Friday, October 12, 2012

11. Assets evolution of the enterprises and the Pareto and power distribution of volumes and slope and duration of the trends in the markets. The statistical link between fundamental and technical analysis.

It has been prove by simulation  (see the book by Bernardo Huberman @The laws of the web@ pages 28-29 chapter 3     http://www.amazon.com/Laws-Web-Patterns-Ecology-Information/dp/0262582252/ref=sr_1_1?s=books&ie=UTF8&qid=1396516209&sr=1-1&keywords=The+laws+of+the+web )
that , if a number of organizations grow multiplicative (in other words exponentially or as logistic growth) , and all have the same growth rate, and start at the same time, then at any time moment, their statistical distribution of sizes, is lognormal. BUT if they start at different times, and/or have different growth rates, then , the resulting distribution, at any time will be a power, or Pareto distribution. 
This explains that the enterprises and financial organizations have as statistical distribution of their asset sizes, a power or pareto distribution (in other words that involves a monomial ax^b of probability density, as a function of the size x.) This means also that if we plot the statistical distribution in logarithmic scale both on the x-axis and the statistical density axis y, it will appear as straight line). 
This statistical law is inherited to the volumes of transactions, and therefore it creates the momentum conservation, or law of trend, that we have analyzed in other posts. 

This inheritance of a power distribution law from the size of the assets, to the volumes of transactions, is the basic mathematical-statistical relation between the fundamental analysis and the technical analysis

If all the organizations would have the same size, then, the markets would behave as random walk (neutral fluctuations up or down) . This is also a proof why the markets do not behave , as random walk , because, of the law of Pareto of economic inequality. And this is also a proof, that the standard models of Black-Scholes of the Options fair pricing are biased in a  systematic way, as their fluctuations are assumed as exponential  random walk.
Of course it could not be otherwise! Would the financial status quo elite give a nobel prize, to a model that would allow , smaller or middle players to gain systematically over the large players? Not probable! The model should be so that all players gain zero in the average, and if we consider the transaction costs, would lose systematically. 


There are 3 contexts of laws required in trading . The appropriate LAWS OF THINKING for trading, the appropriate LAWS OF FEELINGS for trading , and the appropriate LAWS OF ACTIONS for trading. 

The Successful trading is based according to these three laws on
1) POWER OF COLLECTIVE  SCIENTIFIC THINKING: A GREAT AND SIMPLE SCIENTIFIC PERCEPTION OF THE FUNCTION OF THE ECONOMY THROUGH SOME GLOBAL STATISTICAL LAW. E.g. The law of Universal attraction in economy: that big money attracts more big money in the capital markets, and this by the balance of demand and supply makes securities indexes of the companies , that are indeed the big money, to have mainly stable ascending trend, whenever one can observe such one. Valid statistical deductions can be obtained with simple statistical hypotheses tests about the existence or not of a trend, with sample size half the period of a dominating cycle). (STABLE GREAT SCIENTIFIC THOUGHT-FORM  OR BELIEF FACTOR IN TRADING. )

2) POWER OF COLLECTIVE PSYCHOLOGY: A LINK WITH THE POSITIVE COLLECTIVE PSYCHOLOGY.(E.g. that the growth of security indexes also represent the optimism of the growth and success of real business of the involved companies. And we bet or trade only on the ascension of the index, whenever  an ascending trend is observable). (STABLE GREAT POSITIVE COLLECTIVE   EMOTIONAL OR PSYCHOLOGICAL FACTOR IN TRADING. )


3) POWER OF INDIVIDUALS SIMPLE , CONSISTENT AND EASY TO CONDUCT PRACTICE. (e.g. a trading system with about 80% success  rate that utilizes essentially only one indicator in 3 time frames, simple risk management rules of stop loss, take profit, trailing and escalation, and time spent not more than 20 minutes per day. In this way there are not many opportunities of human errors in the conduction of the trading practice. Failed trades are attributed to the randomness and are not to blame the trader). (STABLE SIMPLE AND EASY PRACTICAL  FACTOR IN TRADING)

We may make the metaphor that successful trading is the ability to have successful resonance with the  activities of top minority of those who determine the markets.

