Thesis booklet for Ph.D. dissertation Farhad Kia
Supervisor:
Dr. Levendovszky János, D. Sc.
Doctor of the Hungarian Academy of Sciences
Department of Networked Systems and Services Budapest University of Technology and Economics
1
1. Introduction and scope of the theses
In the advent high speed computation and ever increasing computational power, algorithmic trading has been receiving a considerable interest [1-5]. The main focus of recent research has been to develop real- time algorithms which can cope with portfolio optimization and price estimation within a very small time interval, enabling high frequency, intraday trading. In this way, fast identification of favorable patterns on time series becomes feasible on small time scales, as well, which can give rise to profitable trading where asset prices follow each other in a sec or msec range.
As a result, the present work deals with the problem of finding the best portfolio based on different objective functions, e.g. minimizing the risk or maximizing the predictability by using different stochastic models (e.g. mean revering or Levy processes)
2. Technological motivations and existing results
In this section, an outline of the motivations and practical applications of the selected problems is given, as well as a brief presentation of the existing research results.
2.1 Optimal min loss portfolio selection
Portfolio optimization was first investigated by Markowitz in the context of diversification to minimize the associated risk and maximize predictability[6].
Since the first results, many papers have been dealing with portfolio optimization [7-11], e.g. one of the usual approaches is finding the portfolio which exhibits predictability and minimal risk [12-14]. Another approach is to identify mean reverting portfolios [15-19], where trading actions (e.g. buying or selling) are launched when being out of the mean and complementary actions are taken (e.g.
selling or buying) after reverting to the mean. The traditional strategies are concerned with optimizing the mean reverting parameter of predictability which lends itself to analytical tractability by solving an eigenvalue problem[11].
However, these approaches fail to provide high profits in the presence of bid ask spread or using a large number of assets due to the transactional cost. Therefore, one has to use more sophisticated objective functions, models and trading strategies. In the first thesis group I try to find the best portfolio based on the same objective function (minimizing the probability of loss) but I extended the stochastic models of the underlying time series assuming, Levy model, as well.
2.2
The prediction of financial market indicators is a topic of considerable practical interest [20-29]and, if successful, may involve substantial pecuniary rewards. Neural networks have been used for several years in the selection of investments because of their ability to identify patterns of behavior that are not readily observable. Much of this work has been proprietary for the obvious reason that the users want to take advantage of the insight into the market they gained through the use of neural network technology [23, 30]. As a result, in the second thesis group I focus on estimating the future price of the time series by using an appropriate nonlinear predictor.
3. Models and methods used in the research
In order to develop efficient prediction algorithms for financial time series, I used several models and methods which are outlined in this chapter.
For solving the optimal portfolio selection problem I proposed a novel method which aims at minimizing the loss probability based on two different underlying models which are mean reverting model[11, 31] and Levy model[32, 33]. The traditional strategies are concerned with optimizing the mean reverting parameter of predictability which lends itself to analytical tractability by solving an eigenvalue problem. In this paper I assume that the multidimensional asset price vector out of which the portfolio is to be constructed can be modeled by a VAR (1) process. Then portfolio selection can be broken down in two steps: (i) fast model identification based on the previously observed samples of the corresponding asset prices and; (ii) choosing a portfolio vector which minimizes the probability of negative return. My related results can be found in [J2, J3, and C2].
For solving a prediction based trading strategy I used a suitable nonlinear estimator for predicting the future values of a financial time series provided by a properly trained Feed Forward Neural Network (FFNN) which can capture the characteristics of the conditional expected value. This predicted value can be used directly or it can be used as an estimation of a short term mean of mean reverting model. My related results can be found in [J1, C1].
The implementations are performed on MATLAB [34]and on Metatrader [35]which could give us the opportunity to perform both back-testing and forward- testing and obtaining more promising and reliable results.
