A Simple Model of a Financial Market Based on Herd Behavior

1
The (Mis) Behavior of Markets A Fractal View of
Risk, Ruin and Reward
Discussion by Thomas Lux University of Kiel
WEHIA 05, University of Essex, 13 15 June 2005
Email lux_at_bwl.uni-kiel.de, http//www.bwl.uni-k
iel.de/vwlinstitute/gwrp/team/lux.html
2
Multi-fractality a new stylized fact Structure
of the whole spectrum of temporal dependence
for various powers of returns  
Multi-fractality Nonlinear scaling function of
moments
3
The Empirical Findings

• universal empirical observations power
  transformations of returns exhibit different
  degrees of long-term dependence
• latent consciousness on econometrics literature
  Taylor effect, highest degree of autocorrelation
  for powers around 1
• (Ding et al, J. of Emp. Fin., 1993, Lobato and
  Savin, JBES 1998)
• none of the standard models (GARCH, FIGARCH,
  StoVol) has this feature proper

-gt multi-fractal model (which also is more
parsimonious)
4
The Mechanism of a Multi-Fractal Cascade
_at_ _at_ _at_ _at_ _at_ _at_   _at_ _at_ _at_ _at_
_at_ _at_ _at_ _at_   _at_ _at_ _at_ _at_ _at_ _at_ _at_ _at_ _at_ _at_
_at_ _at_ _at_ _at_ _at_ _at_
Turbulent fluids hierarchical structure of
components Related evidence volatility on fine
time scales (tick-by-tick data) can be better
explained by coarse-grained volatility than vice
versa (Müller et al., 1997) The turbulence in
the financial markets is stronger in this sense
than the turbulence in the fluid (Holdom, 1998)
5
Binomial cascade step 1,4,8 and 12
6
The Multi-Fractal Model of Asset Returns
(Mandelbrot, Calvet and Fisher, 1997)
measure ?(t) from a multi-fractal cascade serves
as a time transformation process. Returns
follow a compound process

• r(t) BH?(t)
• with BH fractional Brownian motion with
  index H.
• (H 0.5 -gt Martingale process)

7
Example Binomial cascade 2nd step 6th
step 12th step Compound process
8
The time transformation
9
Modifications and Extensions

• multi-nomial processes, various distributions
  of multipliers and Brownian increments
• Markov-switching iterative process (Calvet and
  Fisher, 2001)
• grid-free processes (multi-fractal products of
  pulses, Mandelbrot, 1996 Barral and Mandelbrot,
  2001)

a universe of specifications (however, the
details might be less important than the basic
structural set-up)
10
Example Lognormal cascade, MS process 2nd level
only 6th level only 1st to 12th
level Compound process
11
After the Big Picture..... ....The Mundane Tasks
of Financial Economists

• Estimation via scaling function, maximum
  likelihood, GMM, simulated likelihood
• Portfolio management, VaR multi-variate models
  (greater flexibility with fewer parameters than
  GARCH!)
• Option pricing explains deviations from
  Black-Scholes
• Forecasting of Volatility
• ... some examples

12
Forecasting Exercise I Some Major Financial
Markets
In-sample 01/01/1979 to 12/31/1996,
out-of-sample 01/01/1997 to 12/31/1998.
13
In-sample 1 Jan 1979 31 Dec. 1997, out of
sample 1 Jan 1997 31 Dec 1998
14
MSEs of volatility forecasts based on historical
volatility (HV), GARCH (1,1), FIGARCH(1,d,1), and
the Lognormal multi-fractal model (MF).
In-sample 01/01/1979 to 12/31/1996,
out-of-sample 01/01/1997 to 12/31/1998.
15
Forecasting Exercise II The Japanese Stock
Market Selection of stocks from TSE database
(1,200 stocks)     random sample 100
randomly drawn stocks   large volume sample
the 100 stocks with the highest average
  trading volume     in-sample period 1975
1984   out-of-sample period 1985 2001
(17 years covering the   Japanese stock
market bubble, the crash and stagnation  
afterwards!)  
16
Data base 1,200 series of Japanese stock prices
from first section of the Tokyo stock exchange ,
daily data from 1975 to 2001 including volume
Example Nippon Suisan (stock identification
number 1332), a company processing marine food.
Established in 1911 it had 1,534 employees as of
September 2003.
17
Boxplot of MSEs of volatility predictions
(random sample of 100 stocks) MF against GARCH,
FIGARCH, ARFIMA. Boxes show the the
inter-quartile range, whiskers the full range of
results. Outliers 1.42 at lag 1 to 7.75 at
lag 100 for FIGARCH , 1.66 at lag 1 to 21.60 at
lag 100 for GARCH.
18
MSEs of volatility forecasts for four Japanese
stocks historical volatility used for
normalization. In-sample 01/01/1975 to
12/31/1985, out-of-sample 01/01/1986 to
12/31/2001.
19
A Tribute
I feel that when a satisfactory theory for this
area the different degrees of long-term
dependence in various powers! is found, it may
unlock a rush of new results having real
practical importance (Granger, in his comment on
Lobato and Savin, JBES 1998)