In trading there are 3 components in the feelings that must be dealt with. 1) The feeling of MONEY itself, 2) The feeling of the UTILITY of the money 3) The feeling of the RISK of the money each time. What is called usually money management in trading is essentially RISK MANAGEMENT. 



VALID STATISTICS AND PREDICTABILITY
We must make here some remarks about the robust application of statistical predictions in the capital markets.

1) The theory that the efficient markets and in particular that they follow a pure random walk is easy to refute with better statistical experiments and hypotheses tests. The random walk would fit to a market where the sizes of the economic organizations are uniformly random. But the reality is that they follow a Pareto or power distribution, therefore this is inherited in the distribution of the volumes of transactions and also in the emerging trends or drifts. 

2) The statistical models of time series  are more robust , when they apply to the entity MARKET as a whole and are better as  non-parametric , and not when they apply to single stocks and are linear or parametric. The reasons is that  a time series as a stochastic process , requires data of a sample of paths, and for a single stock is available only a single path. While for all the market the path of each stock or security is considered one path from the sample of all paths of all the stocks. 

3) The less ambitious the statistical application the more valid the result. E.g. applying a statistical hypothesis test, or analysis of variance   to test if there is an up or a down trend (drift) or none, is a more valid statistical deduction , than applying a linear model of a time series and requiring prediction of the next step price. 

4) Multivariate statistics, like factor analysis, discriminant analysis , logistic regression,  cluster analysis , goal programming etc are possible to utilize for a more detailed theory of predictability and of portfolio analysis, and sector analysis of the market and not only H. Markowitz theory. 

5) In applying of the above applications of statistics, the researcher must have at first a very good "feeling" of the data, and should verify rather with statistics the result rather than discover it. 

6) The "Pareto rule of complexity-results" also holds here. In other words with less than 20% of the complexity of the calculations is derived more than 80% of the deduction. The rest of the 20% requires more than 80% more complexity in the calculations.


38. RE-POSITIONING THE PORTFOLIO. Optimal adjustments of tradable funds and non-tradable cash.

As I mentioned in the post 3, about the "speculators" they apply an optimal adjustment back-office method that multiplies their profits reduces the risk, and does not need a forecasting of the market. Warren Buffett and many other succesful investors apply this technique as a paramount and basic neccesity in investing and trading.
During 1998 while studying in the University of Portsmouth, I discovered a theorem in the book "Stochastic Differential equations" by B. K.Oksendal (Springer editions) page 223, example 11.5 where he proves through the ITO stochstic calculus that such an adjustment as above is optimal during a constanttrend  against just buy-and-hold , and maximizes the probability to have positive profit 
We may apply optimal adjustments of tradable funds. This means that we divide the funds in to tradable (e.g. 66%) and non-tradable (33%). Al trading is based only on the tradable funds, which in their turn are divided in to usable for margin, and risked e.g. by stop-loss in a trade or excursions, and those reserved for next trades. Then at the start of the trading we mark the equity level E0. For every dE increase from this E0 level or previous level during trading  (e.g. dE=5% over all funds) we adjust by in increasing by 2.5% the non-tradable, and decreasing by 2.5% the tradable, and for every dE decrease from E0 or previous level during trading, we adjust by in decreasing by 2.5% the non-tradable, and increasing by 2.5% the tradable so that the ratio 33% of non-tradable funds to all funds, remains approximately constant. 

Wednesday, April 25, 2012

49. The true rules of volumes and trend


Before reading this post the reader must consult the post 10, of the relation of the power law of volumes (or law of attraction) and the law of action (or trend). Also the post 9, of all the 12 laws of financial markets.


Law of attraction (or power law of the volumes) 


This law explains how the financial populations of demand and supply are shaped. (we mean here not so much population of individuals byt populations of assets, transactions, and money flow.They are shaped much like ecological populations are shaped in biosphere. "Birds of a feather flock together". The shaping of the financial population of the same opinion behavior and decision is also through the mass media news, through web-news, direct gossip, thinking etc. There are three types of financial populations, the money savers, the investors and the traders. These types go from less risk to higher risk in this order. By utilising statistics we may describe the growth and decay of a financial population as a discrete stochastic process. The same law also governs the way that business organisations are shaped. The distribution of the size of populations of demand-supply and of the size of assets of  enterprises follows the Pareto (or a Power) distribution. If we take the logarithm of such a distribution , it will become a straight line. More than 80% of the volume of transactions are made from less than 20% of the organizations (Chebyshev's_inequality , http://en.wikipedia.org/wiki/Chebyshev's_inequality). The freedom of the markets is  more than 80% in favor of the large palyers and less than 20% in favor of the small players. In other words, the FREE MARKETS are the kingdom of the large players, after this law of power of volumes.

