosis 7.40, while for thelatter we got 1.40and 16.88. For all stocks the hypothesis is rejectedat the 1%level. Finally, becauseof the long tails which we observe for financialdata, the range Xmax xminof the sample should be agood indicator for non-normality. Indeed the studentizedrange test turned out to be another usefultool. The corresponding statistic is given by

Xmax - Xmin

i=1ius 12.59 and 14.56 for the two stocks considered above, which means rejectionat the smallest levela = 0.005 which can be found in the tables in David et al. (1954).We will now discuss briefly another class of distributions proposed by Mandelbrot (1963), thestable Pareto distributionswith characteristicexponent a, denoted by SP(a). In the symmetriccasestable Pareto distributions are defined by the log-characteristicfunction

log0(t) = i6t- (cltl)a (7)where 6 denotes the location parameter, c the scale parameter and a the characteristic exponent,defined in the interval 0 < a < 2. For a = 2 it coincides with the normal distribution, for a = 1 itgives the Cauchy distribution. As an immediate consequence, one sees that if (Xi)l


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Hyperbolicdistributions nfinance 287variables - with the same a - if one considers sums. A technique for estimating the characteristicexponent a has been developed by Fama and Roll (1968; 1971). They propose a fractile estimatorgiven by

fsp(a) = 0.827 x - x-f 0.95 2 which is not in thepermitted domain. For sums of length 25 the resulting series have a length of only n25= 29 with our series ofn = 745 data-points for each company. This may lead to considerable sampling errors.


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288 E. Eberleinand U. Keller0

coc;

o aDBK returnssquared eturnst-=0

0 5 10 15 20 25 30lags(days)

Figure5. Autocorrelationsthe picture shows, the autocorrelations of squaredreturns are not higher than those for the returnsthemselves.

3. Hyperbolicdistributions ndfinancialdataHyperbolic distributions are characterized by their log-density being a hyperbola. Recall that fornormal distributions the log-density is a parabola, so one can expect to obtain a reasonablealternativefor heavy tail distributions. The parametrizationof the hyperbolic density, which we willuse later, is given by

hyp(x) = exp(-a62+(x 2 +/(x-p)), (9)2a6KK1(6V/a2/32)where K1denotes the modified Bessel function of the third kind with index 1. The first two of thefour parameters, namely a and 3 with a > 0 and 0 1/31 a, determine the shape of thedistribution, while the other two, 6 and Mu,are scale and location parameters. WithS= (1 + 6 2- /2-)-1/2 and X =- /a one gets a different parametrization hyp(x; X, (, 6,M)


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Hyperbolicdistributions nfinance 289Table2. Estimatedparametersor thehyperbolic istribution

s ftBASF 108.82 1.3550 0.0014 -0.0005BMW 89.72 4.7184 0.0009 -0.0015Daimler Benz 88.19 4.1713 0.0019 -0.0014Deutsche Bank 108.60 0.3972 0.0030 -0.00001Dresdner Bank 110.94 -0.1123 0.0016 0.0002Hoechst 104.78 2.8899 0.0020 -0.0007Preussag 83.51 3.0702 0.0031 -0.0008Siemens 112.65 6.4287 0.0033 -0.0011Thyssen 94.27 0.0792 0.0073 -0.0003VW 83.85 -4.0652 0.0077 0.0009

o exponential Laplace exponential

48o10

c:E0o H(-) H(+)

cHJ

o

o normal-1.0 -0.5 0.0 0.5 1.0chi-domain

Figure 6. The hyperbolic shape triangle.The location of the estimates for the invariant parameters(), Q)is indicated by the numbers 1 to 10, where the numberscorrespond to the order of the shares in Table 2.The locations of the limits of hyperbolic distributions are indicated, where H(-) (or H(+)) means ageneralized inverse Gaussian distribution. The solid line marks the 95%-confidence region for theestimates of VW (number 10)


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290 E. Eberleinand U. KellerTable 3. Frequency distributions

