The Generalized Hyperbolic Model: Financial ... - Semantic Scholar

The Generalized Hyperbolic Model: Financial Derivatives and Risk Measures Ernst Eberlein & Karsten Prause Universitat Freiburg i. Br. Nr. 56

November 1998

Freiburg Center for Data Analysis and Modelling und Institut fur Mathematische Stochastik Universitat Freiburg Eckerstrae 1 D{79104 Freiburg, Germany

[email protected]

The Generalized Hyperbolic Model: Financial Derivatives and Risk Measures

Ernst Eberlein & Karsten Prause November 1998 Abstract

Statistical analysis of data from the nancial markets shows that generalized hyperbolic (GH) distributions allow a more realistic description of asset returns than the classical normal distribution. GH distributions contain as sub-classes hyperbolic as well as normal inverse Gaussian (NIG) distributions which have recently been proposed as basic ingredients for the modelling of price processes. We derive an option pricing formula for GH driven models using the Esscher transform as one possibility to determine prices in an incomplete market. The GH option pricing model is a generalization of the hyperbolic model developed by Eberlein and Keller (1995), Keller (1997). We compare this model with the classical Black-Scholes (1973) model. We also propose a general recipe for the comparison of models for the pricing of derivatives with the market reality. The objectives of this study are to examine the consistency of our model assumptions with the empirically observed price processes and the performance of option pricing models from a practitioner's point of view. Due to the diculty, if not impossibility, of constructing a test in a strict statistical sense for pricing models, we compare the models focussing on criteria which are relevant for practical purposes. Finally, we present a simpli ed approach to the estimation of high-dimensional generalized hyperbolic distributions. We also consider models with stochastic volatility and investigate the application of GH distributions to measure risk in nancial markets.

1 Introduction Generalized hyperbolic (GH) distributions were introduced by Ole E. Barndor-Nielsen (1977) in the context of the sand project as a variance-mean mixture of normal and generalized inverse Gaussian (GIG) distributions. These distributions seem to be tailor-made to describe the statistical behaviour of asset returns. Analyzing nancial data such as stock prices, indices, FX-rates or interest rates, one gets empirical distributions with a rather typical shape. They place substantial probability mass near the origin, have slim anks and a number of observations far out in the tails. The normal distribution on which the classical models in nance are based, fails in all three aspects. How far this deviation from normality goes, depends on the time scale of the underlying data sets. For long term studies based on weekly or even monthly data points the empirical distributions are close to the normal. However the use of scarce data sets means ignoring a large amount of information. Daily data is the minimum one has to consider for most purposes. Analyzing intraday data, i.e. looking at price movements on a microscopic scale, leads to a deeper understanding of the relevant processes.

De nition 1. For x 2 R the generalized hyperbolic distribution is de ned as

?




gh(x; ; ; ; ; ) = a(; ; ;  )  2 + (x ? )2 (?1=2)=2 ?p
?
 K?1=2  2 + (x ? )2 exp (x ? ) ; where

(2 ? 2 )=p2 ?
2 ?1=2   K  2 ? 2

a(; ; ; ) = p

and K is a modi ed Bessel function of the third kind with index . The domain of variation of the parameters is 0  j j < , ;  2 R and  > 0.

Alternative parametrizations used in the literature are

p

 =  2 ? 2; % = =;  = (1 +  )?1=2;  = %;  = ; = :

2nd parametrization: 3rd parametrization: 4th parametrization:

The properties of the Bessel function K (Abramowitz and Stegun 1968) allow to nd simpler expressions for the Lebesgue density if  2 1=2 Z. For  = 1 we get the hyperbolic distribution which is characterized by the fact that the log-density is a hyperbola. This sub-class has the simplest representation of all GH laws, which is favourable from a numerical point of view. For  = ?1=2 we get the normal inverse Gaussian (NIG) distribution; this sub-class is closed under convolution. See Eberlein and Keller (1995), Eberlein, Keller, and Prause (1998), Barndor-Nielsen and Prause (1999) for statistical results concerning the sub-classes of hyperbolic respectively NIG distributions.

2 Estimation of Densities We begin by estimating the generalized hyperbolic, the hyperbolic and the normal inverse Gaussian distributions from daily as well as from high-frequency data. The algorithm and the results concerning German stock prices and NYSE indices are described in detail in Prause (1997, 1999). Analogous results are obtained for the DAX, the German stock index (see Figure 1). We de ne the return of a price process (St )t0 for a time interval
t, e.g. one trading day, as

Xt = log St ? log St?
t:

(1)

Thus the return during n periods is the sum of the one period returns. The numerical estimates for the GH distribution and the sub-classes are given in Table 1. Estimates of the parameter  for daily returns are typically smaller than ?1:0. Looking at the values of the log-likelihood function the proximity of the estimates for generalized hyperbolic, hyperbolic and NIG distributions is obvious. Figure 1 (top) provides a typical plot of empirical and estimated GH densities. The dashed lines represent distributions from the generalized hyperbolic class, the dotted line the normal and the uninterupted line the empirical distribution smoothed by a Gaussian kernel. The plot of the densities shows that GH, hyperbolic and NIG distributions are more peaked and have more mass in the tails than 3

Table 1: Generalized hyperbolic parameter estimates for the daily returns of the DAX from December 15, 1993 to November 26, 1997. The parameter  is xed for the estimation of the hyperbolic and the NIG distribution.