Because of the above law, it holds the law of conservation of the momentum , or law of the trend. As it is obvious, when a large player (e.g. a Central bank etc) is giving the order of a buying or selling, the volumes are so large, that this has to be spanned for a very large number of smaller transactions that would take hours, days or even weeks etc.

In spite of what has been written thousand of times in books and texts of technical analysis, the scientific true rules of the volumes and trend are not simple combinations like "increasing prices" and "increasing volumes" but require not only the sign of the trend but the existence of acceleration or deceleration, otherwise we may easily point out counter-examples (not only sample counter-examples but also counter-examples in the average value too).

The volumes measurement do provide better forecasting. The true rules of volumes are
1. A (statistical) momentum acceleration is a true acceleration, if the volumes are increasing too.
2. A (statistical) momentum deceleration is a true deceleration, if the volumes are decreasing too. 
Of course the converse does not hold: An acceleration can be true, even if the volumes are not increasing. But if they are increasing we are sure it is true acceleration. The same with the deceleration.

Monday, January 16, 2012

47. A method to estimate the suggested leverage from the volatility and the average duration of the position


The next is an article (2001) where the percentage of the funds risked and the leverage of a position is calculated from the volatility and the average duration of the position. It is applied on the futures on indexes of the Arhens Derivatives Exchange (ADEX).


Initial Remark:
 We must be aware that the academic research has accepted and published nay different and often contradictory statistical or probabilistic models of the market, the trend of the market and generally its patterns of moves. Statistical results are very sensitive to the way we handle the sampling over the same market. And each research focuses and handles the sampling of the market in  different ways. It requires special and  wise scientific analysis of the sampling to deduce really sound conclusions that are safe to let us put large amounts of money funds and capital.   The continuous time geometric Brownian motion is a rather awkward model for the trend of the markets, because we have measured  the Pareto distribution rather than log-normal distribution of   the trend of prices , and due to the  Pareto modeled inequalities of the enterprises and their inherited in the markets packets of orders,  as we have mentioned in post 25 , 10 etc. Another difficulty is that it is continuous time model and its statistics even on minutes bars time frames or tick-wise is only approximate. In particular it is a model with much more risk than the actual of the trend in the markets. Still after the Black-Scholes model of option pricing (see e.g. https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model) , that resulted to a Nobel prize, it is standard in academic research to model the trend of the market with a continuous   time geometric Brownian motion. In such models if r is less than (1/2) * s^2  then it is almost certain (in other words with probability equal to 1 ) that given sufficient time the path of the trend will get bankrupt that is it will return to  the initial point! As also the ratio r/s^2 is the optimal exposure and adjustment from deductions of the same theory of ITO calculus (see the book "Stochastic Differential equations" by B. K. Oksendal (Springer editions) page 223, example 11.5  ) then this means that we are led to bankruptcy if the optimal exposure has to be less than 50% !  Other standard also discrete time models of exponential trend of the form y(n)=exp(rt(n))+e(n) with variance(e(n)=s and average(e(n))=0 do not have this bankruptcy property even if r is less than (1/2) * s^2 . It seems to me that such a continuous time model might be approximately  acceptable , but still with many deviations from the reality , for very-very large samples of  period measurements (bars or candlesticks) and only for instruments and time periods with very clear stable trend. If we assume this continuous time model of the trend as acceptable , then the next is an acceptable method of estimation of the required liquidity in buy and hold positions of futures. 





ESTIMATION OF REQUIRED LIQUIDITY FOR INVESTMENT POSITION

IN FUTURES OF THE ATHENS DERIVATIVES EXCHANGE MARKET.