3 Models used
VAR(1) model
Ornstein-Uhlenbeck model
Levy Model
Stationary time series
Methods applied
Modeling(Feed forward neural network)
Identification algorithms(Maximum likelihood estimation, Moore-Penrose pseudo inverse)
Optimization methods (Exhaustive search, greedy search, Simulated annealing)
Learning Algorithms(Back-
propagation(BP) which minimizes the mean squared error (MSE))
Validation
Numerical simulations on real historical data from Forex in MATLAB and Mt4
4. New scientific results
In this chapter I summarized my new scientific results.
Thesis group I. – Portfolio optimization subject to novel objective functions I developed a novel trading strategy and portfolio selection which aims at minimizing the loss probability based on identifying mean reverting portfolios.
After observing historical data for parameter identification, the portfolio selection is performed by minimizing the probability of negative return (loss). Furthermore I developed a new method for calculating the mean and mean reversion parameter as an alternative for traditional way of calculating these parameters. Finally I implemented the proposed method on GPU.
In order to shed light on these results, I will focus on mean reverting portfolios, especially sparse portfolios (in which only a few numbers of assets are used for minimizing the transaction cost).
Following the construction of d’Aspremont [11]and many other experts, we view the asset prices as a stationary, first order, vector autoregressive VAR (1) process. Let si t, denote the price of asset i at time instant t, assume that
1, ,
( ,..., )
t st sn t
s is subject to a first order vector autoregressive process, VAR(1), defined as follows:
1 ,
t t t
s As W (1.1)
Where A is of type nxn matrix and Wt N(0,I) is generated by independent and identically distributed (i.i.d.) noise terms. We can estimate the A as follows:
1 1 1
2 2
ˆ ,
m m
T T
t t t t
t t
A s s s s (1.2)
Mean reverting portfolios can be described by the Ornstein-Uhlenbeck stochastic differential equation
( ) ( ( ) ) ( ),
dP t P t dt d W t
(1.3)Where
P t ( )
is the value of the portfolio, defined as( )
i i t,i
P t n s
wheren
is our portfolio vector.Following the treatment in (Box, G.E., Tiao, G.C., 1977) [36]and (D’Aspremont, 2011), we calculate the predictability of the portfolio (
) asˆ ˆ
T T
n AGA n
Tn Gn
(1.4)
Where
n
is our portfolio vector andG
is the stationary covariance matrix of processs
t.We obtain the following estimate for using Aˆ from (1.5):
2 1 2
1 ˆ
ˆ 1
m
t t
n m t
s As (1.6)5
Where
m
is the length of our observation. For calculating the
, I used sample mean.The solution of (1.3) is given as
( )
0
( ) (0)
t(1
t)
t t s( )
s
P t P e
e
e
d W s
(1.7)I.1. I expressed the new objective function so called the Probability of loss
(P )
L for portfolio selection as purchased portfolio worths less than a financial instrument with the expected returnr
for the given timet ,
This probability can analytically be derived by:
2
1 1 (0)
sgn( (0)) 1 (1 ) (1 ) ,
2 2 (0)
t t t
L t
P P erf P r
e e P e
(1.8)
where
2
0
( ) 2 .
x
erf x e
tdt
(1.9)I.2. I have developed a new numerical method for estimating the optimal time interval during which the portfolio should be held.
It can be seen that the error function is a monotonously increasing function, i.e.
minimizing (1.7) is the same as minimizing/maximizing the argument of the erf function (1.8). Here we have two important parameters, introduced as follows:
As a result, r should be set to zero since it is the lowest possible value in (1.7).
The more r deviates from zero, the larger probability of loss becomes. Thus, there is no sense to optimize the value of r.
However if r is fixed, the investor might be interested to find an optimal time frame for which the portfolio should be held. For the sake of the loss optimization
t
optshould be chosen subject to the following formulaarg min ( (0), , , , , ),
opt
max
t
t f P r t
(1.10)As an example, Figure 2 shows the case, where P(0) 1, 2,10,r0.1,1.
Both axes are given in logarithmic scale. As one can compute, (9) yields
t
opt= 0.4609.