GH ?2:018 46:82 ?24:91 0:0163 0:00336 Hyperbolic 1 158:87 ?29:02 0:0059 0:00374 NIG ?0:5 105:96 ?26:15 0:0112 0:00348

Log-Likelihood 3138.28 3135.15 3137.33

normal distributions. Consequently they are much closer to the empirical distribution of asset returns. Although the dierence between GH, hyperbolic and the NIG distribution is small, it is clear that the generalized hyperbolic distributions are superior to those of the sub-classes. NIG distributions are often a bit closer to the observed returns than hyperbolic distributions. Value-at-Risk (VaR) has become a major tool in the modelling of risk inherent in nancial markets. VaR is de ned as the potential loss given a level of probability  2 (0; 1) P [Xt < ?VaR] = : (2) The plot of VaR as a function of  could also be used to visualize the tail behaviour of distributions. Note, that the concept of VaR applied for only a single  is not satisfactory: VaR does not identify extreme risks appearing with a probability smaller than . Figure 1 (bottom) shows that the tails of the generalized hyperbolic distributions are heavier than the tails of the normal distribution and therefore VaR computed parametrically for the GH distribution and its sub-classes is closer to the empirically observed Value-at-Risk. In the global foreign exchange market (FX market) it is particularly important to look at price movements on an intraday basis. Many traders close their positions over night and try to make a pro t from intraday trading only. Therefore we examine 6 hours returns of USD/DEM exchange rates from the HFDF96 dataset provided by Olsen & Associates. See also J.P. Morgan and Reuters (1996, p. 65) for remarks concerning the leptokurtosis of daily USD/DEM returns. For high-frequency data we follow Guillaume, Dacorogna, Dave, Muller, Olsen, and Pictet (1997) in the de nition of the log-price





p(ti) = log pask (ti) + log pbid(ti) 2

(3)

and the corresponding return

r(ti) = p(ti) ? p(ti ?
t): (4) We estimate the GH parameters for the increments r(ti ) after removing all zero-returns. Although this is only a provisional approach to focus on time periods where trading takes place, the results as plotted in Figure 2 provide a clear picture: The excellent t of generalized hyperbolic distributions and the typical dierence to the normal distribution observed for daily returns repeats for high-frequency data (see also Barndor-Nielsen and Prause 1999). 4

50

Densities empirical 40

normal hyperbolic

density 30

NIG

0

10

20

GH

-0.04

-0.02

0.0

0.02

0.04

Value at Risk 0.08

empirical normal hyperbolic Value at Risk 0.04 0.06

NIG

0.0

0.02

GH

0.0

0.01

0.02 0.03 level of probability

0.04

0.05

Figure 1: DAX from December 15, 1993 to November 26, 1997, daily prices at 12:00h, IBIS data (Karlsruher Kapitalmarktdatenbank). 5

300

Densities

250

empirical normal NIG GH

0

50

100

density 150 200

hyperbolic

-0.006

-0.004

-0.002

0.0

0.002

0.004

0.006

Figure 2: USD/DEM exchange rate from January 1 to December 31, 1996, 6 hours returns, HFDF96 dataset (Olsen & Associates, Zurich).

3 The Generalized Hyperbolic Model We follow Eberlein and Keller (1995), Keller (1997) in the design of a price process (St)t0 and the derivation of an option pricing formula. The rst step is to construct the driving process. Generalized hyperbolic distributions are in nitely divisible (Barndor-Nielsen and Halgreen 1977). Therefore, they generate a Levy process (Xt)t0 , i.e. a process with stationary and independent increments, such that the distribution of X1 is generalized hyperbolic. We call this process (Xt)t0 the generalized hyperbolic Levy motion. The process depends on the ve parameters (; ; ; ; ) and after excluding the drift term it is purely discontinuous. This property follows from the form of the Levy-Khintchine representation of the characteristic function which is given in the appendix. The price process itself is de ned by

St = S0 exp(Xt):

(5)

This process as well does not have a continuous part. In order to give the reader an idea of how the paths of such a process look like, we show in Figure 3 a sample of the intraday price behaviour of stocks. To model the microstructure of asset prices, purely discontinuous processes are more appropriate than the classical diusion type processes with continuous paths. Since we are in an incomplete setting, it is necessary to select a particular equivalent martingale measure. Arbitrage free prices are obtained as expectations under these measures 6

420

440

Share value 460 480

500

520

Intraday asset prices

10

12 14 time scale

16

Figure 3: MAN (Stammaktien), IBIS data from October 28, 1997. (Delbaen and Schachermayer 1994). Note, that it is possible to obtain every price in the full no-arbitrage interval by a particular equivalent martingale measure (Eberlein and Jacod 1997). We choose the Esscher equivalent martingale measure P # given by

?




dP # = exp #Xt ? t log M (#) dP:

(6)

The parameter # is the solution of r = log M (# + 1) ? log M (#) where M is the moment generating function given in Appendix A and r is the constant interest rate. The latter equation ensures that the discounted price process is in fact a P # -martingale. Chan (1999) remarked that in a model very similar to the exponential Levy model in (5) the Esscher transform is the minimal martingale measure in the sense of Follmer and Schweizer (1991), which is a motivation for the choice of the martingale measure obtained by (6) for pricing purposes. Following the arbitrage pricing theory, the price of an option with time to expiration T and payo function H (ST ) is given by e?rT E# [H (ST )]. In particular, for a call option with strike K whose payo is H (ST ) = (ST ? K )+ we obtain for the expectation under P #

S0

Z1

ghT (x; # + 1) dx ? e?rT K

Z1

ghT (x; #) dx;

(7)

where = ln(K=S0) and ght(
; #) is the density of the distribution of Xt under the riskneutral measure. The density ght (
) of the t-fold convolution of the generalized hyperbolic 7

distribution can be computed by applying the Fourier inversion formula to the characteristic function. In the case of NIG distributions one should use the property that this sub-class is closed under convolution. Figure 4 shows that the dierence of generalized hyperbolic option prices to those from the Black-Scholes model resembles the W-shape which was observed for hyperbolic option prices. Note that for options with short maturities the W-shape is more pronounced in the case of the NIG and the GH model.