By Dr COSTAS KYRITSIS
University of Portsmouth UK
Department of Mathematics
and Computer Science
Software Laboratory
National Technical University
of Athens 2001


                                                            Abstract
In this paper we discuss the risk of mark-to-market  loss of positions with leverage, in futures. We make the usual assumptions of Lognormal distribution and geometric Brownian motion for the underlying  as in the Black-Scholes options pricing model. With these assumptions  we estimate tables of  required liquidity for futures on FTSE-20 and FTSE-40 in the Athens Derivative Exchange Market, during the year 2000.
Key words
Derivatives, Geometric Brownian Motion, Stochastic Differential Equations, Simulation, Investment, Liquidity

§1 Introduction  Since August 1999 the Athens Derivative Exchange Market (ADEX) introduced for  the first time futures on the Index FTSE-20, and soon afterwards on the Index FTSE-40. At present the average amount of money that is traded in ADEX is 10-15 billion drachmas that is almost 3% of the traded money in the Athens Stock Exchange Market (ASE, October 2000). The interest is increasing with the introduction of options and stock lending. Nevertheless the peculiarities and risks of investing to futures are not quite clear to the present average investor. For example it is highly risky and not advisable to leave investment positions of futures that have loss till the market has reverse trend and they became profitable, as is done with securities. There is the risk of running out of the budget and eventually closing the position before it becomes profitable. In this paper we analyze the source of this risk and we estimate the required amount of money that should be kept in cash, for probable losses.
§2 Leverage in positions on futures
When investing in positions on futures we do not pay all the money of the investment.
Instead it is calculated daily the profit or loss of the investment position (called mark-to-market) and is paid by the investors and Broker Companies to appropriate clearance bank (Alpha Credit Bank). In addition it is paid a percentage only of the height of the position, as much as it is considered it is risked for 1-2 days for ADEX to close the position if anything goes wrong (default position). The percentage is estimated according to the volatility (standard deviation) of the daily percentage changes of the underlying Index and is called Margin . This percentage at present is 12% for futures on the Index FTSE-20 and 16% for futures on the Index FTSE-40. This makes an advantage for the investor as he must only pay 12%-16% of the height of a position when he opens it. This is called the leverage of the position and is a multiplier of  1/12%=8.33 times and 1/16%=6.25 times respectively. Of course not only the rate of return is multiplied with this numbers but also the Beta (or Elasticity) of the position. The advantage of leverage has also its risks , as the profit or loss is paid daily on 100% of the height of the position and in a reverse trend of the market can easily lead to bankruptcy, something not really possible with investment positions in securities.
§3 Volatility of the price of the underlying .
In order to estimate the required Liquidity we need a forecasting of the underlying Index for the investment horizon, and  the correlation and coupling of the future with the underlying Index .
The most difficult parameter is that of the Market, in other words the forecasting of the Index. This obviously depends on the season and the particular trend of the market at the Investment time.
We shall go around this difficulty in a way that is standard in the pricing of Derivatives and is also used by ADEX in the estimation of the percentages of 12% and 16% for the margin. We shall assume a neutral market, neither growing neither decaying, but with a trend equal to the risk free rate (5.59% in a year base, at present October 2000). So the model of the underlying index is, as in the Black-Scholes option pricing model ,a Geometric Brownian Motion (continuous time random compound interest) of normally distributed rate r and volatility Ïƒ. .For the definition of the stochastic differential equations and the geometric Brownian motion see Oksental p121 Chpt. V p 60 ,exerc.7.9 ,p 121,example 5.1 p 60
The stochastic differential equation of Brownian motion (Ito interpretation)  is:

                                                                                (1)
The exact interpretation of the symbols requires the concepts and definitions of stochastic Integrals and is outside the scope of this paper. For the definitions see Oksental 1995.
            The distribution of the prices Xt is Lognormal.
The solution of this stochastic differential equation is given by the formula:
                                                               (2)
where Bis a Brownian Motion
The logarithm of this process Xt/ X0  is an ordinary Brownian motion with drift:
log(Xt/ X0)=(r-(1/2) s2)t + s Bt .                                                                      (3)
The average time T that it reaches X for the first time starting from x
Is T=log(X/ x0)/(r-(1/2) s2)                                                                               (4)

