1. 24/05/2007 Maria Adler University of Kaiserslautern Department of Mathematics 1 Hyperbolic Processes in Finance Alternative Models for Asset Prices

2. Hyperbolic Processes in Finance 2 24/05/2007 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. Hyperbolic Processes in Finance 3 24/05/2007 The Black-Scholes Model price process of a security described by the SDE volatility drift Brownian motion interest rate price process of a risk-free bond

4. Hyperbolic Processes in Finance 4 24/05/2007 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 The Black-Scholes Model

5. Hyperbolic Processes in Finance 5 24/05/2007 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 Fit of the BS Model to Empirical Data

6. Hyperbolic Processes in Finance 6 24/05/2007 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. Hyperbolic Processes in Finance 7 24/05/2007 Fit of the BS Model to Empirical Data BASF Deutsche Bank

8. Hyperbolic Processes in Finance 8 24/05/2007 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´) Fit of the BS Model to Empirical Data

9. Hyperbolic Processes in Finance 9 24/05/2007 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

10. Hyperbolic Processes in Finance 10 24/05/2007 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 Fit of the BS Model to Empirical Data

11. Hyperbolic Processes in Finance 11 24/05/2007 Fig. 3, E./K., p.6 path, showing price changes of the Siemens stock during a single day Fit of the BS Model to Empirical Data

12. Hyperbolic Processes in Finance 12 24/05/2007 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 Fit of the BS Model to Empirical Data

13. Hyperbolic Processes in Finance 13 24/05/2007 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

14. Hyperbolic Processes in Finance 14 24/05/2007 Density of the Hyperbolic distribution: Hyperbolic Distribution modified Bessel function with index 1 characterized by four parameters: tail decay; behavior of density for skewness / asymmetry shape location scale

15. Hyperbolic Processes in Finance 15 24/05/2007 Hyperbolic Distribution Density-plots for different parameters: 19 0 0 0 0 11 1 0 0 0

16. Hyperbolic Processes in Finance 16 24/05/2007 Hyperbolic Distribution 4 301 3 2

17. Hyperbolic Processes in Finance 17 24/05/2007 Hyperbolic Distribution 41 0 13 0

18. Hyperbolic Processes in Finance 18 24/05/2007 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 Hyperbolic Distribution slopes of the asymptotics location curvature near the mode

19. Hyperbolic Processes in Finance 19 24/05/2007 Hyperbolic Distribution Plots of the log-density for different parameters: 261 0

20. Hyperbolic Processes in Finance 20 24/05/2007 Hyperbolic Distribution 21 210 0

21. Hyperbolic Processes in Finance 21 24/05/2007 Hyperbolic Distribution 4 2 1 8 1

22. Hyperbolic Processes in Finance 22 24/05/2007 setting another parameterization of the density can be obtained Hyperbolic Distribution  and invariant under changes of location and scale  shape triangle

23. Hyperbolic Processes in Finance 23 24/05/2007 Hyperbolic Distribution Fig. 6, E./K., p.13 generalized inverse Gaussian generalized inverse Gaussian

24. Hyperbolic Processes in Finance 24 24/05/2007 Relation to other distributions: Hyperbolic Distribution Normal distribution generalized inverse Gaussian distribution Exponential distribution

25. Hyperbolic Processes in Finance 25 24/05/2007 Representation as a mean-variance mixture of normals: Barndorff-Nielsen and Halgreen (1977) the mixing distribution is the generalized inverse Gaussian with density Hyperbolic Distribution consider a normal distribution with mean and variance : such that is a random variable with distribution the resulting mixture is a hyperbolic distribution

26. Hyperbolic Processes in Finance 26 24/05/2007 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. Hyperbolic Distribution The property of inf. div. is important to be able to define a stochastic process with independent and stationary (identically distr.) increments.

27. Hyperbolic Processes in Finance 27 24/05/2007 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. Hyperbolic Distribution

28. Hyperbolic Processes in Finance 28 24/05/2007 To fit empirical data it suffices to concentrate on the centered symmetric case. Hence, consider the hyperbolic density Hyperbolic Distribution

29. Hyperbolic Processes in Finance 29 24/05/2007 Characteristic function: The corresponding char. fct. to is given by Hyperbolic Distribution All moments of the hyperbolic distribution exist.