4 Rescaling of Generalized Hyperbolic Distributions For the computation of implicit volatilities in the GH model we need to rescale the generalized hyperbolic distribution while keeping the shape xed. An analogous problem occurs when computing GH option prices for a given volatility, e.g. a volatility estimated from historical stock returns. In this section we also give some insights into the structure of GH distributions. The rescaling of generalized hyperbolic distributions is based on the following property concerning scale- and location-invariance. Lemma 2. The terms ,  and  are scale- and location-invariant parameters of the univariate generalized hyperbolic distribution. The same holds for the alternative parametrizations (  , %) and (  , ). Proof. According to Blsild (1981) a linear transformation Y = aX + b of a GH distributed variable X is again GH-distributed with parameters + = , + = =jaj, + = =jaj, + = jaj and + = a + b. Obviously + + =  and + + = . A consequence of Lemma 2 is that the variance of the generalized hyperbolic distribution has the linear structure Var[X1] =  2 C in  2 where C depends only on the shape, i.e. the scale- and location-invariant parameters (Barndor-Nielsen and Blsild 1981). Therefore one can also use  as a scaling parameter. To rescale the distribution for a given variance b 2 one obtains the new e as

2 0 !2 13?1=2 2 b b b b e = b 4 Kb+1(b ) + 2 b2 @ K+2b( ) ? K+1b( ) A5 (8) K( )  K( ) b ? K( ) where (b; b; b ) and consequently b are estimated from a longer time series. To x the shape of the distribution while rescaling with a new e, one has to change the other parameters in the following way

e = b; e = b b ; e = b b and e = b: e e

(9)

Note, that the term in the square brackets is scale- and location-invariant. In order to value German stock options we use shape parameters estimated from stock prices from January 1, 1988 to May 24, 1994 (see Prause (1997) for the parameter estimates) and we rescale the estimated generalized hyperbolic distributions while feeding in volatility estimates from shorter time periods. Figure 5 shows the densities and the corresponding log-densities of hyperbolic distributions. In the rst row we x the shape estimated from Bayer stock prices and rescale the 8

-0.2

0

C.bs-C.gh 0.4 0.2

0.6

BS minus Generalized Hyperbolic Prices

80

60 4 et om 0 atu ri

tim

ty

20

0.8

0.9

1

1.1

1.2

-strike ratio

stockprice

Bayer: -1.79 / 21.34 / 2.67 / 0.01525 / -0

-0.2

0

C.bs-C.hyp 0.4 0.2

0.6

BS minus Hyperbolic Prices

80

60 4 et om 0 20 atu rity

tim

1

0.8

1.1

1.2

0.9 -strike ratio stockprice

Bayer: 1 / 139.01 / 5.35 / 0.00438 / -0.00031

-0.2

0

C.bs-C.nig 0.4 0.2

0.6

BS minus NIG Prices

80

60 4 et om 0 atu ri

tim

ty

20

0.8

1.1 1 0.9 tio ra e -strik stockprice

1.2

Bayer: -0.5 / 81.65 / 3.69 / 0.01034 / -0.00012

Figure 4: Black-Scholes minus GH prices (Bayer parameters, strike K=1000). 9

4

Hyperbolic log-density with const. shape

2

40

50

Hyperbolic density with const. shape

log-density 0

estimated c = 1.5 c = 2.0 c = 2.5

0

-4

10

-2

density 20 30

estimated c = 1.5 c = 2.0 c = 2.5

-0.10

-0.05

0.0

0.05

0.10

-0.06

0.0

0.02

0.04

0.06

4 2

40

log-density 0

zeta = 0.01 zeta = 1 zeta = 10 normal

-4

10

-2

density 20 30

zeta = 0.01 zeta = 1 zeta = 10 normal

0 -0.06

-0.02

Hyperbolic log-density with const. variance

50

Hyperbolic density with const. variance

-0.04

-0.04

-0.02

0.0

0.02

0.04

0.06

-0.05

0.0

0.05

Figure 5: Rescaled hyperbolic densities ( 0 = c b with constant shape parameters b = 0:608; %b = 0:0385 estimated from Bayer stock prices) and hyperbolic densities with constant variance and dierent shapes. distribution as described in (9). The second row of Figure 5 reveals that  describes the kurtosis of the distribution. If  is increased, the density becomes less peaked and converges to the Gaussian distribution. Log-densities give some insight into the tail behaviour of the density. The log-density of the hyperbolic distribution is a hyperbola whereas the normal logdensity is a parabola. Therefore hyperbolic distributions possess (substantially) heavier tails than the normal distribution. Nevertheless, in contrast to stable distributions the moments of GH distributions do exist. Hence, semi-heavy is an appropriate description of the tails of generalized hyperbolic distributions.

5 Smile Eect and Pricing Performance The comparison of generalized hyperbolic prices with Black-Scholes prices in Section 3 hints at the possibility to correct the well-known smiles which appear in Black-Scholes implicit volatilities. Implicit volatilies are computed from observed option prices by inverting the corresponding price formula with respect to the volatility parameter. Usually all parameters 10

