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<(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 modelling of the trend
as acceptable , then the next is an acceptable method of estimation of
the average maximum loss in buy and hold positions.
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<(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 modelling of the trend as acceptable , then the next is an acceptable method of estimation of the average maximum loss in buy and hold positions.
ESTIMATION OF MAXIMUM AVERAGE LOSS FOR INVESTMENT POSITION IN FUTURES OF THE ATHENS DERIVATIVES EXCHANGE MARKET.
By Dr COSTAS
KYRITSIS By
APOSTOLIS KIOHOS
University of
Portsmouth UK (Msc in Risk Management and
Department of Computing and Insurance)
Maths
and
Software
Laboratory
National
Technical University
of Athens 2000
Abstract
In this paper we discuss the risk of
mark-to-market loss of positions with
leverage, of infinite horizon, 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 the tables of required
liquidity for futures on FTSE-20 and FTSE-40 in the Athens Derivatives Exchange
Market and the maximum average Loss of infinite horizon investment positions in
Derivative Exchange Market.
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. The peculiarities
and risks of investing to futures are not quite clear to the present average
investor. In a first publication [Kyritsis C (2001)] we analyzed the required
Liquidity of finite horizon investments in futures. We made use of the
conditional volatility.
In this paper we analyze the liquidity requirement of
infinite horizon investments from the point of view of average maximum loss. Of
course the investments in Derivatives have always an expiration date. But
putting an infinite horizon in the investment makes calculations simpler and at
the same time the real risk is less or equal to the estimated so it is always
safer for more risk averse decision makers.
The main idea is that it should always be possible to
pay the average maximum loss besides the margin reservations. So an estimate of
an average maximum loss, given a margin percentage, leads directly to a
liquidity percentage. The method of average maximum loss is simpler to
calculate, to understand and apply, than the method of conditional volatility.
§2 Leverage and bankruptcy of
positions in 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 Brokerage Companies to an 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 the futures.(March 2001). This makes an advantage for the investor as
he must only pay 12% 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. 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 drowbacks and risks. 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 Average maximum Loss of an
Investment Position.
In
order to estimate the average maximum loss
we have to assume a model of the underlying Index, and the correlation and coupling of the future
with the underlying Index.
We shall proceed 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% for the margin. We shall assume a neutral market, neither
growing neither decaying, but with a trend equal to the risk free rate (2% in a
year base, at present March 2001). 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 σ. 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 Bt is 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 x0
Is T=log(X/ x0)/(r-(1/2) s2) (4)
If β 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 follows
the equation
The way to estimate the average maximum loss of the
investment is the following:
We shall make use of a theorem on the Brownian motion
that can be found in
[Karlin S.-Taylor H.(1975)] Corollary 5.1 Chapter 5 p
361.
The Theorem goes like this:
Let X(t) be a Brownian motion process with drift μ
>0. Let
W=max(X(0)-X(t)).
For all t>=0. (7)
The W has
exponential distribution
Pr(W>w)=exp(-λw), w>=0 (8)
Where
λ=(2|μ|)/(σ2) (9)
The formula (3)
above shows that the logarithm of the prices of the future follows a Brownian
motion with (let us say positive) drift
(r-(1/2) s2)
Therefore the average maximum downward
deviation of the logarithm is
s2 /(2*|(r-(1/2) s2)|)
(10)
As the logarithm is a monotonous
function (respecting order) and the average of a logarithm is the logarithm of
the average, the average downward deviation of the price of the underlying in
other words the average maximum loss of the underlying is (referring to formula
2 above)
X0exp(s2 /(2*|(r-(1/2) s2)|)) (11)
Thus we get the next statement
Theorem
A
The average maximum loss as a percentage is
|1- exp(s2 /(2*|(r-(1/2) s2)|))| (12)
§4 Tables of Average Maximum Loss.
In the next tables we have calculated
the rate ,volatility for a 30 days
sample of the index ftse-20 and
consequently the average maximum loss and liquidity percentage (or percentage
to invest) for Long or short positions for a period of 43 days during 2001. (We assume β=1 for the futures on
them).In the calculations we do not consider the leverage of the positions in
the futures. The percentage to invest is defined by the assumption that the
average maximum loss should be kept in cash . So if the average maximum loss is
say 40%, as the margin now (2001) is a
12% , then the percentage to keep in
cash is 40%/(12% +40%) and the percentage to invest therefore is 60%/(12%+40%).