30. Hyperbolic Processes in Finance 30 24/05/2007 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.

31. Hyperbolic Processes in Finance 31 24/05/2007 Hyperbolic Lévy Motion For the char. fct. of we get The density of is given by the Fourier Inversion formula: only has hyperbolic distribution

32. Hyperbolic Processes in Finance 32 24/05/2007 Fig. 10, E./K., p.19 densities for Hyperbolic Lévy Motion

33. Hyperbolic Processes in Finance 33 24/05/2007 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 Hyperbolic Lévy Motion

34. Hyperbolic Processes in Finance 34 24/05/2007 in the symmetric centered case the hyperbolic Lévy motion is a pure jump process  the Lévy-Khintchine representation of the char. function is Hyperbolic Lévy Motion with being the density of the Lévy measure

35. Hyperbolic Processes in Finance 35 24/05/2007 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

36. Hyperbolic Processes in Finance 36 24/05/2007 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. Hyperbolic Lévy Motion

37. Hyperbolic Processes in Finance 37 24/05/2007 Hyperbolic Model of the Financial Market price process of a risk-free bond interest rate process price process of a stock hyperbolic Lévy motion

38. Hyperbolic Processes in Finance 38 24/05/2007 to pass from prices to returns: take logarithm of the price process Hyperbolic Model of the Financial Market 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

39. Hyperbolic Processes in Finance 39 24/05/2007 Model with exactly hyperbolic returns along time-intervals of length 1: Hyperbolic Model of the Financial Market stock-price process hyperbolic Lévy motion

40. Hyperbolic Processes in Finance 40 24/05/2007 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

41. Hyperbolic Processes in Finance 41 24/05/2007  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. Equivalent Martingale Measure

42. Hyperbolic Processes in Finance 42 24/05/2007 Esscher transforms: The general idea is to define equivalent measures via Equivalent Martingale Measure  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.

43. Hyperbolic Processes in Finance 43 24/05/2007 In the hyperbolic model: Equivalent Martingale Measure 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

44. Hyperbolic Processes in Finance 44 24/05/2007 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 ! Equivalent Martingale Measure

45. Hyperbolic Processes in Finance 45 24/05/2007 If is the density corresponding to the hyp. process, define a new density via Equivalent Martingale Measure  Density corresponding to the distribution of under the Esscher measure

46. Hyperbolic Processes in Finance 46 24/05/2007 To find consider the martingale condition: (expectation w.r.t. the Esscher measure ) This leads to: The moment generating function can be computed as Equivalent Martingale Measure

47. Hyperbolic Processes in Finance 47 24/05/2007 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 Equivalent Martingale Measure

48. Hyperbolic Processes in Finance 48 24/05/2007 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

49. Hyperbolic Processes in Finance 49 24/05/2007 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

50. Hyperbolic Processes in Finance 50 24/05/2007 Option Pricing in the Hyperbolic Model Determine c:

51. Hyperbolic Processes in Finance 51 24/05/2007 Computation of standard hedge parameters („greeks“): E.g. compute the delta of a European call C: Option Pricing in the Hyperbolic Model by using subsequently the definition of and  integral has to be computed numerically  useful for aspects of risk-management

52. Hyperbolic Processes in Finance 52 24/05/2007 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 Fit of the Hyperbolic Model to Empirical Data

53. Hyperbolic Processes in Finance 53 24/05/2007 Fit of the Hyperbolic Model to Empirical Data

54. Hyperbolic Processes in Finance 54 24/05/2007 Fit of the Hyperbolic Model to Empirical Data

55. Hyperbolic Processes in Finance 55 24/05/2007 Fit of the Hyperbolic Model to Empirical Data

56. Hyperbolic Processes in Finance 56 24/05/2007 Fit of the Hyperbolic Model to Empirical Data

57. Hyperbolic Processes in Finance 57 24/05/2007  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 Fit of the Hyperbolic Model to Empirical Data The hyperbolic distribution fits empirical data better than the normal distribution.

58. Hyperbolic Processes in Finance 58 24/05/2007 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 Fit of the Hyperbolic Model to Empirical Data 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.

59. Hyperbolic Processes in Finance 59 24/05/2007 Fit of the Hyperbolic Model to Empirical Data

60. Hyperbolic Processes in Finance 60 24/05/2007 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.

61. Hyperbolic Processes in Finance 61 24/05/2007 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 Conclusion

62. Hyperbolic Processes in Finance 62 24/05/2007 Conclusion 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 Fig. 7, E./K., p.15 Normal Distribution Limit

63. Hyperbolic Processes in Finance 63 24/05/2007 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

64. Hyperbolic Processes in Finance 64 24/05/2007 Thank you for participating!