An intelligent statistical arbitrage trading system
Nikos S. Thomaidis1? and Nick Kondakis1,2
1 Management & Decision Analysis Laboratory, Dept. of Financial Engineering & Management,
University of the Aegean, 31 Fostini Str., GR-821 00, Chios, Greece, tel: +30-2271-0-35454 (35483), fax:
+30-2271-0-35499, {nthomaid, kondakis, g.dounias}@fme.aegean.gr, URL:
http://decision.fme.aegean.gr
2 Kepler Asset Management, 100 Wall Street, New York, NY 10005, nick@kondakis.com
Abstract. This paper proposes an intelligent combination of neural network theory and fi-
nancial statistics for the detection of statistical arbitrage opportunities in specific pairs of
stocks. The proposed intelligent methodology is based on a class of neural network-GARCH
autoregressive models for the effective handling of the dynamics related to the statistical mis-
pricing between relative stock prices. The performance of the proposed intelligent trading
system is properly measured with the aid of profit & loss diagrams, for a number of different
experimental settings (i.e. sampling frequencies). First results seem encouraging; neverthe-
less, further experimentation on the optimal sampling frequency, the forecasting horizon and
the points of entry and exit is necessary, in order to achieve highest economic value when
transaction costs are taken into account.
Keywords: statistical arbitrage, intelligent trading systems, neural networks, GARCH models.
JEL codes C14, C22, G11
1 Introduction
In the last few years, a substantial amount of computational intelligent methodologies have been applied to
the development of financial forecasting models that attempt to exploit the dynamics of financial markets.
A great majority of intelligent approaches employ a network learning technique, such as feedforward, radial
basis function or recurrent NN [13, 17], although certain paradigms such as genetically-evolved regression
models [5, 8, 11, 14] or inductive fuzzy inference systems [9] are also encountered in the literature. Forecasting
experience has shown that predictability in data can be increased if modelling is directed to a combination
of asset prices rather than (raw) individual time series. Combinations can be seen as a means of improving
the signal-to-noise ratio and hence enhancing the predictable component in the data[3].
Statistical arbitrage is an attempt to profit from pricing discrepancies that appear in a group of assets.
The detection of mispricings is based upon the identification of a linear combination of assets, or else a
“synthetic” asset, whose time series is mean-reverting with finite variance. For example, given a set of assets
X1,...,Xn, a statistical mispricing can be considered as a linear combination ω = (w1, w2, ..., wn) such that
w1X1 + w2X2 + · · ·+ wnXn ∼ mean reverting (0, σ2t ), σ2t <∞
? This research is funded by the Public Benefit Foundation “Alexander S. Onassis” and by a grant from
“Empeirikion” Foundation.
2
where in the above definition we also allow for time-dependent volatility σ2t . The vector ω represents actual
proportions of each asset to be held in the trading portfolio (the minus sign means selling the corresponding
asset). The requirement of mean-reversion is to ensure that mispricings eventually “die out” and do not
grow indefinitely. If they did, it would be impossible to control the risk exposure of the trading portfolio.
The standard approach to identify statistical mispricings is to run a regression of the values of one asset,
say X1t, against the others X2t, ..., Xnt and test the residuals for mean-reversion. Several tests have been
developed for this purpose in the econometric literature, the most famous of which are the Dickey-Fuller
and Phillips-Perron (see e.g. [6]). Note that the residuals of the regression model represent the mispricing at
each time t of X1t relatively to {X2t,...,Xnt}. The next step is to create a model that describes the dynamics
of mispricings, i.e. how errors of different magnitude and sign (positive/negative) are corrected over time.
To take advantage of predictability, price forecast need to be incorporated into a dynamic trading strategy.
An arbitrage trading system identifies the “turning points” of the mispricings time-series and takes proper
positions on the constituent assets when mispricings become exceptionally high (i.e. ω for a positive and -ω
for a negative mispricing). An arbitrage strategy as described above is not without risk; although profitable
in the long run, its instant profit depends heavily on the ability of market prices to return to the historical
or predicted norm within a short period of time. Generally, the weaker the mean-reversion the higher the
probability of observing adverse movements of the synthetic.
