Maria Adler

1
Hyperbolic Processes in Finance
Alternative Models for Asset Prices
2
Outline

• The Black-Scholes Model
• Fit of the BS Model to Empirical Data
• Hyperbolic Distribution
• Hyperbolic Lévy Motion
• Hyperbolic Model of the Financial Market
• Equivalent Martingale Measure
• Option Pricing in the Hyperbolic Model
• Fit of the Hyperbolic Model to Empirical Data
• Conclusion

3
The Black-Scholes Model
price process of a security described by the SDE
volatility
drift
Brownian motion
price process of a risk-free bond
interest rate
4
The Black-Scholes Model

• Brownian motion has continuous paths
• stationary and independent increments
• market in this model is complete
• ? allows duplication of the cash flow of
• derivative securities and pricing by
• arbitrage principle

5
Fit of the BS Model to Empirical Data

• statistical analysis of daily stock-price data
  from 10 of the DAX30 companies
• time period 2 Oct 1989 30 Sep 1992 (3 years)
• ? 745 data points each for the returns
• Result assumption of Normal distribution
  underlying the
• Black- Scholes model does not
  provide a good fit to
• the market data

6
Fit of the BS Model to Empirical Data
Quantile-Quantile plots density-plots for the
returns of BASF and Deutsche Bank to test
goodness of fit Fig. 4, E./K., p.7
7
Fit of the BS Model to Empirical Data
BASF
Deutsche Bank
8
Fit of the BS Model to Empirical Data

• Brownian motion represents the net random effect
  of the various factors of influence in the
  economic environment
• (shocks price-sensitive information)
• actually, one would expect this effect to be
  discontinuous, as the individual shocks arrive
• indeed, price processes are discontinuous looked
  at closely enough (discrete shocks)

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Fit of the BS Model to Empirical Data

• Fig. 1, E./K., p.4
• typical path of a Brownian motion
• continuous
• the qualitative picture does not change if we
  change the time-scale,
• due to self-similarity property

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Fit of the BS Model to Empirical Data

• real stock-price paths change significantly if we
  look at them on different time-scales
• Fig. 2, E./K., p.5 daily stock-prices of
  five major companies over a period of three years

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Fit of the BS Model to Empirical Data

• Fig. 3, E./K., p.6
• path, showing price changes of the Siemens stock
  during a single day

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Fit of the BS Model to Empirical Data

• Aim to model financial data more precisely than
  with the BS model
• ? find a more flexible distribution than the
  normal distr.
• ? find a process with stationary and
  independent increments (similar
• to the Brownian motion), but with a more
  general distr.
• this leads to models based on Lévy processes
• ? in particular Hyperbolic processes
• B./K. and E./K. showed that the Hyperbolic model
  is a more realistic market model than the
  Black-Scholes model, providing a better fit to
  stock prices than the normal distribution,
  especially when looking at time periods of a
  single day

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Hyperbolic Distribution

• introduced by Barndorff-Nielsen in 1977
• used in various scientific fields
• - modeling the distribution of the grain size
  of sand
• - modeling of turbulence
• - use in statistical physics
• Eberlein and Keller introduced hyperbolic
  distribution functions into finance

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Hyperbolic Distribution

• Density of the Hyperbolic distribution

modified Bessel function with index 1
characterized by four parameters
tail decay behavior of density for
shape
skewness / asymmetry
location
scale
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Hyperbolic Distribution

• Density-plots for different parameters

• 9 0
• 0 0 0
• 1 1
• 0 0 0

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Hyperbolic Distribution

• 4
• 0
• 1
• 3 2

17
Hyperbolic Distribution

• 1
• 0 0
• 3
• 0 0

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Hyperbolic Distribution

• the log-density is a hyperbola (? reason for the
  name)
• this leads to thicker tails than for the normal
  distribution, where the log-density is a parabola

slopes of the asymptotics
location
curvature near the mode
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Hyperbolic Distribution

• Plots of the log-density for different parameters

• 6
• 1
• 6
• 0 0

20
Hyperbolic Distribution

• 2
• 1
• 10
• 0 0

21
Hyperbolic Distribution

• 4 4
• 2 2
• 1 8
• 1 1

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Hyperbolic Distribution

• setting
• another parameterization of the density can
  be obtained

• and invariant under changes of
  location and scale

? shape triangle
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Hyperbolic Distribution
generalized inverse Gaussian
generalized inverse Gaussian
Fig. 6, E./K., p.13
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Hyperbolic Distribution

• Relation to other distributions

Normal distribution
generalized inverse Gaussian distribution
Exponential distribution
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Hyperbolic Distribution

• Representation as a mean-variance mixture of
  normals
• Barndorff-Nielsen and Halgreen (1977)
• the mixing distribution is the generalized
  inverse Gaussian with density

• consider a normal distribution with mean
  and variance

• such that is a random variable with
  distribution

• the resulting mixture is a hyperbolic
  distribution

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Hyperbolic Distribution

• Infinite divisibility
• Definition
• Suppose is the characteristic
  function of a distribution.
• If for every positive integer ,
  is also the power of a char.
• fct., we say that the distribution is infinitely
  divisible.

