=Paper=
{{Paper
|id=Vol-1941/MIDAS2017_paper3
|storemode=property
|title=Machine Learning for Multi-step Ahead Forecasting of Volatility Proxies
|pdfUrl=https://ceur-ws.org/Vol-1941/MIDAS2017_paper3.pdf
|volume=Vol-1941
|authors=Jacopo De Stefani,Olivier Caelen,Dalila Hattab,Gianluca Bontempi
|dblpUrl=https://dblp.org/rec/conf/pkdd/StefaniCHB17
}}
==Machine Learning for Multi-step Ahead Forecasting of Volatility Proxies==
https://ceur-ws.org/Vol-1941/MIDAS2017_paper3.pdf
Machine Learning for Multi-step Ahead
Forecasting of Volatility Proxies
Jacopo De Stefani, Olivier Caelen1 , Dalila Hattab2 , and Gianluca Bontempi
Machine Learning Group, Departement d’Informatique, Université Libre de
Bruxelles, Boulevard du Triomphe CP212, 1050 Brussels, Belgium
Worldline SA/NV R&D, Bruxelles, Belgium1
Equens Worldline R&D, Lille (Seclin), France2
{jacopo.de.stefani,gianluca.bontempi}@ulb.ac.be
olivier.caelen@worldline.com
dalila.hattab@equensworldline.com
Abstract. In finance, volatility is defined as a measure of variation of
a trading price series over time. As volatility is a latent variable, several
measures, named proxies, have been proposed in the literature to repre-
sent such quantity. The purpose of our work is twofold. On one hand, we
aim to perform a statistical assessment of the relationships among the
most used proxies in the volatility literature. On the other hand, while
the majority of the reviewed studies in the literature focuses on a uni-
variate time series model (NAR), using a single proxy, we propose here
a NARX model, combining two proxies to predict one of them, show-
ing that it is possible to improve the prediction of the future value of
some proxies by using the information provided by the others. Our re-
sults, employing artificial neural networks (ANN), k-Nearest Neighbours
(kNN) and support vector regression (SVR), show that the supplemen-
tary information carried by the additional proxy could be used to reduce
the forecasting error of the aforementioned methods. We conclude by
explaining how we wish to further investigate such relationship.
Keywords: financial time series, volatility forecasting, multi-step ahead
forecast, machine learning
1 Introduction and problem statement
In time series forecasting, the largest body of research focuses on the prediction
of the future values of a time series, with either a single or a multiple steps ahead
forecasting horizon, given historical knowledge about the series itself. In statisti-
cal terms, such problem is equivalent to the forecast of the expected value of the
time series in the future, conditioned on the past available information. In the
context of stock market, the solution to the aforementioned problem could allow
to determine the future valuation of a company, thus giving an information to
the traders about how to act upon such change in the valuation. However, from
the traders’ standpoint, price is not the only variable of interest.The knowledge
of the intensity of the fluctuations affecting this price (i.e. the stock volatility)
2 Machine Learning for Multi-step Ahead Forecasting of Volatility Proxies
allow them to assess the risk associated to their investment. Since volatility is not
directly observable given the time series, according to the granularity and the
type of the available data, one could compute different measures, named volatil-
ity proxies [21]. Although volatility proxies based on intraday trading data exist
[17], due to the restrictions on the access to such fine grained data, the rest of our
analysis will be focused on proxies for daily data. A standard approach to volatil-
ity forecasting, once a given proxy has been selected, is to apply either a statis-
tical Generalized AutoRegressive Conditional Heteroskedasticity (GARCH)-like
model [2], or to apply a machine learning model. In addition, several hybrid
approaches are emerging [16, 10, 19], including a non-linear computational com-
ponent into the standard GARCH equations. In all the aforementioned cases,
we deal with a univariate problem, where a single time series is used to predict
the future values of the series itself. An exception is represented by the work of
[30] where a volatility proxy is combined with external information (namely the
volume of the queries to a web search engine for a given keyword). This paper
proposes a method for multiple step ahead forecast of a volatility proxy incorpo-
rating the information from a second proxy in order to improve the prediction
quality. The purpose of our work is twofold. First, we aim to perform a statistical
assessment of the relationships among the most used proxies in the volatility lit-
erature. Second, we explore a NARX (Nonlinear Autoregressive with eXogenous
input) approach to estimate multiple steps of the output, where the output and
the input are two different proxies. In particular, our preliminary results show
that the statistical dependencies between proxies can be used to improve the
forecasting accuracy. The rest of the paper will be structured as follows: Section
2 will introduce the notation and provide a unified view on the different volatil-
ity proxies. Section 3 will introduce the formulation of the volatility forecasting
problem as a machine learning task and will described the different tested mod-
els. Section 4 concludes the paper with a discussion of the results and the future
research directions.
