To scroll down the pages, left click on the number of pages and then double left lick in the middle of the page. Then use the page down-up buttons of the keyboard.
\STATISTICAL RISK IN THE MARKETS. THE LAWS OF GROWTH, CYCLES , DEMAND-SUPPLY, INEQUALITIES.
A popularized, scientific investors risk management research with a) new non-Marshalian demand-supply laws, b) new price-volumes cyclic patterns and c) market behavior due to the economic inequalities d) best speculative (vanila) options strategy e) backoffice risk management systems, in the capital and inter-bank markets.
Thursday, August 17, 2023
Friday, December 18, 2020
69. OPTION PRICING BASED ON THE CONCEPT OF INSURANCE: MARKET MODELS-FREE METHODS THAT GIVE AS SPECIAL CASE THE BLACK-SCHOLES OPTION PRICING.
OPTION PRICING BASED ON THE CONCEPT OF INSURANCE: MARKET MODELS-FREE METHODS THAT GIVE AS SPECIAL CASE THE BLACK-SCHOLES OPTION PRICING.
Sunday, January 15, 2017
65. THE IMPACT OF THE CONVERGENCE OF THE GREEK ECONOMY TO EMI IN THE STOCKMARKET: BAYES, NESTED ESTIMATION OF THE STOCK TRENDS
THE IMPACT OF
THE CONVERGENCE OF THE GREEK ECONOMY TO EMI IN THE STOCKMARKET: BAYES, NESTED
ESTIMATION OF THE STOCK TRENDS
By
Dr. Costas Kyritsis
National
Technical University of Athens 1999
1. Introduction
The time when an economy enters the first world
economy is a very interesting time. It is even more interesting if the group of
nations where it enters, in this case European Union, becomes gradually, with
respect to some parameters, the strongest economy in the world.
Although the Greek economy is by far not perfect or
advanced, there is the firm decision to handle its indices, as much as
possible, so as to qualify according to the standards of European Monetary
Integration. These standards are set, for the Greek economy, mainly in the next
profile:
a) The Inflation rate less than 1.5%
b) The deficit of the Government less than 0.9% of the
Gross National Product
c) The national debt less than 100% of the Gross
National Product
d) Growth rate of the Gross National Product at least
4.5%.
2. Macroeconomics
factors influencing the prices in the Athens Stockmarket
There is no doubt that the previous standards of EMI
make a profile of a mature economy and also no doubt that a young state like
the Greek (less than 2 hundred years
old) has major difficulties in qualifying in the profile of EMI, before 2001
.It is worth trying nevertheless, even only for the benefit of eliminating the
continuous currency devaluation of the national wealth through the exchange
rates.
Experience has showed that the basic magnitudes of
Macroeconomics that have significant impact on the changes of prices of stocks
in the Stockmarket are:
a) The average rate of deposit in the banks, or the
rate of change of the time-value of money.
b) The exchange rates
c) Mass-media information about other economies and
changes of prices in other Stockmarkets.
The procedures with which the previous factors
influence the changes of prices in Stockmarket is always through the aggregate
demand and supply for each stock:
1) Surplus of demand to purchase stocks in the
computer waiting lines creates growth of the price of the stock (Bull-market)
2) Surplus of supply to sell stocks in the computers
waiting lines creates falling of the price of the stock (Bear-market)
The exact equations of how stochastic demand and
supply results in to the random variables of price and volume and their
changes, is not an issue to cover in the present paper. It is not of
intractable difficulty to formulate though.
We shall state, nevertheless, the basic equations of competition
of demand and supply for each stock. The equations of two populations in
competition have been a topic of systematic study. It may not be surprising
that such equations have been studied and solved not in the science of
Economics but in Ecology. They are a standard topic in an area initiated by
Volterra and his equations for populations.
Let us denote by x (tn) and y(tn) the average
value, at time
tn of the random variable of the volume of orders of
the demand to buy and of the volume of
orders of the supply to sell a
stock. The next equations describe the interplay of demand and supply:
(1) x(tn+1)= x (tn)(a-b x (tn)-c y(tn))
(2) y(tn+1)= y (tn)(e-f x (tn)-g y(tn))
The symbols a, b, c, d, e, f. g are constants defining
the competition.