BS minus Hyperbolic implicit volatility

1.2

50 to m 100 150 aturit y

0 -0.05

0.8

0.9 1 1 stockpri ce-strike .1 ratio

1.2

mean= 0.0002 / sd= 0.00553

BS minus NIG implicit volatility

BS minus GH implicit volatility

50 to m 100 150 aturit y

0.8

mean= 0.0002 / sd= 0.00749

0.9 1 1.1 stockpri ce-strike ratio

1.2

time

1.2

-0.05

50 to m 100 150 aturit y

0.9 1 1 stockpri ce-strike .1 ratio

time

-0.05

0.8

0

0

0.05

0.05

0.1

0.1

mean= 0.214 / sd= 0.06077

time

0.9 stockpri 1 ce-strik 1.1 e ratio

time

0.8

50 to ma 100 150 turity

0.05

0.8 0.4 0.6 0 0.2

1

0.1

Black-Scholes implicit volatility

mean= -0.0253 / sd= 0.02336

Figure 6: Black-Scholes implicit volatilities and comparison of the implicit volatilities of Black-Scholes, hyperbolic, NIG and GH prices (Daimler Benz calls from July 1992 to August 1994, 62504 observations). necessary for option pricing are known to traders except the volatility. In the GH model we rely on the rescaling mechanism described in Section 4 to obtain a single volatility parameter. In this section we compute the implicit volatilities and analyze the pricing performance of the GH model. The study is based on intraday option and stock market data of Bayer, Daimler Benz, Deutsche Bank, Siemens and Thyssen from July 1992 to August 1994. The option data set contains all trades reported by the Deutsche Terminborse (DTB) during the period above. The preparation of the data sets is described in detail in Eberlein, Keller, and Prause (1998, Chapter IV); the latter article also includes a discussion of implicit volatilities in the hyperbolic model and of the dierent approaches to reduce the smile. Implicit volatilities in the Black-Scholes model typically follow a pattern denoted as smile, i.e. they are low for options at the money and the highest implicit volatilities are observed for options with short maturities in and out of the money. Figure 6 (top left) illustrates the implicit volatilities of Daimler Benz calls in the Black-Scholes model. To compare these with implicit volatilities in the GH model, we built the dierences and plotted them in Figure 6. The pattern re ects the W-shapes from Figure 4. We observe a more pronounced correction of the smile eect in the GH model|due to the heavier tails of the distribution. 11

Table 2: Fitted coecients for call options from July 1992 to August 1994. SC marks the results for the symmetric centered versions of the models. Daimler Benz Black-Scholes Hyperbolic Hyperbolic SC NIG NIG SC GH GH SC

b0

b1

0:2177 0:2186 0:2184 0:2191 0:2189 0:2207 0:2201

?0:00029 ?0:0003 ?0:000293 ?0:000305 ?0:000296 ?0:000321 ?0:000306

b2

40:53 36:89 36:33 35:11 34:48 32:81 31:98

R2

0:5416 0:4972 0:4951 0:4746 0:4716 0:4378 0:4343

A dierent approach to analyse the smile behaviour of a particular option pricing model is to t a linear model for the implicit volatilities Imp;i = b0 + b1Ti + b2(%i ? 1)2=Ti + ei ; (10) where ei is the random error term, %i the stockprice-strike ratio S=K and i the number of the trade in the option data set. The cross-term (% ? 1)2=T re ects the degression of the smile eect with increasing time to maturity T . Table 2 shows the regression coecients for the Black-Scholes, the GH model and the respective symmetric centered versions. The values of the coecient for (% ? 1)2=T are smaller for hyperbolic, NIG and GH prices compared to those from Black-Scholes prices. Hence these new models reduce the smile eect. The largest correction is observed for the symmetric centered GH model. A criterion complementary to implicit volatilities in testing option pricing models is the computation of pricing errors. The same change in volatility has a greater eect on a price with a longer maturity. Hence the smile eect hints at an incorrect modelling of options with a short time to maturity whereas pricing errors are more meaningful for long-term derivatives. The average pricing error is dominated by errors from options with longer maturities; this can be seen in Figure 7. A better option pricing model has, in particular, to improve the pricing performance of options with longer maturities. Two dierent out-of-sample volatility estimators are used to examine the pricing performance: Hist30 is de ned as the standard deviation of daily asset returns 30 calendar days prior to a trading day. For Imp.median20 (resp. Imp.median30) we took the running median of the implicit volatilities of the 20 (resp. 30) options quoted prior to a trade. The construction of the estimator Imp.median allows on one hand for quick adaptation to new information in the secondary market, on the other hand the estimator is robust enough against outliers in observed implicit volatilities. These outliers are likely to occur because prices can change only in tick-size steps and small changes in the price have a large impact on implicit volatilities for options with short maturities. In Table 3 we show the mean (and the standard deviation) of the pricing errors for Bayer and Thyssen calls, i.e. 1

N

N  X i=1

Cmodel,i ? Cbi 12



(11)

1.2 ratio

1.4

time

0.8 stockpric1 e-strike

50 to ma 100 150 turity

.quoted C.BS - C 2.0 1.0 .0 0 .0 -1

Black-Scholes Pricing Errors

mean= 0.062 / sd= 0.7449

Figure 7: Pricing performance of the Black-Scholes model (Thyssen calls from July 1992 to August 1994, volatility estimator Imp.median20). where Cbi denotes the observed option price of trade i = 1; : : :; N . The hyperbolic, the NIG and the GH model lead to smaller errors for the historical volatility estimator.

6 When do we Observe Mispricing? In the last section we analysed mispricing patterns with respect to time-to-maturity and moneyness. For instance, we observed small pricing errors for at-the-money options with short maturities. A rst look at the average daily pricing errors as a time series in Figure 8 reveals that Black-Scholes prices computed with intraday implicit volatilites are more accurate than prices based on historical volatility estimates. The same observation applies for the pricing errors given in Table 3 above. In an ecient secondary market new information aecting the volatility of the underlying will cause immediate reactions in the prices of derivatives. Clearly, volatility estimators based on historical data will react more slowly. Nevertheless, Beckers (1981) observed that historical volatilities add information to implicit volatilites. Therefore, he concluded that at that time the secondary market was not informationally ecient. This has changed after the crash of 1987. See Christensen and Prabhala (1998) for an up-to-date study of implicit and realized volatility. To give a more explicit answer to the question posed in the title of this section, we look at the correlations of the average pricing errors AvErr (resp. the standard deviations SdErr) computed for each trading day with other observable parameters. All results are given in Table 4. In the text below we concentrate on those results which hold for both, Bayer and Thyssen options and both volatility estimators Imp.median20 and Hist30. First we look at the correlation of errors with the following three volatility estimates: the average daily implicit volatility (denoted by ImpVol), the historical volatility estimator 13

Table 3: Pricing performance: mean and standard deviation of the dierence of the model price minus quoted price (July 1, 1992 to August 19, 1994). estimator

Black-Scholes Hyperbolic NIG GH mean st. dev. mean st. dev. mean st. dev. mean st. dev.