Several authors have suggested approaches that attempt to take advantage of price discrepancies by
taking proper transformations of financial time-series; see e.g. [2, 3, 16] for stocks of FTSE 100 Stock Index,
[4, 10] for equity index futures and [12] for exchange rates. Amongst them, [3, 4, 12] employ a neural net-
work model to describe the dynamics of statistical mispricings. In this paper, we propose a new intelligent
methodology for the identification of statistical arbitrage opportunities. Our approach deviates from the
main trend in that it attempts to detect nonlinearities both in the mean and the volatility dynamics of the
statistical mispricing. For this purpose, we use a newly proposed class of combined neural network -GARCH
volatility models. The methodology is applied to the detection of statistical arbitrage opportunities in a pair
of two Indian stocks.
The rest of the paper is organised as follows: In section 2 we describe the application data, including
intraday quotes of stock prices. Section 3 presents the methodology for detecting price discrepancies between
stocks and section 4 details the NN-GARCHmodel used to forecast the dynamics of the statistical mispricing.
In section 5 we present two arbitrage trading systems based on a high- and low-frequency predictive model.
Section 6 concludes the paper and discusses directions for further research.
2 Sample data
For the application and testing of the trading strategy we chose the stocks of Infosys Technologies Ltd and
Wipro Ltd, both Application Software companies from the Indian stock market. We did so for two reasons:
1. We plan to further deploy the system onto a larger set of stocks with sector neutrality so we chose
two active names from the Technology/Software sector. Choosing stocks from the same industry sector
usually results in better mean-reversion behavior. In addition, both companies have active ADR issues
in the US which adds some interesting interactions and influences.
2. We are in the process of investigating the extension of statistical arbitrage equity strategies into devel-
oping and emerging markets. We are also interested in studying in detail the execution intricacies of the
various markets and thus we will be paying special attention to trading costs in follow-up work.
Both stocks trade in the National Stock Exchange of India, headquartered in Mumbai, India. The NSE
trading system is a fully automated order-driven market. The market trades from 09:55 to 15:30 hours with
3
a closing session held between 15.50 hours and 16.00 hours. We are using tick-by-tick data, time and sales
as well as best bid and offer and corresponding sizes for the historical period from February 1, 2005 until
November 8, 2005. We have run a specialized filtering algorithm to remove any spurious trades that deviate
significantly from the actual market. Subsequently the tick data is consolidated into 1-minute bars that
include the open, high, low and close price, the total share and tick volume and the VWAP price. We have
appropriately adjusted the price and volume data for dividend and split actions. We plan to use the tick
data information to work out our trading-cost models in subsequent studies.
3 Identifying statistical mispricings
Figure 1 shows hourly closing prices of Infosys and Winpro from February 2 to November 83. As a first
attempt to construct a synthetic asset, we ran a regression of Infosys against Winpro, hence forth X2
and X1 respectively, over the first 200 sample observations and we then used the regression coefficients to
compute the statistical mispricing. The resulting series is depicted in figure 2. Observe that the estimated
combination is weakly mean-reverting especially in the first 600 observations. The Phillips-Perron (PP)
test statistic over the whole sample period is -2.0183, which is below (in absolute terms) the 1, 5 and 10%
critical levels (-3.8803, -3.3585 and -3.0380 respectively). Hence, the hypothesis of mean reversion cannot be
accepted.
Dec04 Feb05 Mar05 May05 Jul05 Aug05 Oct05 Dec05
1500
2000
2500
3000
Dec04 Feb05 Mar05 May05 Jul05 Aug05 Oct05 Dec05
200
300
400
500
NINFO NWPRO
Fig. 1. Hourly data of Infosys and Winpro from 02/02/2005 to 08/11/2005.
3 Prices in this diagram are adjusted for splits and dividends.
4
0 200 400 600 800 1000 1200
−60
−40
−20
0
20
40
60
Time (trading hours)
m
is
pr
ici
ng
Fig. 2. The synthetic asset constructed from a static regression.
In order to control the non-stationarity of the synthetic asset, we adopt an adaptive estimation scheme
in which the coefficients of the linear combination are periodically re-calculated. In particular, we define the
mispricing as
Zt = X2,t − αt−1 − βt−1X1,t
where αt, βt are estimated on the basis of a window of length W , {X1,j ,X2,j , j = t−W + 1, ..., t}. Instead
of using linear regression, we adopt a slightly different procedure for calculating α’s and β’s: we define β as
the mean price ratio between the two stocks over the specified window and we subsequently choose α so as
to minimize the total variation of Zt within the window, i.e.
