advanced-algorithmic-trading

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I would like to thank you for being patient so far as it might seem that these chapters on
time series analysis theory are far away from the "real action" of actual trading. However true
quantitative trading research is careful, measured and takes significant time to get right. There
is no quick fix or "get rich scheme" in quant trading.
We are very nearly ready to consider our first trading model, which will be a mixture of
ARIMA and GARCH. Hence it is imperative that we spend some time understanding the ARIMA
model well.
Once we have built our first trading model we are going to consider more advanced models in
subsequent chapters including state-space models (which we will solve with the Kalman Filter)
and Hidden Markov Models, which will lead us to more sophisticated trading strategies.
11.2 Autoregressive Integrated Moving Average (ARIMA)
Models of order p, d, q
11.2.1 Rationale
ARIMA models are used because they can reduce a non-stationary series to a stationary series
using a sequence of differencing steps.
We can recall from the previous chapter on white noise and random walks that if we apply
the difference operator to a random walk series {x t } (a non-stationary series) we are left with
white noise {w t } (a stationary series):
∇x t = x t − x t−1 = w t (11.1)
ARIMA essentially performs this function but does so repeatedly d times in order to reduce
a non-stationary series to a stationary one. In order to handle other forms of non-stationarity
beyond stochastic trends additional models can be used.
Seasonality effects such as those that occur in commodity prices can be tackled with the
Seasonal ARIMA model (SARIMA). Unfortunately we will not be discussing SARIMA in this
book. Conditional heteroskedastic effects such as volatility clustering in equities indexes can be
tackled with ARCH and GARCH, which we will discuss later in this chapter.
We will first consider non-stationary series with stochastic trends and fit ARIMA models to
these series. We will also finally produce forecasts for our financial series.
11.2.2 Definitions
Prior to defining ARIMA processes we need to discuss the concept of an integrated series:
Definition 11.2.1. Integrated Series of order d. A time series {x t } is integrated of order d, I(d),
if:
∇ d x t = w t (11.2)
That is, if we difference the series d times we receive a discrete white noise series.