Bayer

Hist30 0.798 2.389 Imp.median20 0.279 1.475 Imp.median30 0.276 1.399

0.78 2.379 0.28 1.462 0.276 1.393

0.773 2.379 0.281 1.457 0.281 1.457

0.752 2.373 0.283 1.449 0.283 1.449

Hist30 0.208 1.894 Imp.median20 0.062 0.745 Imp.median30 0.055 0.696

0.191 1.891 0.065 0.744 0.058 0.7

0.189 1.89 0.066 0.745 0.058 0.701

0.19 1.89 0.065 0.744 0.057 0.7

Thyssen

Mispricing Black/Scholes, Hist30

Mispricing Black/Scholes, Imp.median30 Average Error 90% Quantile

AvErr.BS.Imp.median30

Average Error 90% Quantile

5

AvErr.BS.Hist30

5

0

0

-5

-5

0

100

200 300 calendar days

400

0

100

200 300 calendar days

400

Figure 8: Pricing performance of Thyssen calls from June 1, 1993 to August 19, 1994. Hist30 and the volatility index VDAX of the Deutsche Terminborse. The index VDAX is constructed as the average of implicit volatilities of at-the-money options on the DAX with 45 days to expiration (Deutsche Borse 1997). It is obvious that the standard deviation of the pricing errors SdErr has a positive correlation with all three volatility series. Looking at the correlation of daily trading volume with SdErr reveals that at those days with increased trading it is also more dicult to compute prices which are close to the market. Moreover, the standard deviation of the pricing errors increases when the stockprice (Kassakurs) increases.

7 Alternative Testing Approaches Lo (1986) examined a procedure which allows for a test of an option pricing model in a classical statistical sense. He proposed to nd con dence intervals for the estimates of the parameters in the model, especially the volatility. The variance may be estimated by the 14

Table 4: Correlation of the average pricing errors AvErr (resp. standard deviations SdErr) of Black-Scholes prices computed for each trading day (with volatility estimator Imp.median20) and other parameters. The p-values are computed using Pearson's test with a 2-sided alternative. ImpVol is the average value (weighted by the trading volume) of the Black-Scholes implicit volatilities during one trading day and Kassakurs the stockprice at noon (12:00h). Bayer calls Correlation p-value

Hist30

AvErr SdErr AvErr SdErr AvErr SdErr AvErr SdErr AvErr SdErr

ImpVol ImpVol Hist30 Hist30 VDAX VDAX Kassakurs Kassakurs Volume Volume

Imp.median20

AvErr SdErr AvErr SdErr AvErr SdErr AvErr SdErr AvErr SdErr

ImpVol ImpVol Hist30 Hist30 VDAX VDAX Kassakurs Kassakurs Volume Volume

Thyssen calls Correlation p-value

?0:13

0:124 0:582 0:379 0:357 0:452 0:206 0:626 0:062 0:34

0:0025 0:0040 0 0 0 0 0 0 0:0846 0

?0:108

0:0149 0 0 0 0:0029 0 0:5664 0 0:6020 0

0:076 0:242 ?0:087 0:035 0:024 0:221 0:251 0:56 0:198 0:38

0:0785 0 0:0468 0:4219 0:5771 0 0 0 0 0

?0:2

0 0 0:7710 0:0011 0:8327 0 0:0479 0 0:8050 0

15

0:19 0:684 0:502 ?0:126 0:353 0:02 0:615 0:019 0:412

0:285 ?0:013 0:148 ?0:009 0:267 0:069 0:653 0:009 0:518

usual ML-estimator which is a consistent and uniformly asymptotically normal distributed estimator. Since option prices are a monotone function of volatility, this results in con dence bounds for the option price given by a speci c model. The test, built on large-sample theory, has the following de ciency: due to the frequent changes in the volatility of the underlying, it is not appropriate to use arbitrarily large data sets for the estimation. Another de ciency arises from the fact that option prices re ect the trader's expectation on the behaviour of stock prices during the period from the time point of the trade to expiration. This is not the period of time used to estimate historical parameters. The trader's forecasts may be biased in an unknown way, e.g. in expectation of a crash which will never occur. Hence a rigorous statistical test as described above is dicult if not impossible. Moreover, the dierences in the underlying and the risk-neutral density were used to test for crash fears in the secondary market (Bates 1991; Bates 1997). Therefore this asymptotic approach seems not to be adequate to test the performance of option pricing models. Bakshi, Cao, and Chen (1998) observed that in contrast to the monotonicity of call prices with respect to stock prices, which is assumed in all models with one source of randomness, stock prices and call prices often move in opposite directions. The same result holds for the German market. We zoomed into single option series and calculated the increments
Ci and
Si of stocks and call options from quote to quote. Overnight-increments were removed to avoid decreasing call prices due to a shorter time to maturity. For roughly 10% of the quoted increments the product
Ci
Si is negative (type-I violations in Bakshi, Cao, and Chen 1998). From Table 5 we can also see that the (absolute) magnitude of the call and stock price violating the monotonicity of the increments is clearly above tick-size. These violations hint at possible diculties for the practical implementation of hedging strategies and for tests of option pricing models based on dynamical hedging. Table 5: Do call prices and stock prices move in dierent directions? Percentage and magnitude of violations of the monotonicity in Daimler Benz call series. The last two columns refer only to those quotes violating the monotonicity condition. option series strike maturity quotes 550 650 700 800 800 800 900

1 7 12 9 12 10 12

93 93 92 92 92 93 94

641 336 254 138 142 511 133

violations # %

average for violations j
C j j
S j

64 28 26 14 16 70 19

0:4125 0:2393 0:4885 0:2071 0:425 0:2743 0:5579

9:98 8:33 10:23 10:14 11:26 13:69 14:28

1:0422 0:875 1:2308 1:1286 0:9875 1:1229 1:1474

There are several possible explanations for this eect. Bakshi, Cao, and Chen (1998) concluded from their empirical results that a stochastic volatility model may explain this eect, but they also mention that the stochastic volatility model of Hull and White (1987) explains only 60% of the observed violations of the monotonicity. But if these violations could be explained by stochastic volatility models the volatility of the average volatility during the lifetime of the call must be rather high. A more appropriate explanation of these violations is 16

the in uence of trading at the secondary market. Prices of derivatives are not only determined by the behaviour of the underlying but also by supply and demand (Kallsen 1998).