< lc < 2c < 3o < 4c < 5c > 5rStandard normal 0.683 0.955 0.997 1.000 1.000 0.0000006BASF 0.774 0.944 0.987 0.996 0.997 0.0026846 (2)hyp (a = 108.82, f = 1.36, 6 = 0.0014, p = 0) 0.769 0.948 0.988 0.997 0.999 0.0006383 (0.5)BMW 0.796 0.949 0.987 0.996 0.996 0.0040268 (3)hyp (a = 89.72, / = 4.72, 6 = 0.0009, p = 0) 0.779 0.951 0.989 0.998 0.999 0.0005342 (0.4)Daimler Benz 0.785 0.954 0.987 0.995 0.999 0.0013423 (1)hyp (a = 88.19, -= 4.17, 6 = 0.0019, j = 0) 0.774 0.950 0.989 0.998 0.999 0.0005826 (0.4)Deutsche Bank 0.797 0.964 0.988 0.993 0.997 0.0026846 (2)hyp (a = 108.60, -= 0.38, 6 = 0.003, p = 0) 0.777 0.954 0.990 0.998 1.000 0.0004267 (0.3)Dresdner Bank 0.807 0.950 0.988 0.995 0.997 0.0026846 (2)hyp (a = 110.94, 3 = -0.11, 6 = 0.0016, = 0) 0.781 0.953 0.990 0.998 1.000 0.0004497 (0.3)Hoechst 0.785 0.944 0.989 0.996 0.997 0.0026846 (2)hyp (a = 104.78, 3= 2.89, 6 = 0.002, p = 0) 0.771 0.950 0.989 0.998 0.999 0.0005439 (0.4)Preussag 0.792 0.957 0.991 0.995 0.997 0.0026846 (2)hyp (a = 83.51,4 = 3.07, 6 = 0.0031, u = 0) 0.778 0.953 0.990 0.998 1.000 0.0004472 (0.3)Siemens 0.792 0.953 0.989 0.995 0.997 0.0026846 (2)hyp (a = 112.65, / = 6.43, 6 = 0.0033, p = 0) 0.768 0.950 0.989 0.998 0.999 0.0005139 (0.4)Thyssen 0.762 0.960 0.989 0.993 0.996 0.0040268 (3)hyp (a = 94.27, 3= 0.07, 6 = 0.0073, p = 0) 0.788 0.957 0.991 0.998 1.000 0.000 3547 (0.3)VW 0.772 0.962 0.991 0.995 0.997 0.0026846 (2)hyp (a = 83.85, P = -4.07, 6 = 0.0077, , = 0) 0.741 0.938 0.984 0.996 0.999 0.0010790 (0.8)In thecolumns he relative requenciesf the returnsn theinterval(-kcr,kr) arecomparedwith theprobabilities f thefittedhyperbolicdistributions.The tails startingat 5orare givenin the column abelledby > 5or,wherethe numbersnbracketsgive the absolutefrequenciesound in this regionand the expectedvalues for the corresponding yperboldistribution.

which has the advantage that ( and Xare invariant under transformations of scale and location. Thenew invariant shape parametersvary in the triangle 0 < IXI ( < 1, which was therefore called theshape triangleby Barndorff-Nielsenet al. (1985). For ( -+ 0 the normal distributionis obtained as alimiting case; for ( -+ 1 one gets the symmetricand asymmetric Laplace distribution;for X -+ ? itis a generalizedinverse Gaussian distribution (in fact, a distributionhaving density of the form (10));and, finally, for IX -- 1 we will end up with an exponential distribution.It was pointed out by Barndorff-Nielsen (1977) that the hyperbolic distribution can berepresented as a normal variance-mean mixture where the mixing distribution is generalized


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Hyperbolicdistributions nfinance 2910

o1

04

(0Co v 7Eo

10

6d 13162od

-1.0 -0.5 0.0 0.5 1.0chi-domain

Figure7. Parameter stimates or the returns f Commerzbanks time-lagsncrease

inverse Gaussian with density___ _ expr 1 -1dIG(x)= exp (-QyxI+ 'x) , x > 0. (10)2K1 ) 2

Let 7 = 62 and b,= a2 -_2 and consider a normal distribution with mean p +?a02 and variance a-2such that a2 is endowed with the distribution(10). Then the mixtureis a hyperbolicdistribution withdensity (9).To estimate the parameters of the hyperbolic distribution given by (9) we used a computerprogram described in Blkesild and Sorensen (1992). Assuming independent and identicallydistributed variables, a maximum likelihood estimation is performed by the program and theresults for our data set are given in Table 2.Figure 6 shows the estimates (X,2) which are all towards the top of the triangle, thus far fromnormality. They are well centred, which means that the distributions are nearly symmetric.Moreover, the indicated 95% confidence region for the estimates of VW gives first evidence thathyperbolic distributions are appropriatefor daily returns. To assess the goodness of fit, in Table 3we provide the frequency distributions in the same form as in Section 2 (see also Fama 1965).


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(C

0)

W

SCuCO0

CUEcd

0

o

d

-4 -2 0 2 4Standardquantilesof the fitted heoreticaldistribution

BASF6

Hyperbolic

un6..C StandardNormal

-4 -2 0 2 4Standard quantiles

Figure 8. QQ and density plots for BASF

0-d(UCO0)

Eo

Cu

O

E00C?