βt = mean (X2j/X1j , j = t−W + 1, ..., t)
and
αt = mean(X2j , j = t−W + 1, ..., t)− βtmean(X2j , j = t−W + 1, ..., t)
This procedure has been experimentally found to give more reasonable estimates of the synthetic vector
that also show more stability over time. In figure 3 we show the synthetic time series resulting from a different
choice of the window length. Observe that the more often the values of α and β are updated, the stronger is
the mean-reversion of the synthetic and hence the more abrupt are the corrections of mispricings. All series
are found mean-reverting; the PP test statistic over the whole sample is -10.0852 for W = 10, -6.4869 for
W = 30 and -4.5622 for W = 100, which are above common critical levels. In subsequent experiments, we
report results obtained for a synthetic calculated on the basis of a window of 10 observations.
4 Modelling the dynamics of the statistical mispricings
To describe the dynamics of the statistical mispricings we use autoregressive models relating the current value
of Zt to its own lags. This gives us an idea of how mispricings of different size and sign (positive/negative)
5
0 200 400 600 800 1000
−30
−20
−10
0
10
20
30
Window length=10
0 200 400 600 800 1000 1200
−30
−20
−10
0
10
20
30
Window length= 30
0 200 400 600 800 1000 1200
−30
−20
−10
0
10
20
30
Window length=100
Fig. 3. The synthetic time series obtained from an adaptive estimation scheme for a window length of 10,
30 and 100, respectively.
are corrected over time. We also go one step further to model both the mean and the volatility structure
of the statistical mispricings. This is because in high sampling frequencies (intra-day data), we find that
the volatility of Zt (i.e. the average uncertainty about the realised value) is not constant over time but
strongly depends on the history of Zt. In particular, large (positive or negative) shocks to Zt are on average
followed up by large shocks of either sign. This “clustering” of volatility, typical in most financial time
series, is termed in the literature as Autoregressive Conditional Heteroskedasticity (ARCH). Any changes in
the short-term volatility level of Zt deserve special attention from a modelling point of view, as they have
important implications for the risk control of the statistical arbitrage. Until today, the most popular models
for the volatility dynamics of a time series are the class of ARCH/GARCH models[1, 7]. A GARCH model
shows how a sudden negative or positive mispricing affects the future volatility of mipricings.
In our approach, we attempt to model both nonlinearities in the correction of mispricings as well as
volatility spill-overs. For that purpose, we use a recently proposed class of joint neural network-GARCH
models[15] that is intended to capture both effects. In this framework, an autoregressive model for the
6
conditional mispricing takes the general form:
Zt = φ0 + φ
′Zt + f(Zt; θ) + ²t (4.1a)
²t|It−1 ∼ N
�
0, σ2t
(4.1b)
σ2t = α0 +
p∑
i=1
αiσ
2
t−i +
q∑
j=1
βj²
2
t−j (4.1c)
where φ0 ∈ R, Zt is a vector of lags of Zt (Zt = (Zt−1, Zt−2, ..., Zt−n)′ ∈ Rn), ²t is the innovation term,
φ ∈ Rn and f(Zt; θ) is a feedforward neural network with a single hidden layer and l neurons, i.e.
f(Zt; θ) =
l∑
j=1
λjF (w
′
jZt − cj)
where F (z) = 1
1+e−z is the logistic function, cj , λj ∈ R and wj ∈ Rn. With It−1 we denote the information
available up to time t, including the history of mipricings Zt, shocks ²t and volatilities σ
2
t . Note that in the
above specification, ²t is assumed conditionally normally distributed with volatility σ
2
t that depends on past
σ2t ’s as well as (the magnitude of) past unexpected mispricings. Equation (4.1c) is called a GARCH(p,q)
model.
In [15] we propose a complete model-building cycle for NN-GARCH models that comprises the following
stages: a) specification of the model (e.g. determining the number of neurons in the hidden layer, the con-
nections from inputs to hidden neurons, etc), b) estimation and c) in- and output-of-sample evaluation. This
is a simple-to-complicate modelling approach that starts from a linear model and gradually complicates the
specification if the data indicate so. The procedure is roughly as follows:
1. Estimate a linear model with no GARCH component and choose the optimal set of input variables by
means of AIC of SBIC.
2. Test the null hypothesis that the true data-generating process is a linear model against the alternative
of a neural network model with a single hidden neuron. If linearity is not rejected at a given confidence
level then stop. Otherwise, estimate a NN model with a single neuron and test it against a NN model
with an additional neuron. Repeat the above procedure until first acceptance of the null.