The property of inf. div. is important to be able
to define a stochastic process with
independent and stationary (identically
distr.) increments.
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Hyperbolic Distribution

• Barndorff-Nielsen and Halgreen showed that the
  generalized inverse Gaussian distribution is
  infinitely divisible.
• Since we obtain the hyperbolic distribution as a
  mean-variance mixture from the gen. inv. Gaussian
  distr. as a mixing distribution, this transfers
  infinite divisibility to the hyperbolic
  distribution.
• ? The hyperbolic distribution is infinitely
  divisible and we can define the hyperbolic Lévy
  process with the required properties.

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Hyperbolic Distribution

• To fit empirical data it suffices to concentrate
  on the centered
• symmetric case.
• Hence, consider the hyperbolic density

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Hyperbolic Distribution

• Characteristic function
• The corresponding char. fct. to
  is given by

• All moments of the hyperbolic distribution exist.

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Hyperbolic Lévy Motion

• Definition
• Define the hyperbolic Lévy process corresponding
  to the inf. div.
• hyperbolic distr. with density

stoch. process on a prob. space
starts at 0
has distribution and char.
fct.
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Hyperbolic Lévy Motion

• For the char. fct. of
  we get

The density of is given by the
Fourier Inversion formula
only has hyperbolic distribution
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Hyperbolic Lévy Motion
Fig. 10, E./K., p.19 densities for
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Hyperbolic Lévy Motion

• Recall for a general Lévy process
• char. fct. is given by the Lévy-Khintchine
  formula
• characterized by
• a drift term
• a Gaussian (e.g. Brownian) component
• a jump measure

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Hyperbolic Lévy Motion

• in the symmetric centered case the hyperbolic
  Lévy motion
• is a pure jump process
• ? the Lévy-Khintchine representation of the char.
  function is

with being the density of the Lévy
measure
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Hyperbolic Lévy Motion

• Density of the Lévy measure

• and are Bessel functions
• using the asymptotic relations about Bessel
  functions, one can deduce that
• behaves like 1 / at the origin
  (x ? 0)

- Lévy measure is infinite, - hyp. Lévy motion
has infinite variation, - every path has
infinitely many small jumps in every finite
time-interval
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Hyperbolic Lévy Motion

• The infinite Lévy measure is appropriate to model
  the everyday movement of ordinary quoted stocks
  under the market pressure of many agents.
• The hyperbolic process is a purely discontinuous
  process but there exists a càdlag modification
  (again a Lévy process) which is always used.
• The sample paths of the process are almost surely
  
• continuous from the right and have limits
  from the left.

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Hyperbolic Model of the Financial Market
price process of a risk-free bond
interest rate process
price process of a stock
hyperbolic Lévy motion
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Hyperbolic Model of the Financial Market

• to pass from prices to returns take logarithm of
  the price process

results in two terms
hyperbolic term
sum-of-jumps term

• after approximation to first order the remaining
  term is
• since Lévy measure is infinite and there are
  infinitely many small jumps,
• the small jumps predominate in this term
  squared, they become even
• smaller and are negligible

the sum-of-jumps term can be neglected and to a
first approximation we get hyperbolic returns
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Hyperbolic Model of the Financial Market

• Model with exactly hyperbolic returns along
  time-intervals of length 1

stock-price process
hyperbolic Lévy motion
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Equivalent Martingale Measure

• Definition
• An equivalent martingale measure is a
  probability measure Q, equivalent to P such that
  the discounted price process
• is a martingale w.r.t. to Q.
• Complication in the Hyperbolic model
• financial market is incomplete
• no unique equivalent martingale measure (infinite
  number of e.m.m.)
• we have to choose an appropriate e.m.m. for
  pricing purposes

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Equivalent Martingale Measure

• Two approaches to find a suitable e.m.m.
• 1) minimal-martingale measure
• 2) risk-neutral Esscher measure
• In the Hyperbolic model the focus is on the
  risk-neutral Esscher measure. It is found with
  the help of Esscher transforms.

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Equivalent Martingale Measure

• Esscher transforms
• The general idea is to define equivalent measures
  via

• choose to satisfy the required
  martingale conditions
• The measure P encapsulates information about
  market behavior
• pricing by Esscher transforms amounts to choosing
  the e.m.m.
• which is closest to P in terms of information
  content.