The figure shows well how the curve behaves, starting from0.5
and tending to
1
yielding a global minimum atlog (0.4609)
10 0.3364.
Figure 1. An example with the parameter setP(0) 1,
2,
10,r0.1,
1.I.3. I have developed a portfolio selection trading strategy based on minimizing the loss probability for mean reverting model.
I assume that parameters
,
could be derived for all possible portfolios and the investor defines a time framet
which limits the holding time of the portfolio and an expected returnr .
Finally, the best portfolio must be chosen, where the „best” means the one with the lowest probability of loss.
7
Figure 2. The process of portfolio selection based on loss probability minimization.
My new „loss-probability-minimization” strategy has been tested on FOREX data series. For the sake of comparable performance, I tested the lambda maximization strategy against the proposed method. In my comparisons I tried to be fair: I did not apply any sophistication, or plug-in that to optimize the models.
Figure 3. The performance of the two algorithms – Equity in case of existing stop loss and in case of no-stop loss
The poor performance of the traditional maximum lambda strategy is due to the fact that there are many positions closed in loss, since the stop loss limit is hit. For fair comparison we also show the performance, when stop loss limit is switched off in figure 5. Although the strategy “maximizing lambda” is better without stop loss in Figure 5, one should not forget that the lambda maximization algorithm yields worse balance in the flat periods, when the equity remains constant. That is, if there is a leverage applied, this strategy could easily yield bankruptcy.
I.4. I have applied a new method for estimating the mean and
parameter of mean reverting model which can be calculated without estimating the mean reversion basic parameters such asA
andG
in traditional way.Assume that prompt prices are available, so we can measure Portfolio value P (t ), where K is the number of points available for the calculation. In other words, the length of the time window, where measured results are available, equals K. ∆t .After a few pages of analysis one can get,
1 2 1
2
2 1 1 1
ˆ ,
( ) S S S S
µ S S S S S
(1.11)and
2
1 1
2
2 1
1 ln
2
S S S
t S S
(1.12)Where
1 1 1
0
1 1
2 2
0
2 2
1 1
0
1 ( . )
1 ( . )
1 ( ( . ))
1 ( ( . ))
1 ( . ) (( 1). )
K
i K
i K
i K
i K
i
S P i t
K
S P i t
K
S P i t
K
S P i t
K
S P i t P i t
K
For the sake of comparable performance, I tested the old traditional and common way of
and
calculation against these novel methods. In my comparisons I tried to be fair: I did not apply any sophistication, or plug-in that to optimize the models.9
Figure 4. The performance of traditional parameter estimation and proposed parameter estimation method.
The numerical results obtained on FOREX data have been demonstrated that higher profit can be achieved by the traditional parameter estimation than the proposed parameter estimation. However one shouldn’t forget that the complexity of the proposed method is much lower than the traditional method.
I.5. I implemented the proposed method on GPU.As I used an exhaustive search for portfolio selection therefore having big pool of portfolios which, in turn, makes this algorithm cumbersome and slow. Using GPGPU can solve this problem.
Using the new proposed methods for calculating the mean and
parameters, I implemented the min loss portfolio selection using GPGPU.Figure 5. The performance of CPU based implementation and GPU based implementation
I.6. I developed a new technique to obtain safer and smaller pool of portfolios than the fully hedged portfolio pool, which can be calculated much faster and could enhance the performance of the loss-minimal algorithmic trading.
Suppose the number of assets is denoted by n=1…..N. we have cardinality constraint denoted by k which should be small in compare to n (we want to have small transaction cost therefor we would like to choose only few of them).
The solution is first we pick one asset and then we try to find the best linear combination of (k-1) assets which gives us the best estimate of the chosen asset in a given period. With this asset and the linear combination which we achieved we can construct our portfolio.
The size of our new pool would be
! (k 1)!(n )!
n
k
.In this case we have much smaller pool of portfolios in compare to pool of all possible portfolios and also because we have only special portfolios which had smallest deviation in the past we can expect that they might be safer than the other portfolios.