3. Once the mean model (4.1a) is specified, test the null hypothesis of no GARCH effects in the volatility
of the residuals of the model against the hypothesis that residuals follow a GARCH process of a given
order. If null is not rejected then stop. Otherwise, jointly estimate a NN-GARCH model.
There are two important things to note about the above procedure. First, the decision of whether to add
an extra neuron or not is not based on heuristic arguments but rather on statistical hypothesis testing of
“neglected nonlinearity”. Such tests are available in the framework of Maximum Likelihood Theory. Second,
the procedure presented above does not directly estimate a NN-GARCH model but chooses the most suitable
specification given the data complexity (linear or nonlinear, with or without a GARCH component). Hence,
it is aimed to produce non-redundant models that are less likely to overfit the data.
5 Application: statistical arbitrage between the two Indian stocks
Our methodology for exploiting statistical arbitrages consists of the following steps:
1. Constructing a “synthetic asset” and testing for mean-reversion in the price dynamics. Synthetics are
calculated for various sampling frequencies.
7
2. Modelling the mispricing-correction mechanism between relative prices. For this purpose we use the
general framework of NN-GARCH models (4.1).
3. Obtaining 1- and h-step-ahead forecasts for the future value of the mispricing. Forecasts are given in
the form of a conditional probability density from which confidence bounds on the future value of the
mispricing are derived. The estimation of an h-step ahead conditional density is based on the simulation
of 800 error scenarios. Errors are calculated as
√
htut, where ut are sampled with replacement from
model’s standardised residuals (the residuals divided by the estimated variance obtained by the GARCH
model). In that way, we avoid restrictive assumptions on the distribution of the error-generating process.
4. Implementing a trading system to exploit the predictable component of the mispricing dynamics. The
trading strategy is roughly as follows: buy (long) the synthetic asset if Zt < Zˆ
L,α
t+h and sell (short) the
synthetic asset if Zt > Zˆ
H,α
t+h , where Zˆ
L,α
t+h Zˆ
H,α
t+h denote the (1− α)% low and high confidence bound on
the value of the mispricing h steps ahead4. In our approach the confidence interval is a decision variable,
which has to be adjusted so that profit is maximized.
5.1 Experiment A: a high-frequency model
In this section, we give an example of a microscopic model, which is calculated from 1-minute closing prices
of stocks. The values of the synthetic from August 2 to 19, a total of 5000 observations approximately, are
used for the specification of the mispricing-correction model and the sample observations from August 29 to
September 22 for out-of-sample evaluation. In this sampling frequency, we find both nonlinear corrections
of mispricings and ARCH volatility effects. The specified NN-GARCH model includes lags 1-10 in the linear
part, 1 hidden neuron with variable Zt−10 connected to it and a GARCH(1,1) equation.
We report results from a trading system that is based on 5-minute-ahead forecasts. Figure 6 shows the
performance of arbitrage trading strategies by varying the confidence level. Observe that as the interval gets
narrower (1 − α is decreased) the accumulated profit becomes higher, although the number of trades put
in the synthetic is almost exponentially increased. Hence, the average profit per trade gets lower. Figure 7
shows several trading instances of a system with bounds set at 40% confidence. Circles represent entering
points of trades. Note the effect of the GARCH component of the model, which is to dynamically adjust
the confidence bounds, or else the uncertainty about the future realised value, whenever large unexpected
mispricings occur. This in general prevents trading in periods of high volatility and risk (see e.g. the first
50 observations of the lower “snapshot” of figure 7). In figure 4 we give the profit& loss diagram of the
afore-described trading system for the period September 23 - October 11. The total number of trades is 823
and the average trade per profit is 78.793/823=0.096. Although from a forecasting point of view the system
is statistically correct, it still remains unprofitable as the average profit per trade is below the transaction
costs, which in this market are of the order of 3 rupies. The profitability of this high-frequency system is
severely limited by the fact that positions are necessarily closed at the end of the 1-minute interval5. It is
important to note that keeping a trade open for a time interval grater than 1 minute is equivalent to not
adjusting α’s and β’s until the trade is closed. The final outcome of such trades is strongly based upon how
well the two synthetic time series, the every 1-min adjusted and that calculated from unchanged estimates
of α’s and β’s, locally resemble each other. It is certain that as the values of the mispricing calculated from
unchanged estimates of α’s and β’s have not been “seen” by the model, the performance of op
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