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Equivalent Martingale Measure

• In the hyperbolic model

moment generating function of the hyperbolic Lévy
motion

• The Esscher transforms are defined by

• The equiv. mart. measures are defined via

is called the Esscher measure of parameter
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Equivalent Martingale Measure

• The risk-neutral Esscher measure is the Esscher
  measure of parameter such that
  the discounted price process
• is a martingale w.r.t. (r is the
  daily interest rate).
• Find the optimal parameter !

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Equivalent Martingale Measure

• If is the density corresponding to
  the hyp. process,
• define a new density via

? Density corresponding to the distribution of
under the Esscher measure

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Equivalent Martingale Measure

• To find consider the martingale condition
• (expectation w.r.t. the Esscher measure
  )
• This leads to
• The moment generating function can be computed as

47
Equivalent Martingale Measure

• Plug in, rearrange and take logarithms to get
• Given the daily interest rate r and the
  parameters this
• equation can be solved by numerical methods
  for the martingale
• parameter .
• ? determines the risk-neutral Esscher
  measure

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Option Pricing in the Hyperbolic Model

• Pricing a European call with maturity T and
  strike K ,
• using the risk-neutral Esscher measure
• A usefull tool will be the Factorization formula
• Let g be a measurable function and h, k and t be
  real numbers, then

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Option Pricing in the Hyperbolic Model

• By the risk-neutral valuation principle (using
  the risk-neutral Esscher
• measure) we have to calculate the following
  expectation

Pricing-Formula for a European call with strike K
and maturity T
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Option Pricing in the Hyperbolic Model
Determine c
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Option Pricing in the Hyperbolic Model

• Computation of standard hedge parameters
  (greeks)
• E.g. compute the delta of a European call C

by using subsequently the definition of
and

• integral has to be computed numerically
• useful for aspects of risk-management

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Fit of the Hyperbolic Model to Empirical Data

• E./K. performed the same statistical analysis for
  the hyperbolic model as for the Black-Scholes
  model (to fit empirical data)
• E.g. consider again the QQ-plots and density
  plots
• Fig. 8, E./K., p.16 BASF
• Fig. 9, E./K., p.17 Deutsche Bank

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Fit of the Hyperbolic Model to Empirical Data
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Fit of the Hyperbolic Model to Empirical Data
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Fit of the Hyperbolic Model to Empirical Data
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Fit of the Hyperbolic Model to Empirical Data
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Fit of the Hyperbolic Model to Empirical Data

• ? QQ-plots almost no deviation from
  straight line
• assumption of
  hyperbolic distribution is supported
• ? density plots hyperbolic distribution provides
  an almost excellent fit
• to the empirical data,
  esp. at the center and tails

The hyperbolic distribution fits empirical data
better than the normal distribution.
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Fit of the Hyperbolic Model to Empirical Data

• B./K. performed a similar study
• - daily BMW returns during Sep 1992 Jul
  1996 (100 data points)
• - standard estimates for mean and variance
  of normal distribution
• - computer program to estimate parameters of
  the hyp. distr.
• ? maximum likelihood estimates
• Fig. 2, B./K., p.14 Density plots

Comparison of option prices obtained from the
Black-Scholes model and the hyperbolic model
with real market prices shows, that the
hyperbolic model provides a better fit.
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Fit of the Hyperbolic Model to Empirical Data
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Conclusion

• The hyperbolic distribution provides a good fit
  for a range of financial data, not only in the
  tails but throughout the distribution
• ? more accurate model for stock prices /
  returns
• The hyperbolic model should esp. be preferred
  over the classical Black-Scholes model, when
  modeling daily stock returns, i.e. when looking
  at time periods of a single day.

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Conclusion

• For longer time periods the Black-Scholes model
  is still appropriate
• E./K. estimated the parameters of
  the hyperbolic distr.
• (2nd param.) for the stock returns of
  Commerzbank, considering
• different time periods, i.e. 1, 4, 7, .,
  22 trading days

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Conclusion
Fig. 7, E./K., p.15
the pairs ( , ) are given in the shape
triangle and one can see, that the parameters
tend to the normal distribution limit as the
number of trading days increases
Normal Distribution Limit
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References

•  Bingham, Kiesel (2001) Modelling asset returns
  with hyperbolic
• distributions. In "Return Distributions in
  Finance", Butterworth-
• Heinemann, p. 1-20 
• Eberlein, Keller (1995) Hyperbolic
  distributions in finance.
• Bernoulli 1, p. 281-299
•  Barndorff-Nielsen, Halgreen (1977) Infinite
  divisibility of the
• hyperbolic and generalized inverse Gaussian
  distributions. 
• Wahrscheinlichkeitstheorie und verwandte
  Gebiete 38, p. 309-311
• Hélyette Geman Pure jump Lévy processes for
  asset price modelling. Journal of Banking
  Finance 26, p. 1297-1316

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