Such equations, formulated in continuous time and
deterministic mode are the well known equations of competition in Ecology (see
e.g. [Maynard S.J] p 59 formula 36).We notice that they are non-linear
equations. They have been solved numerically,
studied and applied in many situations of populations in competition
.The populations involved here are of the investors who want to buy and those
who want to sell .The equations describe the
effect in demand and supply of the automatic negotiation algorithm in
the computers waiting lines . These equations if formulated in continuous time
they do not involve oscillations. But when formulated in discrete time and as
stochastic processes or time series, they do involve (non-linear) oscillations which is the common
experience for anyone that has spent some time in front of a monitor of a
Stockmarket company. If we make use of the prey-predator or host-parasitoid ,
Volterra equations that different from the equations (1), (2) only at a plus
sign instead of a minus sign at he coefficient f in (2), then we get larger scale oscillation.
During 1997
there was a major impact on the price growth in the Athens Stockmarket of the
size, at year base, close to 50% .It is supposed that it was created by the
fall of the deposit rates of the banks
(factor a) mentioned in this paragraph ).
During 1998 there was an even larger impact on the
price growth of a size close to 70%. It is supposed that it was created mainly
by the currency devaluation in the exchange rates decided by the government in
March 1998.
As the latter case was the most dramatic, we shall try
to analyze it with a new statistical method.
3. Bayes
fractal-like nested estimation of time series
As it is known there is a topic in statistics called Bayes estimators. (See e.g. [Mood A.-Graybaill A.F.-Boes D.C.] pp 339-351). The main idea is that
when we have a parameter in a distribution that we must estimate, we may assume
as a meta-level that it is already a random variable with an a priori given
distribution . For example if we are estimating a Gaussian (normal) random
variable N(m,s) we may assume that we have a double variation and a second
stochastic level and that the parameters m, s are already Gaussian (normal)
random variables with means mm , ms and variances Sm Ss . It is not that we want to make the computations more
complicated but that we need to fit a more flexible model to the real
situation.
For doubly stochastic time series see [Tong
H.] pp 117-118. We shall describe a general method to refine autoregressive
time series models, such that at each refinement, it appears higher order
variability and higher Bayes order as discussed above. For the sake of clarity
we shall apply it to the Black-Scholes
lognormal model of the prices of stocks .The model is known in
stochastic processes and stochastic differential equations as the geometric
Brownian motion . (see [Oksendal B.] pp
59-61 ,198-199 and 223-225 or [Karlin S-taylor H.M.] pp 267-269 ,357
,363,385 and [Mallaris A.G.-Brock W.A.] pp 220-223. It is a linear SDE of constant
coefficients and multiplicative «noise»
or innovation.
Although much
popularity is related to this model, it
cannot describe but the «buy-and-hold» situation in the Stockmarket . We may
try to vary this model with the idea of Bayes so as to include reversal
patterns and price motion with or without resistance. We supplement the idea of
Bayes by corresponding to each new stochastic or Bayes level a finer grid of
the argument .In this way different models appear to different scale regimes,
but still something is repeated thus we follow also the basic idea of
self-similarity introduced graphically
by Mandelbrot with fractals and
multi-fractals .
Mandelbrot
has applied his idea of self-similar fractals to the Stockmarkets, arguing that
much of the oscillating effects of stock prices are not observed in the
Black-Scholes model.
There are many new results of qualitative dynamics of
dynamic systems under the term «chaos». The ideas are not irrelevant but in
order to apply them in a professional way to Stockmarkets we require
them in stochastic differential equations or time
series (see [Tong H.])
The idea of nested patterns of «tides» (trend of a
year or more) ,«waves» (in seasonal horizon) and «ripples» (day or intra-day
oscillations ) goes back to the theory of Dow and Elliot in the Technical
Analysis of stocks (see [Murphy J.J.] pp
24-35 ,371-414). [Murphy J.J.] . It
is also obvious the relevancy of the Elliot wave theory with Spectral Analysis
and fast Fourier transformation in time series.
The way to enhance the «buy-and-hold» model of
Black-Scholes is as follows:
1) We define a nested system of grids in the time
argument .For example starting with an horizon of a year we partition it to
smaller seasonal horizons (e.g. 60 Stockmarket days). We may continue in this
way to monthly, weekly and finally daily horizons .
2) For the first one year horizon we perform an
ordinary estimation of the Black-Scholes model .It gives the buy-and-hold
trend.