8 Multivariate Generalized Hyperbolic Distributions In the previous sections we have discussed univariate generalized hyperbolic distributions as a model for stock prices focussing on option pricing. We shall now investigate the estimation of multivariate GH distributions and the application to measure risk. De nition 3. The d-dimensional generalized hyperbolic distribution (GHd) is de ned for x 2 Rd by the Lebesgue density
? ?p
K?d=2   2 + (x ? )0
?1 (x ? ) ghd (x) = ad ? p exp 0 (x ? ) ;
d= 2 ?  ?1 2 + (x ? )0
?1(x ? ) ?p2 ? 0

 ad = ad (; ; ; ;
) = ?p
(2 )d=2 K  2 ? 0
These parameters have the following domain of variation1 :  2 R; ;  2 Rd ;  > 0; 0
< 2. The positive de nite matrix
2 Rdd has a determinant j
j = 1. For  = (d + 1)=2 we obtain the multivariate hyperbolic and for  = ?1=2 the multivariate normal inverse Gaussian distribution. Alternatively Blsild and Jensen (1981) introduced a second parametrization (; ; ; S ) where p 2  =   ? 0
,  =
1=2(2 ? 0
)?1=2 and S = 2
. Generalized hyperbolic distributions are closed under forming marginals, conditioning and ane transformations (Blsild 1981). For the mean and the variance of X  GHd one gets E X =  + R( )
1=2; (12)  ?
?
2 ? 1 1 = 2 0 1 = 2 Var X =   R ( )
+ S( ) 

; (13)





where R(x) = K+1 (x)= K(x) and S(x) = K+2 (x) K (x) ? K2+1 (x) K2(x) were introduced to simplify the notation. A maximum likelihood estimation of all parameters in higher dimensions is computationally too demanding since the number of parameters 3 + d(d + 5)=2 increases rapidly with the dimensions. Therefore we propose a simpli ed algorithm for symmetric GH distributions which allows for an ecient estimation also in higher dimensions. The rst step of the estimation follows a method of moments approach: we estimate the sample mean b 2 Rd and the sample dispersion matrix  using canonical estimators. Since  = 0 in the symmetric case, E X =  and from (13) we get the following estimate for


b = 2  :
 R ( )

(14)



b by norming the sample dispersion matrix such that j
j = 1. Consequently we compute
The second step is to compute yi = (xi ? b)0
b ?1(xi ? b) (15) We omitted the limiting distributions obtained at the boundary of the parameter space; see e.g. Blsild and Jensen (1981). Generalized hyperbolic distributions are symmetric i = (0; : : : ; 0) . 1

0

17

from observations xi 2 Rd ; 1  i  n. Then the log-likelihood function for the second step is given as

  L(x; ; ; ) = n  log(=) ? d2 log(2) ? log K() +

n X i=1

log K?(d=2)

n ?p ?p2 + y
+  ? d  X 
log  2 + y  : i

2

i=1

(16)

i

The last step is to maximize this log-likelihood function with respect to (; ;  ). We have implemented the estimation algorithm successfully for hyperbolic and NIG distributions, i.e. for xed  = 1 and  = ?1=2. In the case of an arbitrary  we encounter numerical problems due to extremely small values of the involved Bessel functions K . As in the univariate case the log-likelihood function simpli es for  2 1=2 Z. For NIG distributions, i.e.  = ?1=2, the number of Bessel functions K which needs to be computed for the log-likelihood function is reduced by one. In the case of hyperbolic and hyperboloid distributions we have to compute only one Bessel function instead of n + 1 in the general GH case. Since the evaluation of Bessel functions is the time-consuming part of the third step, computation is much simpler for hyperbolic distributions. For xed  it is also possible to estimate only  in the second step. Nevertheless, we have chosen to estimate the covariance structure in the rst \method of moments" step and the parameters (;  ) characterizing the kurtosis and the scale in the likelihood step. For a price process St 2 Rd we de ne percent returns xt 2 Rd by





x(ti) = St(i) ? St(?i)
t St(?i)
t  log St(i) ? log St(?i)
t; 1  i  d;

(17)

which are approximated by the log-returns de ned in (1). The motivation to choose this de nition is that the return of a portfolio described by a vector h 2 Rd is then simply given by h0 xt . See J.P. Morgan and Reuters (1996, Section 4.1) for a discussion of temporal and cross-section aggregation of asset returns. The marginal densities of the GH distributions can be derived using a theorem of Blsild (1981). Typically we obtain the pattern shown in Figure 9 for the densities and log-densities: The marginal distributions of hyperbolic and NIG distributions are closer to the empirical distribution than normal distributions. In the center marginals of hyperbolic distributions are closer to the empirical distribution but in the tails marginals of NIG distributions provide a better t.

9 Market Risk Measurement Let us start with a general result on densities.

Theorem 4. Let X be a d-dimensional random variable with symmetric generalized hyperbolic distribution, i.e. with = (0; : : :; 0)0, and let h 2 Rd where h = 6 (0; : : :; 0)0. The dis-

tribution of h0 X is univariate generalized hyperbolic GHd ( ;  ;  ;  ;  ), where  = ;  = jh0
hj?1=2;  = 0;  = jh0
hj1=2 and  = h0.

