Introduction
Stationarity is fundamental to the use of standard statistical
techniques to forecast expected future outcomes. However, it is a topic
that receives surprisingly little mention within trading
literature. This bulletin shares some thoughts on the importance of
considering stationarity when developing and analysing trading
algorithms.
Definition
The term stationarity is used in different ways in different contexts.
Where it is encountered in economics it is frequently used in relation
price. In this context a stationary market is effectively non-trending.
However, for a broader definition a stochastic variable can be
described as
weak-sense stationary if the following properties are time invariant:
1) The mean.
2) The variance
3) The auto-correlation structure.
Hence, the standard random walk model of stock prices is stationary in
that the distribution of log returns has a mean of zero,
constant
variance and auto-correlation that is zero across all time.
Stationarity and Markets
Stationarity is an implicitly assumed in many applications
where statistics are used to model future expected outcomes.
Fixed
physical systems or gambling games with fixed rules are expected to
have stationary statistics. In these cases, the expected value of
future outcomes can be modelled based on the prior distribution. Market
prices are the result of the individual actions of a large number of
market participants. The number of participants and their
behaviour can change significantly with time and so stationarity cannot
be assumed. However, fundamental economic factors will act to
stabilise market behaviour in the medium to long term. In the short
term, market statistics change significantly during the
trading
session. Although there is no reason to expect stationarity in
financial
markets the use of summary statistics and standard statistical
techniques is common place in trading.
For example:
1) Summary statistics for trading systems
2) Montecarlo draw-down analysis
3) Position sizing algorithms
4) VAR based risk analysis
However, in the absence of of
anything better, standard
statistical techniques are likely to find continued
application in trading.
What to do in a changing world?
Methods of coping with the additional uncertainty that arises
from non-stationarity fall into three
main categories:
1) Fudge factors.
2) Re-optimisation.
3) Adaptive algorithms.
Examples of fudge factors include expecting draw-downs twice as large
as historical maximum draw-down and fudging optimal-f to make it safe.
For risk analysis, VAR has a fudge factor built into it's
definition. The failure of trading systems to cope with changing market
statistics frequently leads to re-optimisation. Walk-forward testing
has developed from the technique of regular re-optimisation to help
analyse system stability.
Adaptive Algorithms
The design of adaptive algorithms attempts to either model or
compensate
for the non-stationary behaviour of markets. Trading systems may
include a range of linear and non-linear algorithms to adapt
trading patterns according to the recent market statistics. Simple
examples
include volatility based position sizing and volatility based indicator
look-back periods. A range of statistical techniques is also available
to model or remove non-stationary behaviour. For example the GARCH
model attempts to model stochastic volatility and
includes mean-reverting characteristics.
The figure shows the arrangement of a
trading system designed to
compensate for non-stationary market behaviour. The non-linear
trading
algorithm is designed to exploit a predictable market inefficiency
present in the market time-series data. In practice, the inefficient
behaviour will be a small part of the overall efficient price
behaviour. Since the the broad market behaviour is non-stationary, the
trading algorithm is made adaptive to compensate. The output
of
the
trading algorithm is a sequence of orders which could be
executed
in the
market. The profit distribution of notional trades at this
point
should
be stationary with a positive mean. However, orders at this
point
are normalised for constant return. A position sizing algorithm is
used to scale the normalised orders to achieve optimum safe equity
growth.
In this arrangement, the effectiveness of the adaptive algorithm at
ensuring stationarity of the trade distribution has a direct impact on
overall profitability. Residual non-stationarity of the trade
distribution requires that the position sizing algorithm is backed off
to maintain safe limits on draw-down.
Conclusion
Unlike gambling games, markets can not be expected to have stationary
statistics. Standard statistical techniques are still useful but
allowance needs to be made for greater uncertainty in expected
outcomes. Adaptive algorithms can be used to compensate for
non-stationary behaviour and optimise equity growth.