3) In the seasonal horizon we increase the Bayes
stochastic order. For each season in the one year horizon we estimate a second
Bayes order model. The four seasonal models are pasted automatically to a more
flexible overall model than the Black-Scholes
4) We continue to increase the Bayes order by one for
each finer horizon, of a month, a week or a day and we estimate a new model for
each smaller horizon.
The resulting time series fits pretty well to the real
life surprises of the Stockmarket .
The method resembles the splines in numerical analysis
only that it is not performed on polynomials and the models are not
deterministic but stochastic.
A good question is how we increase the Bayes order. A
simple method is to consider the constant coefficients of the initial model as
varying linearly relative to time. This introduces for estimation new constant
parameters .At each finer grid we assume the previous constant parameters as
varying linearly and we estimate the new constant parameters.
In the next paragraph we shall perform the method at
two only horizons of one year and a
seasonal of 60 Stockmarket days .
4. An example:
The impact of the currency
devaluation in the spring of 1998.
As we mentioned in the previous paragraph the
Black-Scholes model of the prices of stocks is the geometric Brownian motion in
other words defined in continuous time by the stochastic differential equation:
(3) dx=rxdt+σxdz.
Where x is the price of the stock and z is a Brownian
motion.
In this example we implement the discrete time,
non-homogeneous time-series version defined by the equation
(4)
xn+1=(r+s en )xn
We make use of a close relative to it, which is the next time series in explicit form:
(5) xn=exp(rn+sen)
Where en is
a normal error or innovation. We do not insist on any stationarity assumption.
We make the
assumption that the «noise» or innovation term is additive in the exponent instead of multiplicative and of constant
variance, that is, an homoskedasticity assumption that makes the variance of
the residual, in the exponent, constant in time.
This simplifies the estimation of the parameters of
the time series
The application of the original model of constant
coefficients for an one year horizon is straightforward and is very well known.
We proceed with the nested Bayes estimation that we described in the previous
paragraph .We assume for the four seasonal (3-months) horizons of one year that
the model has variable coefficients
and that the coefficients vary linearly with respect to time. This introduces
new constant coefficients a, b in (5) :
(rn= an+b)
The exponent becomes now quadratic with respect to
time.
(6) xn=exp((an+b)n+sen)
More generally we estimate the equation
(7) xn=exp((an+b)n+c+ sen)
We notice that the equation is almost the normal curve
except of a linear term or sign reversal.
To estimate it we take the logarithm of the prices and
apply polynomial regression.
The exponent is in general an at most quadratic
polynomial .If the coefficient of the quadratic term is negative, we have an
instance of an almost Gaussian (normal) curve, which is interpreted as follows:
1) Increase of the prices with an asymptotic upper
resistance, which becomes a reversal pattern (first part of the curve)
2) Decrease of the prices with an obvious asymptotic
lower resistance at zero, thus practically without resistance (second part of
the curve)
If the coefficient of the quadratic term of the
exponent is positive then the probable cases are:
3) Increase of the prices very fast (faster than the
simple exponential growth) without upper resistance (second part of the curve)
4)
Decrease of the prices with lower asymptotic resistance that becomes a reversal
pattern (first part of the curve)
Thus the qualitative dynamics of the stock at each
time are described by the above four dynamic states
The results of the least squares estimation of this
linear model with time variable
coeficients are given below.
The estimated model between the
dates 10/03/1998 (n=1) and 05/06/1998
(n=60),that is 60 Stockmarket days is
(8) xn=exp(((-0,00025)n+0,023223)n+7,317873+ en)
The maximum of the normal curve occurs in the day n=47 that is in 19/05/98.
In this date the model gives a clear selling signal .Of course we cannot
trade with the general index .But it would give one if we had applied it for a
particular stock . The author scored
code in visual basic in Excell in order to analyse the buying and
selling signals during the year.The results were quite positive for forecasting
.For further
analysis of optimal trading se bibliography below from BREIMAN L.1961 to GENCAY
R. 1998.
The variance of the residual and the goodness of fit are given below:
(9) S=
8409,733584
(10) R=
92,62713729
The reader should be warned nevertheless, that a high goodness of fit of
a forecasting model, for a particular short time interval, as the above, is not
adequate for a repetitive, trading based
on it and for a long time (years). For a model to be used for repetitive
trading and for a long time (years), it should be tested that for the goodness
of fit at repetitive forecasting does remains high for long times intervals,
that must me at least 2 to 5 years, but even better 20-25 years.