If  Î² is the beta (elasticity)  of the future Yt over the index Xt we  assume that the futures also, follows the equation
                                                                                        (5)

If l is the leverage of the investment the value of the investment position on the future follwos actually the equation

                                                                                      (6)

The conditional variance of the Index price X is calculated to have the formula
(see Karlin S-Taylor H.M.(1975)  p  357)

Var(X(t)/X(0)=X)=X2Exp(2t(r+σ2/2))*(Exp(tσ2)-1)                           (7)       
§4 Required Liquidity tables .
For the required liquidity of position of horizon t we may take the conditional volatility (standard deviation) of the price of the future .We should not forget that these formulas assume a neutral market and the liquidity is the average and not a worse scenario.
From the above  formulae and assuming Î²=1 ,r=5.59% ,σ=40% (historic volatility of 1 year) for the FTSE-20 and Ïƒ=47% for the FTSE-40 (values that are used by ADEX at present and are estimated by the daily data) we compute the next tables.
Days-FTSE-20
Liquidity Percentage
Days-FTSE-20
Liquidity Percentage
Days-FTSE-20
Liquidity Percentage
Days-FTSE-20
Liquidity Percentage
1
2.53%
21
11.77%
41
16.68%
61
20.65%
2
3.58%
22
12.06%
42
16.90%
62
20.83%
3
4.39%
23
12.34%
43
17.11%
63
21.01%
4
5.07%
24
12.61%
44
17.32%
64
21.19%
5
5.68%
25
12.88%
45
17.53%
65
21.37%
6
6.22%
26
13.14%
46
17.74%
66
21.55%
7
6.73%
27
13.40%
47
17.94%
67
21.73%
8
7.20%
28
13.66%
48
18.14%
68
21.91%
9
7.64%
29
13.91%
49
18.35%
69
22.09%
10
8.06%
30
14.16%
50
18.54%
70
22.26%
11
8.46%
31
14.40%
51
18.74%
71
22.44%
12
8.84%
32
14.64%
52
18.94%
72
22.61%
13
9.21%
33
14.88%
53
19.13%
73
22.78%
14
9.56%
34
15.12%
54
19.33%
74
22.95%
15
9.90%
35
15.35%
55
19.52%
75
23.13%
16
10.24%
36
15.58%
56
19.71%
76
23.30%
17
10.56%
37
15.80%
57
19.90%
77
23.47%
18
10.87%
38
16.03%
58
20.09%
78
23.63%
19
11.18%
39
16.25%
59
20.28%


20
11.48%
40
16.47%
60
20.46%



Days/FTSE-40
Liquidity Percentage
Days/FTSE-40
Liquidity Percentage
Days/FTSE-40
Liquidity Percentage
Days/FTSE-40
Liquidity Percentage
1
2.98%
21
13.88%
41
19.75%
61
24.53%
2
4.21%
22
14.22%
42
20.01%
62
24.75%
3
5.16%
23
14.56%
43
20.26%
63
24.98%
4
5.97%
24
14.88%
44
20.52%
64
25.20%
5
6.68%
25
15.20%
45
20.77%
65
25.42%
6
7.32%
26
15.52%
46
21.02%
66
25.63%
7
7.91%
27
15.83%
47
21.26%
67
25.85%
8
8.47%
28
16.13%
48
21.51%
68
26.07%
9
8.99%
29
16.43%
49
21.75%
69
26.28%
10
9.49%
30
16.73%
50
21.99%
70
26.49%
11
9.96%
31
17.02%
51
22.23%
71
26.71%
12
10.41%
32
17.31%
52
22.47%
72
26.92%
13
10.84%
33
17.59%
53
22.70%
73
27.13%
14
11.26%
34
17.87%
54
22.94%
74
27.34%
15
11.67%
35
18.15%
55
23.17%
75
27.55%
16
12.06%
36
18.42%
56
23.40%
76
27.76%
17
12.45%
37
18.70%
57
23.63%
77
27.96%
18
12.82%
38
18.96%
58
23.86%
78
28.17%
19
13.18%
39
19.23%
59
24.08%


20
13.54%
40
19.49%
60
24.31%






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