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Figure 6. The performance of fully-hedged pool and the new pool
The numerical results obtained on FOREX data have been demonstrated that higher profit can be achieved by the new pool than simply using the fully-hedged pool.
I optimized portfolios assuming that the value of the portfolio follows a Levy process. First I identify the parameters of the underlying Levy process and then portfolio optimization is performed by maximizing the probability of positive return. The method has been tested by extensive performance analysis on Forex.
The sell price of a given asset
i
at a given time instantt
is denoted bys t
i
. The same asset can be bought at a price denoted byb t
i , b t
i s t
i ,
where( ) ( )
i i
b t s t
is the so-called bid-ask spread.An investor is supposed to have a combination of assets in which the numbers of different assets at hand are denoted by n( )t (which is a sparse integer vector
portfolio.
As a result, the prompt value of the portfolio held by the investor is given as ( ) max( , ( )) ( ) min( , ( )) ( ),
P t 0 nt st 0 n t b t (1.13)
where vector 0 refers to the all zero vector (each component is zero).
As a generalization of the Samuelson model, one can describe financial markets' security prices as Lévy processes. Lévy process X t( ) is characterized by the following properties:
The paths of X t( ) are right continuous, with left limits, i.e.,
0
, lim Pr X t ( ) X t ( ) 0.
The increments have identical distribution, i.e., for 0 s t X t, ( )X s( ) has the same distribution as
X t s .
( )
X t has independent increments, that is, for 0 s t X t, ( )X s( ) is independent of { ( ) :X u us}.
I.7. I introduced and implemented the methods for identifying and modeling the financial time series as Levy Process.
I will focus on the Levy jump process [37], which is constructed from the linear drift and the Poisson point process and given as
1
( ) (0) .
Nt
k k
P t P t J
(1.14)Here
is the drift parameter,{ } N
t is a Poisson point process, with mean N
t t ,
andJ
k is a sequence of independent, identically distributed discrete random numbers with zero mean, i.e., J
k 0
. In financial markets,J
k represents the jumps between different ticks in subsequent transactions.Under the conditions described above, the mean of (1.13) can be calculated as
13
1
1
( ) (0)
.
t
t
N k k N
k k
k
P t P t J
t J t t J t
(1.15)
Thus, the statistical mean of
P t ( ) P t ( ) /
yields parameter
, for anyT
1
1 ( ) (( 1) ) ( ) (0)
ˆ .
K
i
P iT P i T P KT P
K T KT
(1.16)In most cases
proved to be a very small number, hence we assume that its effect can be neglected in subsequent transactions i.e., t 0.
Since
{ } N
t is a Poisson point process, its mean,
can be estimated by the average number of transactions,
( 1)
( 1)1
ˆ 1 ,
K
K T T
i T iT
i
N N
N N
KT KT
(1.17)for any
T .
Finally, the statistics of
J
k must be simply counted by relative frequencies,1
ˆ 1 ( )
K
i k
k
p I J i
K
(1.18)where I(.) is the indicator function and
p ˆ
i estimates the probability J
k i .
I.8. I developed a portfolio selection strategy based on maximizing the probability of positive return after modeling the time series as Levy Model.
We assume that parameters, i.e.
p ,
could be derived for all possible portfolios and the investor defines a time framet
which limits the holding time of the portfolio and an expected return.1
1
(0) (1 ) (0) , if P(0), long position is taken,
(0) (1 ) (0) , if P(0),short position is taken.
t
t
N
t k k
L N
t k k
P t J r P
P
P t J r P
(1.19)We have considered only the first case, the other case can be similarly constructed (the limits of the sumwill change). Which can be rewritten as
1
(0) (1 ) 1
Nt
t
L k
k
P J P r t
(1.20)To calculate the probability of the portfolio value remaining under a certain limit, the discrete probability values must be summed up accordingly. Thus, this value can be calculated as
0
(0) (1 ) 1 ( ) 0
(0) (1 ) 1 ( )
· .