In figure 2 we have an superimposed form the general index and the
estimated “normal” curve for the
seasonal horizon of 60 days .
In table 1 they are given the
numerical data of the chart .As soon as we have estimated the model by
continuing it in a resaonable forward horizon we have an effective forecasting
.The forecasting is corrected at best every day so that the buing or selling
signals are with minimum time delay .
We have used data of closing daily prices and not intra-day data .
The Bayes nested estimation can be extended for shorter horizons and the
exponent becames a polynomial of
order higher than the quadratic .
Figure 2
Table 1
Date
|
General Index
|
Normal Smoothing
|
Date
|
General Index
|
Normal Smoothing
|
10.03.1998
|
1542,017
|
1517,54
|
24.04.1998
|
2437,958
|
2473,98
|
11.03.1998
|
1577,069
|
1531,26
|
27.04.1998
|
2456,469
|
2300,71
|
12.03.1998
|
1612,116
|
1543,62
|
28.04.1998
|
2473,89
|
2445,80
|
13.03.1998
|
1647,124
|
1537,37
|
29.04.1998
|
2490,196
|
2511,56
|
16.03.1998
|
1682,055
|
1649,69
|
30.04.1998
|
2505,364
|
2621,44
|
17.03.1998
|
1716,873
|
1737,37
|
04.05.1998
|
2519,372
|
2602,82
|
18.03.1998
|
1751,541
|
1754,93
|
05.05.1998
|
2532,2
|
2634,54
|
19.03.1998
|
1786,021
|
1861,73
|
06.05.1998
|
2543,827
|
2582,62
|
20.03.1998
|
1820,275
|
1919,91
|
07.05.1998
|
2554,239
|
2509,78
|
23.03.1998
|
1854,263
|
1950,75
|
08.05.1998
|
2563,418
|
2450,16
|
24.03.1998
|
1887,948
|
1922,86
|
11.05.1998
|
2571,351
|
2358,15
|
26.03.1998
|
1921,289
|
1992,81
|
12.05.1998
|
2578,028
|
2438,39
|
27.03.1998
|
1954,248
|
2063,32
|
13.05.1998
|
2583,437
|
2494,66
|
30.03.1998
|
1986,784
|
2083,89
|
14.05.1998
|
2587,571
|
2494,70
|
31.03.1998
|
2018,857
|
2005,80
|
15.05.1998
|
2590,423
|
2469,84
|
01.04.1998
|
2050,429
|
1988,78
|
18.05.1998
|
2591,99
|
2500,44
|
02.04.1998
|
2081,46
|
1995,00
|
19.05.1998
|
2592,269
|
2493,70
|
03.04.1998
|
2111,91
|
2063,50
|
20.05.1998
|
2591,259
|
2547,01
|
06.04.1998
|
2141,741
|
2135,31
|
21.05.1998
|
2588,963
|
2573,98
|
07.04.1998
|
2170,914
|
2129,08
|
22.05.1998
|
2585,383
|
2606,48
|
08.04.1998
|
2199,391
|
2124,76
|
25.05.1998
|
2580,525
|
2669,76
|
09.04.1998
|
2227,134
|
2157,39
|
26.05.1998
|
2574,396
|
2621,33
|
10.04.1998
|
2254,106
|
2158,12
|
27.05.1998
|
2567,005
|
2523,03
|
13.04.1998
|
2280,27
|
2255,81
|
28.05.1998
|
2558,364
|
2549,07
|
14.04.1998
|
2305,593
|
2266,35
|
29.05.1998
|
2548,484
|
2591,03
|
15.04.1998
|
2330,037
|
2339,28
|
01.06.1998
|
2537,381
|
2536,09
|
16.04.1998
|
2353,571
|
2448,55
|
02.06.1998
|
2525,071
|
2551,47
|
21.04.1998
|
2376,161
|
2627,90
|
03.06.1998
|
2511,571
|
2581,24
|
22.04.1998
|
2397,776
|
2623,39
|
04.06.1998
|
2496,903
|
2567,21
|
23.04.1998
|
2418,384
|
2618,65
|
05.06.1998
|
2481,086
|
2562,82
|
References
[Azariadis C.] Intertemporal Macroeconomics
Blacwell 1993
[BLACK f.sCHOLES m.] The pricing of
Options and Corporate Liabilities
Journal of Economic Theory 10 pp 239-257
[BREIMAN L.].(1961) Optimal gambling Systems for
Favorable Games Proc.Fourth Berkeley
Sympos. on Mathematics Statistics and Probability ,University of California
Press ,Berkeley 1,pp65-78.