Date
|
Unsystematic
Risk
|
BETA
for 30 days
|
Average
rate for 30 days
|
Variance of General index
|
Daily
Variance of Ftse
|
AverageMaximumLoss
|
Percentage to
Invest
|
Year
Volatility of Ftse
|
3/1/2001
|
0,0002
|
1,1952
|
-0,026
|
0,0005
|
0,000854
|
37,8118%
|
24,09%
|
0,461946
|
4/1/2001
|
0,0002
|
1,1885
|
-0,027
|
0,0005
|
0,000835
|
37,6409%
|
24,17%
|
0,456838
|
5/1/2001
|
0,0002
|
1,1885
|
-0,027
|
0,0005
|
0,000835
|
37,6409%
|
24,17%
|
0,456838
|
8/1/2001
|
0,0002
|
1,2338
|
-0,028
|
0,0005
|
0,000944
|
45,5163%
|
20,86%
|
0,485736
|
9/1/2001
|
0,0002
|
1,2509
|
-0,029
|
0,0005
|
0,000975
|
50,2929%
|
19,26%
|
0,493766
|
10/1/2001
|
0,0002
|
1,257
|
-0,03
|
0,0005
|
0,000954
|
49,9603%
|
19,37%
|
0,488253
|
11/1/2001
|
0,0003
|
1,1246
|
-0,03
|
0,0006
|
0,001084
|
59,1191%
|
16,87%
|
0,520627
|
12/1/2001
|
0,0003
|
1,1065
|
-0,03
|
0,0006
|
0,001059
|
56,6757%
|
17,47%
|
0,51456
|
15/1/2001
|
0,0003
|
1,1506
|
-0,03
|
0,0007
|
0,001224
|
68,4562%
|
14,91%
|
0,553205
|
16/1/2001
|
0,0003
|
1,1606
|
-0,029
|
0,0007
|
0,00124
|
69,2779%
|
14,76%
|
0,556877
|
17/1/2001
|
0,0003
|
1,1309
|
-0,029
|
0,0006
|
0,001048
|
55,6878%
|
17,73%
|
0,511829
|
18/1/2001
|
0,0002
|
1,1485
|
-0,031
|
0,0006
|
0,00091
|
48,9313%
|
19,69%
|
0,476949
|
19/1/2001
|
0,0002
|
1,1549
|
-0,031
|
0,0006
|
0,000907
|
49,5285%
|
19,50%
|
0,47631
|
22/1/2001
|
0,0013
|
0,0576
|
-0,032
|
0,0006
|
0,001275
|
78,7285%
|
13,23%
|
0,564572
|
23/1/2001
|
7E-05
|
1,3764
|
-0,032
|
0,0004
|
0,000872
|
49,1165%
|
19,63%
|
0,466904
|
24/1/2001
|
7E-05
|
1,3704
|
-0,035
|
0,0004
|
0,000857
|
52,6611%
|
18,56%
|
0,462762
|
25/1/2001
|
0,0001
|
1,3017
|
-0,035
|
0,0004
|
0,00085
|
52,8936%
|
18,49%
|
0,460912
|
26/1/2001
|
0,0001
|
1,2687
|
-0,036
|
0,0004
|
0,000834
|
53,1531%
|
18,42%
|
0,456526
|
29/1/2001
|
0,0002
|
1,2655
|
-0,037
|
0,0004
|
0,000816
|
53,7830%
|
18,24%
|
0,451718
|
30/1/2001
|
0,0002
|
1,2636
|
-0,038
|
0,0004
|
0,000843
|
57,4241%
|
17,29%
|
0,459009
|
31/1/2001
|
0,0002
|
1,1922
|
-0,039
|
0,0005
|
0,000902
|
64,1609%
|
15,76%
|
0,474985
|
1/2/2001
|
0,0002
|
1,1714
|
-0,039
|
0,0005
|
0,000879
|
62,3608%
|
16,14%
|
0,468833
|
2/2/2001
|
0,0002
|
1,1765
|
-0,04
|
0,0005
|
0,000898
|
65,9084%
|
15,40%
|
0,473774
|
5/2/2001
|
0,0002
|
1,2137
|
-0,039
|
0,0005
|
0,000937
|
68,0976%
|
14,98%
|
0,484022
|
6/2/2001
|
0,0002
|
1,201
|
-0,038
|
0,0005
|
0,000969
|
69,0006%
|
14,81%
|
0,492119
|
7/2/2001
|
0,0002
|
1,207
|
-0,038
|
0,0005
|
0,000973
|
69,1451%
|
14,79%
|
0,493107
|
8/2/2001
|
0,0003
|
1,1904
|
-0,039
|
0,0005
|
0,000979
|
71,1249%
|
14,44%
|
0,494666
|
9/2/2001
|
0,0003
|
1,1763
|
-0,04
|
0,0005
|
0,000979
|
73,9884%
|
13,96%
|
0,494813
|
12/2/2001
|
0,0003
|
1,173
|
-0,04
|
0,0005
|
0,001047
|
81,2309%
|
12,87%
|
0,511608
|
13/2/2001
|
0,0003
|
1,168
|
-0,04
|
0,0005
|
0,001042
|
81,8545%
|
12,79%
|
0,510414
|
14/2/2001
|
0,0004
|
1,1669
|
-0,041
|
0,0005
|
0,001083
|
87,6833%
|
12,04%
|
0,520409
|
15/2/2001
|
0,0004
|
1,1851
|
-0,04
|
0,0005
|
0,001078
|
85,4915%
|
12,31%
|
0,519044
|
16/2/2001
|
0,0004
|
1,15
|
-0,041
|
0,0005
|
0,000996
|
78,0027%
|
13,33%
|
0,499067
|
19/2/2001
|
0,0004
|
1,1568
|
-0,025
|
0,0005
|