!
t
n t
L t k
n
P r t
t n
n i
n i
P N n J P r t
e t
n p
(1.21)
I.9. I have developed a portfolio selection trading strategy based on loss probability minimization for Levy model.
In our test we used 10 days of waiting time. We closed the trades at the end of each day if our profit was positive. If not, at the end of the next day the same comparison was done ....etc. After 10 days, the portfolio is closed if still remained in the negative profit region.
We did not use any leverage for back-testing. The results are indicated in Figure 7.
15
Figure 7. The performance of the proposed method in back-testing mode: balance and equity of the proposed method (Lévy) and the mean reverting algorithm.
As one can see from the figure, the profit was approximately 11%; the maximum drawdown was less than 5% in this three year period.
Thesis group II. New results for prediction based trading.
I developed a prediction based trading strategy using feed forward neural network. Furthermore I attached this predictor to a traditional mean reverting trading model as a filter which I show the practical viability of the proposed method.
Let us assume that we trade on the mid prices, the corresponding asset price time series is denoted by
x
nand follows a nonlinear AR(J) process
1,...,
n n n J n
x F x x (1.22)
where F is a Borel measurable function and
nN 0,
i.i.d.r.v.-s, being independent ofx
n.II.1. I used FFNN based prediction for trading on financial time series. The optimal trading strategy has been derived by using the fact that FFNN can represent the conditional expected value. Furthermore I extended the model to exploit the information of bid/ask.
For trading, I construct an estimator
1,..., , 1,...,
,
n n n J M
x Net x x w w Net x w (1.23)
where
, ( )L (L1)... (1) ...
n i ij nm n m
i j m
x Net x w
w
w
w x (1.24) is a Feedforward neural Network (FFNN) depicted by Figure 8 and vector w denotes the free parameters subject to training.Figure 8. The structure of feed forward neural network (FFNN).
In case I use mid-price for prediction but trade on Bid/Ask, The result is not so good. However, as expected this is due to the fact I have not exploited the information given in the bid and ask series.
17
Figure 9. Balance with respect to time (single asset) trading on Mid price and trading on Bid/Ask.
In the Simple Model our training set is constructed only by mid-price which is the only option in some conditions but in this Model, we use two separate networks(1.25,1.26) to include the bid-price which denoted by
x
and ask-price denoted byy
to construct our new training sets where
1,...,
n n n J n
x F x x (1.25)
1,...,
n n n J n
y F y
y
(1.26)In this case, I use following model to cover the spread.
x
k NetBid(y, x, w),x
k y
k1BUY(1.27)
y
k Net
Ask(y, x, u),
y
k x
k1SELL(1.28) In the figure bellow we have smaller number of trades on horizontal axis in comparison to previous cases because it might happen that the predicted value is not greater than Ask Price or it is not even less than Bid Price, therefore in some cases we do not trade. Again on the vertical axis we have account balance but this time we have some positive growth.
Figure 10. Balance with respect to time (single asset).
The achieved profit 1.23 %( Profit: $12.30) which is good in the presence of bid-ask spread. MAX Drawdown=0.85%. One can see that even in the presence of bid-ask spread the method can materialize profit.
II.2. I developed an effective filter based on proposed method which can be used with other trading strategies e.g. Mean reverting strategy to improve the performance of those strategies.
In figure bellow you can see how good this method improved the result of traditional mean reverting portfolio selection model with using this prediction based filter.
19
Figure 11. Performance of Mean reverting portfolio selection method with and without using the prediction-based filter.
II.3. I extended the model to predict more step away to better cover the spread and possibly extend the profit, as I let the price series change more dominantly to get out of the spread and materializing more profit. Here the goal is to find the optimal step parameter.
The step is defined as how many candles in the future we predict. The next figure shows the result regarding the step parameter, i.e. the account profit in percentage (optimization period is 6 months) is plotted as a function of the step parameter.
Figure 12. Profit as a function of prediction step.
II.4. I introduced a novel method to have better logical length for observation period and to have more important and meaningful points which can help us to avoid range market noises.