18

Thyssen logdensity and marginals (DAI-DBK-THY)

Thyssen density and marginals (DAI-DBK-THY)

10

empirical normal sym. hyperbolic sym. NIG

0

0.001

0.01

10

density 20

log-density 0.1 1

30

empirical normal sym. hyperbolic sym. NIG

-0.06

-0.04

-0.02

0.0 x

0.02

0.04

0.06

-0.10

-0.05

0.0 x

0.05

0.10

Figure 9: Marginals density for Thyssen obtained from the 3-dimensional estimate from Daimler Benz { Deutsche Bank { Thyssen. Proof. Let h1 6= 0 without loss of generality. Apply Theorem Ic) of Blsild (1981) with

0h1 h2


hd1 01 0 B C B .C 0 1 0 A=B ... C @ ... A and B = B @ .. C A: 0

0

1

0

(18)

Then project the d-dimensional GH distribution onto the rst coordinate using Theorem Ia). The latter theorem may be used to calculate risk measures for a portfolio of d assets with investments given by a vector h 2 Rd . As an example we look at a portfolio consisting of three German stocks: Daimler Benz, Deutsche Bank and Thyssen from January 1, 1988 to May 24, 1994. We choose h = (1; 1; 1)0 and show the empirical density of the returns h0 xt of the portfolio in Figure 10. Theorem 4 gives the corresponding densities obtained from the d-dimensional estimates of symmetric hyperbolic and symmetric NIG distributions. Figure 10 shows also the direct estimate from h0xt for the univariate GH distribution. The densities and log-densities in Figure 10 indicate that symmetric GH distributions lead to a rather precise modelling of the return distribution of the portfolio. As a consequence one can get more realistic risk measures than the traditional ones based on the normal distribution. Figure 11 shows the Value-at-Risk over a 1-day horizon with respect to a level of probability  2 (0; 1) and the shortfall which we de ne as





Shortfall;t = ? E h0 xt h0 xt < q () ;

(19)

where q : [0; 1] ! R is the corresponding quantile function. Note: the shortfall goes clearly beyond the concept of VaR because it takes into account the extreme negative returns. The log-density of the empirical distribution in Figure 10 shows 19

15

Density: 1 DAI + 1 DBK + 1 THY empirical normal sym. hyperbolic 10

sym. NIG

0

5

density

1-dim GH

-0.15

-0.10

-0.05

0.0

0.05

0.10

0.15

10

Log-density: 1 DAI + 1 DBK + 1 THY empirical normal 1

sym. hyperbolic sym. NIG

0.001

0.01

log-density 0.1

1-dim GH

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

Figure 10: Distribution of the returns of a portfolio consisting of Daimler Benz, Deutsche Bank and Thyssen. 20

0.20

VaR: 1 DAI + 1 DBK + 1 THY empirical normal 0.15

sym. hyperbolic sym. NIG

0.0

0.05

VaR 0.10

1-dim GH

0.0

0.05

0.10 level of probability

0.15

0.20

0.20

Shortfall: 1 DAI + 1 DBK + 1 THY empirical normal 0.15

sym. hyperbolic sym. NIG

0.0

0.05

Shortfall 0.10

1-dim GH

0.0

0.05

0.10 level of probability

0.15

0.20

Figure 11: Risk measures for Daimler Benz{Deutsche Bank{Thyssen. 21

the magnitude of the negative returns of multi-asset portfolios in relation to the more frequent small returns. The Basel Committee on Banking Supervision (1995, IV.23) has proposed a backtesting procedure to investigate the quality of Value-at-Risk estimators. We follow this procedure in the comparison of standard VaR estimation approaches and VaR estimators based on GH distributions.2 After computing the VaR for each day in the time period from January 1, 1989 to May 24, 1994 we count the losses greater than the Value-at-Risk. Since Value-atRisk is essentially a quantile, the percentage of excess losses should correspond to the level of probability . One standard method to compute VaR is to simulate the return of the portfolio by the preceding 250 observed returns and to use the quantile of this distribution. The Basle Committee calls this approach historical simulation. A second simulation technique proposed to forecast VaR is by Monte Carlo methods. It is computationally intensive for large portfolios. Therefore, it is less widely used to measure the exposure to (linear) market risk (Basel Committee on Banking Supervision 1995, I.7). We have not applied this method because distinct dierences to the Variance-Covariance approach are only obtained for nonlinear portfolios (Buhler, Korn, and Schmidt 1998). However, a full valuation approach based on GH distributions, for instance for portfolios with derivative contracts, might easily be implemented. The mixture representation of GH distributions allows to generate random numbers eciently. We also apply the Variance-Covariance approach which is based on the multivariate normal distribution. The results given in Table 6 show that the standard estimators for Value-at-Risk underestimate the risk of extreme losses. This eect is visible in the percentage of excess losses in the historical simulation. In the Variance-Covariance approach we observe too high values for the level of probability  = 1% and too small values for  = 5%. This corresponds to Figure 11 (top). The percentages of realized losses greater than VaR are closer to the level of probability in both cases  = 1% and  = 5% for the symmetric hyperbolic and the symmetric NIG distribution than in the standard approaches. An approach similar to the rescaling mechanism proposed above for the univariate case is to estimate the shape from a longer time period and to use an up-to-date covariance matrix . This allows us to incorporate the risk of extreme events, even if they do not occur in the preceding 250 trading days, which is the minimum time period proposed by the Basel Committee on Banking Supervision (1995). Thus it is necessary to choose a sub-class, i.e. a parameter  2 R, and to x a long-term estimate for  . We compute the matrix S in the second parametrization using

S = 2
= R ( ) : 

(20)

A further re nement is possible by choosing an appropriate estimate for the covariance matrix. We select the multivariate IGARCH model of Nelson (1990) in which variance 12;t The Basel Committee on Banking Supervision (1995, IV.3) has chosen a holding period of 10 days to put more weight on options. Nevertheless, we use a 1-day horizon because our portfolio contains no options and the increased number of returns in the observation period allows more accurate statistical results. 2