76.a y

-4 -2 0Standardquantiles of the fit

Deutscheto6-LOdH

00)0m2

S

0)0

cu660

-4 -2 0Standardqu

Figure 9. QQ and density p


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Hyperbolicdistributions nfinance 293For a quantitative analysiswe will again make use of x2 tests with the same estimation proceduresmentioned above (see Section 2 for a closer description). In this case it is correct to accept thehypothesis of a hyperbolic distribution at significancelevel a whenever 2 < 141 i = 1,2,3,since we were estimating the four parametersof the distributions. For all stocks the hypothesis of ahyperbolic distribution is accepted at the level a = 0.01. We give the explicit values for two cases.

,2 2 ,2 2 ,2 2X1 Xk-5;0.99 X2 Xk-5;0.99 X3 Xk-5;0.99BASF 26 33.41 5.45 13.28 10.7 26.22Deutsche Bank 17.5 33.41 9.27 13.28 15.94 26.22

Kolmogorov-Smirnov test values vary between 0.70 and 1.20 for all sharesand are well below 1.63,representinga lower bound for the 1% level for this number of observations. Let us also note thatthe values for the normal distribution are, with one exception, all above 1.8.Using a largerdata set, namely stock pricesfrom 1January 1988 to 24 May 1994,we also appliedthe fitting procedure to returns for time-lags of 1, 4, 7, ..., 22 trading days. The resultingestimatesfor Xand ( for the data for Commerzbank aregiven in the shape triangle in Fig. 7. It is strikinghowthe parameters tend to the normal distribution limit as the lags increase.Finally, we return to the graphicalmethods used above to underline our conclusion. In Figs 8 and9 we see the QQ plots for two of the shares. Because of the strong effect outliers have on QQ plots,for these plots the two outliers produced by the crash in October 1989 and by the Moscow coup wereremoved. To stay consistent we did the same for the density pictures and the plots of Fig. 4 inSection 2, although there is no effect as far as the density plots are concerned.Figures 8 and 9 show an excellent fit. Comparison with Fig. 4 suggests a strong preferencefor thismodel over the classical one. So we arrive at the conclusion that daily stock returns should bemodelled by hyperbolic distributions.

4. HyperbolicL vy motionHyperbolic distributionsare infinitelydivisible. This was shown by Barndorff-Nielsenand Halgreen(1977) by proving infinitedivisibility of the generalizedinverse Gaussian distribution, which is usedin the representation of the hyperbolic distribution as a mixture of normals. Given the empiricalfindings on stock returns in the previous section we concentrate now on the symmetriccentredcase,that is, where / = p = 0. This means also X = 0 in the second parametrizationmentioned above.Consequently in the symmetric centred case, using = -2 - 1 for notational convenience, thedensity (9) can be written as

hyp(,6(x) exp -( 1+ . (11)Let (Z,')t,>o be the Levy process definedby the infinitelydivisiblehyperboliclaw with densityhyp(,that is, the process with stationary independent increments such that Z4' = 0 and Y(Z?'6) has


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294 E. Eberlein and U. Keller

.

P0

---- -,

06

C~j

xo ?0

10-10Figure10. Convolution emigroup ensitiesdensity hyp,b6.We call (Z('6)t>ohyperbolic Levy motion. Since (Z4'6)t>o is a process with centredindependent increments it is a martingale. By stationarity and independence the second momentsare given by E{(Zt'6)2} = tE{(ZC'6)2}< oc with

E{(Z',6)2} 621 K2() (12)K() (12)As we will see later, all moments of (Z('6)t>oexist, therefore (ZF'6)t>os in fact an LP-martingale(p> 1).Using the mixture representation, one can easily compute the characteristic function of thehyperbolic distribution given by hyp,b6,namely

" K1 2 ?62u2)(U; , 6)=K(V/2-622)(13)1(() /( 62u22so one can see that hyperbolic distributions are not closed under convolution. By construction itfollows that E(eiuz~' = ,t(u; (, 6) = {0(u; C,6)}t for t

> 0. Thereforethe density of IY(Zt'6)is givenby the Fourier inversion formula1 0ff'6(x) = - cos(ux)Ot(u; , 6)du. (14)

The integral on the right-hand side can be computed numerically.The results are shown in Fig. 10,


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Hyperbolicdistributions nfinance 295where the densities of the convolution semigrouparerepresentedfor values of t varyingbetween 0.2and 2. The shape and the scale parameterswere chosen as ( = 6 = 1.From the explicit form of b one can derive the fact that the process (Zf')t>0odoes not have aGaussian part, since the first cumulant of the characteristic function given by (13) is asymptoticallylinear in u. Therefore it is a purely discontinuous process. We can choose a cadlag version. Theprocess can be representedin the form