To find important points (IPs) I used Moving Average Convergence/Divergence indicator (MACD) and with the help of it I found the maximum value and minimum value in each MACD phase change. The procedure is indicated in Figure 13.
Figure 13. The Process of finding the important points (IPs).
Here as it is indicated in the Figure 13, the observation period is not static anymore and it depends on the number of IPs chosen which is based on the market condition.
Figure 14. Performance of dynamic vs. static observation period
21
As one can see from the figure 14, with dynamic observation period we can achieve higher and more stable results than using the static observation period.
5. Conclusions and applications of the results
In this dissertation I have examined two different approaches of computational finance: optimal portfolio selection based on min loss probability and prediction based trading. These methods were tested by back- and forward testing using MATLAB[34] and Meta trader[35] and their performance has indeed far outdone other traditional methods. In each case, I have managed to come up with novel approaches which
can capture a wild class of random portfolio value sequence,
The methods presented here can pave the way towards high frequency, intraday trading,
with these low drawdowns which we had, we can easily use bigger leverages to magnify our profit,
Furthermore, I managed to improve the model identification, as well. In the case of the sparse, mean reverting portfolio selection I have made significant improvements to the parameter estimation of the
and
. In the prediction based trading I introduced a novel method which make the observation period dynamic and more logical.Considering the above results, I have achieved the aims of the dissertation.
Finally, in each case I have implemented a proof of concept and have run extensive simulations on real world data. The results on real world data are convincing in each case.
6. Summary
The thesis was concerned with developing fast trading algorithms and portfolio optimization for implementing high frequency trading in real-time. The first thesis group dealt with portfolio optimization with extended objective functions over mean reverting processes. Namely, not only the predictability factor is maximized (leading to a generalized eigenvalue problem) but an analytical expression has been derived for the loss probability (the probability of the value of the portfolio hitting negative value). Based on this formula, a minimum risk portfolio can be selected by minimizing the loss probability. In this way, investment
been extended to Levy processes providing a more general framework for portfolio optimization. In order to perform optimization fast methods are worked out for the model-parameter identification, as well. The methods have been extensively tested on the Forex database.
The second thesis group treats the issue of prediction based trading.
Neural based prediction is developed to cope with the bid-ask spread.
The optimality is proven analytically, while the performance is tested on historical time series of Forex. As a general conclusion, de results developed in the thesis can pave the way toward less-risk trading and a safer financial world in general.
7. Acknowledgements
I am very grateful to my supervisor Dr. Levendovszky Janos and my mentor Dr. Jeney Gábor for their support and guidance throughout my work on this thesis and my research work. I would also like to thank Robert Sipos, Fogarasi Norbert, Attila Ceffer and Thai Hoc Nguyen for useful discussions and collaboration on some of the problems presented in the theses.
Many thanks also belong to my wife and my entire family who supported me while working on this thesis. It would have been very difficult without their support.
8. Publications of the author
Journal Publications
J1. Kia, F., Levendovszky, J. (2012) Prediction based – high frequency trading on financial time series. Periodica Polytechnica 56/1 (2012) 29–34 doi: 10.3311/PPee.7165
J2. Kia, F., Gábor, J., Levendovszky, J. (2014) Loss-minimal
Algorithmic Trading Based on Levy Processes. TEMJOURNAL. vol. 3, no.3, pp. 210-215, 2014
J3. Kia, F., Gábor, J., Levendovszky, J. (2014) Minimum Probability of Loss Trading Strategy for Mean Reverting Portfolios. International Interdisciplinary Journal of Scientific Research Vol. 1 No. 3
23 Conference Presentations
C1. Kia, F. (2013) Prediction Based-High Frequency Trading On Financial Time Series 5th International Conference on Neural Computation Theory and Applications, Portugal, Sep 2013
C2. Kia, F. (2013) Minimum Probability Of Loss Trading Strategy for Mean Reverting Portfolios, 10th International Symposium on Business Information Systems, Hungary, Nov 2013
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