0,00098
|
42,1816%
|
22,15%
|
0,495096
|
20/2/2001
|
0,0004
|
1,1319
|
-0,053
|
0,0005
|
0,00099
|
110,0591%
|
9,83%
|
0,497466
|
21/2/2001
|
0,0004
|
1,3202
|
-0,053
|
0,0003
|
0,000898
|
95,8285%
|
11,13%
|
0,473785
|
22/2/2001
|
0,0003
|
1,3257
|
-0,053
|
0,0003
|
0,000829
|
86,7790%
|
12,15%
|
0,455127
|
23/2/2001
|
0,0003
|
1,3502
|
-0,053
|
0,0002
|
0,000732
|
74,0986%
|
13,94%
|
0,427887
|
27/2/2001
|
0,0003
|
1,3861
|
-0,054
|
0,0002
|
0,000783
|
81,4854%
|
12,84%
|
0,442465
|
28/2/2001
|
0,0003
|
1,3574
|
-0,054
|
0,0002
|
0,000731
|
75,3731%
|
13,73%
|
0,427452
|
1/3/2001
|
0,0003
|
1,3592
|
-0,055
|
0,0002
|
0,000726
|
76,2499%
|
13,60%
|
0,426166
|
2/3/2001
|
0,0003
|
1,4426
|
-0,055
|
0,0002
|
0,000762
|
80,4376%
|
12,98%
|
0,43642
|
5/3/2001
|
0,0003
|
1,5486
|
-0,056
|
0,0002
|
0,000777
|
85,2209%
|
12,34%
|
0,440694
|
Bibliography
Azariadis C.(1993) Intertemporal
Macroeconomics Blackwell
Black F.Scholes M.(1976)
The pricing of Options and
Corporate Liabilities Journal of
Economic Theory 10 pp 239-257
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 .
Gencay R.(1998) Optimization of technical trading
strategies and the
profitability in security markets.Economic
Letters 59 pp249-254.
Elton E.J.-Gruber M.J. (1991) Modern portfolio theory
and investment
analysis Wiley.
Elliot,R.N. (1980) The major workd of R.N. Elliot ,Chappaqua
(edited by
Robert Prechter) NY:New Classics Library .
Hamilton J.D.(1994) Time Series Analysis ,Princeton University
Press
Kloeden E.P.-Platen E.-Schurtz h.(1997) Numerical Solutions of SDE
Through Computer Experiments Springer
Karlin S-Taylor H.M.(1975)
A first course in stochastic processes. Academic press .
Kyritsis C.
(2001) Estimation of required liquidity
for investment position
in futures of the athens derivatives exchange market.
(To appear)
Magee J.(1992)
Technical Analysis of Stock Trends New York Institute of Finance
Mallaris A.G.-Brock W.A.(1982) Stochastic Methods in Economics and
Finance North-Holland
Merton R.C (1968) Lifetime portfolio selection under
uncertainty:The
continuous time case. Rev. Econ.Statist. II pp247-257
Mertzanis Ch.(1998) Limit of prices and variance of
the general index in the
Athens Stocmarket Proceedings of the Conference of the
Institute of Statistics .
Millionis A.E..-Moschos D.(1998) Information
Efficiency and application in
the Athens Stockmarket. Proceedings of the Conference
of the Institute of Statistics
Murphy J.J. Technical Analysis of the Futures Markets
.New York
Institute of Finance
Oksendal B (1995)
Stochastic Differential equations Springer 1995
Papadamoy S.-Papanastasiou D.-Tsopoglo (1998) Research
on the predictability
of the stocks in the Athens Stockmarket. Proceedings
of the Conference of the Institute of Statistics .
Roy S.(1995)
Theory of dynamic choice for survival under uncertainty
Mathematics of social sciences 30 pp171-194
Spanoudaki J.-Kampaneloy A.-Antonogiorkakis P.(1998)
Volume indices in
the Technical analysis of stocks.Monte-Carlo empirical
results. Proceedings of the Conference of the Institute of Statistics
Samuelson P.A.(1968)
Lifetime portfolio selection by
dynamic programming
Rev. Econ.Statist. II pp239-246