22

2 are given by and covariance 12 ;t

12;t = (1 ? ) 122 ;t = (1 ? )

X t1 X t1

t?1(rt ? r);

(21)

t?1(r1;t ? r1)(r2;t ? r2);

(22)

where 0 <  < 1 is a decay factor, rt ; r1;t; r2;t returns of nancial assets and r; r1; r2 the corresponding mean values. To allow for a comparison, we have used the decay factor  = 0:94 applied in J.P. Morgan and Reuters (1996) for daily returns. Finally we propose to reduce the risk measurement problem to one dimension by computing quantiles for the return h0 xt of the whole portfolio. We estimate hyperbolic and NIG distributions and calculate the corresponding quantiles. Table 6: Ex post evaluation of risk measures: percentage of losses greater than VaR. Each trading day the Value-at-Risk for a holding period of one day is estimated from the preceding 250 trading days (Daimler Benz, Deutsche Bank and Thyssen from January 1, 1989 to May 24, 1994, Investment of 1DM in each asset). VaR Estimation Method

 = 1%  = 5%

Historical Simulation Variance-Covariance RiskMetrics / IGARCH

2:08 1:63 1:34

5:79 4:45 4:75

Symmetric hyperbolic Symmetric NIG Symmetric hyperbolic, long-term  Symmetric NIG, long-term  Hyperbolic IGARCH, long-term  NIG IGARCH, long-term  1-dimensional Hyperbolic 1-dimensional NIG

1:48 1:26 1:26 1:04 1:11 1:11 1:41 1:41

4:9 4:75 4:45 4:9 4:82 5:34 4:9 4:97

In the backtesting experiment all described standard and GH-based approaches are used to calculate Value-at-Risk estimates. The results of the study for a linear portfolio are shown in Table 6. Taken together, the use of a long-term shape parameter incorporates the possibility of extreme events, even if there was no crash in the preceding 250 trading days, whereas the GH-IGARCH approach describes the volatility clustering observed in nancial markets. This yields more accurate results of GH-based models in the ex-post evaluation of the risk measures.

10 Conclusion In the rst part of this paper we presented the generalized hyperbolic distributions respectively their sub-classes and estimation results concerning daily as well as high-frequency returns. 23

The greater exibility of this distribution allows an almost perfect t to asset return distributions. Consequently we introduce the generalized hyperbolic Levy motion and derive an option pricing formula following the Esscher approach proposed by Eberlein and Keller (1995). Using the rescaling mechanism of generalized hyperbolic distributions we analyze the implicit volatilities and the prices obtained in the GH model. We observe a distinct correction of the smile eect in the GH model. After that we give some insights into typical patterns of mispricing. The rst part ends with some general comments on the testing of option pricing models. Risk measures are used in banks and investment rms with two objectives. Internally they provide a possibility for the management to allocate risk: Setting limits in terms of risk helps business managers to allocate risk to those areas which they feel oer the most potential, or in which their rms' expertise is greatest. This motivates managers of multiple risk activities to favor risk reducing diversi cation strategies.3 On the other hand banking regulators as well as the management want to reduce the probability of bankruptcy. Therefore, they set limits to the exposure to market risk relative to the capital of the rm. Is Value-at-Risk the adequate measure to meet these two requirements? Quantile-based methods like Value-at-Risk have the disadvantage that they do not consider losses occuring with a probability below a given level of probability. Stress testing oers a partial solution of this problem focussing on extreme scenarios. To avoid bankruptcy one has to forecast the distribution of the maximum expected loss. Therefore regulators should use other risk measures than VaR. However, for the internal allocation of risk in a rm VaR is more appropriate. Nevertheless, a better modelling of the extreme events especially for nonlinear portfolios is desirable. We also would like to mention the axiomatic concept of coherent risk measures developed by Artzner, Delbaen, Eber, and Heath (1999). In the last two sections we have shown that it is possible to estimate symmetric generalized hyperbolic distributions in an ecient way and to construct more accurate risk measures for multivariate price processes. Symmetric hyperbolic and symmetric NIG distributions are characterized by the covariance matrix and a shape parameter. This simple structure allows a further sophistication of GH risk measures by xing a long-term shape parameter, which describes the probability of rare events, and choosing the favourite estimate of the covariance matrix. A statistical study in accordance with the backtesting concept required by the Basel Committee on Banking Supervision recon rms the excellent results concerning Value-at-Risk estimation for multivariate price processes. Moreover, we have shown that generalized hyperbolic distributions might also be used as a building block for risk measures beyond VaR.

Acknowledgement We thank Deutsche Borse AG, Frankfurt, for a number of data sets concerning stock and option prices. We also used IBIS data from the Karlsruher Kapitalmarktdatenbank and the high-frequency data set HFDF96 provided by Olsen & Associates, Zurich. 3

J.P. Morgan and Reuters (1996, p. 33), see also Chart 3.1.

24

Appendix A Moment Generating and Characteristic Function

Lemma A.1. The moment generating function of the generalized hyperbolic distribution is  2 ? 2 =2 K(p2 ? ( + u)2 ) u p ; j + uj < : M (u) = e 2 ? ( + u)2 K( 2 ? 2 ) Lemma A.2. The characteristic function of the generalized hyperbolic distribution is  2 ? 2 =2 K (p2 ? ( + iu)2 )  iu p : '(u) = e 2 ? ( + iu)2 K( 2 ? 2 ) Theorem A.3. The Levy-Khintchine representation of '(u) is ln '(u) = iu +

Z?
eiux ? 1 ? iux g(x)dx;

with Levy density

! ? ? p2y + 2 jxj
x Z 1 exp e ?  j x j ? p g(x) = jxj ; p
dy + 1f0g e 0  2y Jj2j ( 2y ) + Yj2 j ( 2y ) see Wiesendorfer Zahn (1999) for the proof in the case of  < 0. Here J and Y denote Bessel functions of the rst and second kind.

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