Z ' = x(Iz (., du,dx) - du v(dx)), (15)oJR\{o}where pz is the random measure of jumps

Aiz(w,dt, dx) = l{Zzs()#o}(s,Az,(w))(dt, dx) (16)s>Oassociated with the process (Z4'6)t>o.ts compensator du v(dx) contains the Levy measure v of thedistribution given by hyp,b6.Note that the compensator is deterministic, because of the indepen-dence of the increments.Deriving the Levy-Khinchine formula for (Z4'6)t>O,he densityg(x; (, 6) ofthe L6vy measure v can be computed as follows. Again from the representation of hyp(,6 as amixture of normals one gets

(u)= exp t ) (17where K(s) = log E{exp(-sY)} and Y is an inverse Gaussian variable with density given by (10).Using the integral representation (5) given in Halgreen (1979) and a modified expansion forlog(1 + u2/2) given on p. 16 of the same paper, which in our case is written

u2 C eiUx-1- iuX-log 1+ - e xdx(18)one gets

( = J (eiux 1- iux)g(x;,6)dx. (19This is in view of (17) the LUvy-Khinchine representation for the integrable process (15) with

g(x;1 exp- xl 2y + ((16) exp(-x)20)E2X1 y(J2(6 2-)Y)+yde(6x p)) 0XI)

Here J1 and Y1are the Bessel functions of the first and second kind, respectively.Using well-knownasymptotic relations about J1 and Y1(see formulae 9.1.7, 9 and 9.2.1, 2 in Abramowitz and Stegun1968) the denominator of the integrandis asymptotically equivalent to a constant for y -- 0 and toy-- for y --+o. Therefore, one can deduce that g(x) behaves like 1/x2 at the origin, which meansthat there is an infinite number of small jumps in every finite time-interval.As a model for stock prices the naturalcandidate is now the process driven by a hyperbolic Levy


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Hyperbolicdistributions nfinance 297Table4. Optionprices or DeutscheBankTime to expiration Share value Hyperbolicprice Black-Scholes priceS= 2 650 0.01 0.00700 5.28 5.56

750 50.46 50.457= 5 650 0.10 0.06700 8.82 9.00750 51.24 51.207 = 10 650 0.65 0.58700 12.94 13.07750 52.87 52.82

7 = 30 650 5.25 5.26700 24.00 24.09750 60.63 60.65

Thesearetheprices or a European alloptionwith an assumed trikeof F = 700and anassumed nterestrate of r = 0.08 (p.a.). The values for ( and the volatilitya are theestimates escribed bove.Now we choose 0 by

So = e-rtEO(St). (27)such that (e-rtSt)t>ois a martingale.Let M(' (u, t) = E{exp(uZt'6)} denote the moment generating function of the hyperbolic Levymotion indexed by ( and 6 and consider

M'6(u, t;0) = euf;ff'(x; 8)dx, (28)the corresponding function under PO.Since by stationarity M '6(u, ;0) - M '6(u,1;O8), rom (27)we get

er - M


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298 E. Eberleinand U. Keller6 and (. If the derivative security is, for example, a European call option with exerciseprice F andtime to expiration 7, the value at time 0 is given by

EO[e-rr(S - F)+] (32)where x+ denotes max(x, 0). This expectation under P0*can be written asSof"6(x;0*+1dx- e-rrFf"'((x;0*)dx (33where c = In(F/So). Sinceff'6(x; 0*) is related to the original densityftf'6(x) by (25), whereftC'6(x)sgiven by (14), this value can be computed numerically (see Table 4).Some practitionersmight prefer to see the volatility a as an explicit parameter, although 6 is themore natural parameter here. By (12), o2 = 62K2(()/K1 (). Thus for

6 =: 6, ( K, (() (34)we get a process (Ztto= (Z '6 )t>o with E{(Zf)2)= 1. Now one can replace (Zt,'6)t>oby theprocess (aZt)t>o.

For example, equation (21) readsdY,= pY,_dt aY,_dZ(, (35

where a denotes the daily volatility. In order to get 0* from (31) one has to replace 6 by a(5 in thatformula, and the same remark holds for the pricing formula (33).

AcknowledgementsOur warmest thanks go to Ole E. Barndorff-Nielsenfor introducing us to hyperbolic distributionsand to Preben Blaesildand Michael K. Sorensen, also of the Department of Theoretical Statistics,University of Aarhus, for providing sophisticated software for parameterestimation. Special thanksgo to Frankfurter Wertpapierb6rseAG for providing high-frequency stock-price data for severalvalues traded on the Frankfurt Stock Exchange. The daily stock-price data for the 30 DAX valuesused in this paper were delivered by KarlsruherKapitalmarktdatenbank(KKMDB).

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