[BROER D.P.-JANSEN W.J.](1998) Dynamic Portfolio Adjustment
and Capital Controls: An Euler Equation Approach Southern Economic Journal 64(4) pp 902-921.
[Constantinides G.M.] (1979) Multiperiod consumption and investment behavior with
convex transaction costs Management
Science Vol. 25 No 11 Nov.,pp1127-1137
[DAVIS M.H.-NORMAN A.R.] (1990) Portfolio Selection
with transaction costs Mathematics of
Operations Research Vol. 15 No 4 PP 676-713
[DIMOPOULOS D
(1998)] . Technical analysis ,Eurocapital Pubications
[DUFFIE D.-SUN T.](1990) Transaction costs and portfolio choice in a
discrete-continuous time setting Journal
of Economic Dynamics and Control 14 pp. 35-51.
[DUMAS B.-LUCIANO EL.](1991) An exact Solution to a
Dynamic Portfolio Choice Problem under Transaction Costs.The Journal of Finance Vol. XLVI No2 pp 577-595.
[ELLIOT,R.N.] (1980) The major works of R.N. Elliot ,Chappaqua (edited by Robert Prechter) NY:New
Classics Library .
[ELTON E.J.-GRUBER M.J.] (1991) Modern portfolio
theory and investment analysis Wiley.
[FROST ,A.J.-PRECHTER R.R.](1978) Elliot Wave
Principle key to stock market profits.
Chappaqua NY:New Classics Library .
[GENCAY R.](1998) Optimization of technical trading
strategies and the profitability in security markets. Economic Letters 59 pp249-254.
[Hair-Anderson-Tatham-Black] Maltivariate Analysis
5th edition Prentice Hall 1998
[Hamilton J.D.] Time Series Analysis ,Princeton University
Press 1994
[Kloeden E.P.-Platen
E.-Schurtz H.]
Numerical Solutions of SDE Through Computer Experiments Springer 1997
[Karlin S-taylor H.M.] A first course in stochastic processes. Academic
press 1975
[Lambert H.Koopmans] The Spectral Analysis of time series Academic Press
Probability and mathematical Statistics Vol 22 1995
[Lutkepohl H.] Introduction to Multiple Time Series Analysis
Springer 1993
[Magee J.] Technical Analysis of Stock Trends New York Institute
of Finance 1992
[Mallaris A.G.-Brock W.A.] Stochastic Methods in Economics and Finance
North-Holland 1982
[MAynard S.J.] Models in Ecology Cambridge
University Press 1979
[MERTZANIS CH.] Limit of prices and variance of the
general index in the Athens Stocmarket Proceedings of the Conference of the
Institute of Statistics 1998
[MILLIONIS A.E..-MOSCHOS D.] Information Efficiency
and application in the Athens Stockmarket. Proceedings of the Conference of the
Institute of Statistics 1998
[Mood A.-Graybaill A.F.-Boes
D.C.] Introduction to
the theory of Statistics McGraw-Hill 1974
[Murphy J.J.] Technical Analysis of the Futures Markets .New York
Institute of Finance
[Nicholis D.F.-Quinn B.G.] Random coefficients autoregressive models:an
introduction .Lecture Notes in Statistics No 11 Springer NY.
[Oksendal B.] Stochastic Differential equations Springer 1995
[PAPADAMOY S.-PAPANASTASIOU D.-TSOPOGLOU] Research on
the predictability of the stocks in the Athens Stockmarket. [ScarthM.W.] Macroeconomics Hancourt Brace &Company Canada 1996
[SPANOUDAKI I.-KAMPANELOY A.-ANTONOGIORKAKIS P.]
Volume indices in the technical analysis of stocks.Monte-carlo empirical
results. Proceedings of the Conference of the Institute of Statistics 1998
[Tong H.] Non-Linear Time series :A dynamic System Approach
Clarendon Press Oxford 1990
Subscribe to:
Posts (Atom)