2 Volatility proxies: definition and notation
In this paper we consider univariate time series whose value at time t is denoted
by the scalar value yt . Let us consider the following quantities of interest, each of
(o) (c) (h) (l)
them on a daily time scale: Pt , Pt , Pt , Pt , respectively the stock prices at
the opening, closing of the trading day and the maximum and minimum value
for each trading day; vt being the volume 1 . We will assume the availability of a
training set of T past observations of each univariate series.
In the absence of detailed information concerning the price movements within
a given trading day, stock volatility becomes directly unobservable [27]. To cope
with such problem, several different measures (also called proxies) have been pro-
posed in the econometrics literature [21, 12, 20, 13] to capture this information.
However, there is no consensus in the scientific literature upon which volatility
1
Number of traded stocks in a given day.
Machine Learning for Multi-step Ahead Forecasting of Volatility Proxies 3
proxy should be employed for a given purpose. We will proceed by reviewing the
different types of proxies available in the literature: σ SD,n ,σ i and σ G .
Volatility as variance The first proxy corresponds to the natural definition of
volatility [21], that is a rolling standard deviation of a given stock’s continuously
compounded returns over a past time window of size n:
v
u n−1
SD,n
u 1 X
σ t =t (rt−i − r̄n )2 (1)
n − 1 i=0
where !
(c)
Pt
rt = ln (c)
(2)
Pt−1
represents the daily continuously compounded return for day t computed from
(c)
the closing prices Pt and r̄n represents the returns’ average over the period
{t, · · · , t − n}. In this formulation, n represents the degree of smoothing that is
applied to the original time series.
Volatility as a proxy of the coarse grained intraday information The σti
family of proxies is analytically derived in [12] by incorporating supplementary
information (i.e. opening, maximum and minimum price for a given trading day)
and trying to optimize the quality of the estimation.
The first estimator σt0 , which the authors propose as benchmark value, simply
consists of the squared value of the returns (i.e. the ratio of the logarithms of
the closing price time series):
" (c)
!#2
Pt+1
σt0 = ln (c)
= rt2 (3)
Pt
The second proposition σt1 is able to reduce the variance of the estimator,
by including the opening price, and computing a weighted average between two
components, representing respectively the nightly and daily volatility:
" (o)
!#2 " !#2
(c)
1 Pt+1 1 Pt
σt1 = · ln (c)
+ · ln (o)
(4)
2f Pt 2(1 − f ) Pt
| {z } | {z }
Nightly volatility Intraday volatility
The value of f ∈ [0, 1] represents the fraction of the trading day in which the
market is closed. In the case of CAC40, we have that f > 1 − f , since trading is
only performed during roughly one third of the day. In this case, the weighting
scheme proposed in (4) will give higher weight to the intraday volatility, with
respect to the nightly one.
4 Machine Learning for Multi-step Ahead Forecasting of Volatility Proxies
The third estimator, derived in [20] through the modeling of the price evo-
lution as a stochastic diffusion process with unknown variance, is a function of
the variation range (i.e. the difference between maximum and minimum value
for the current trading day):
" !#2
(h)
1 Pt
σt2 = · ln (l)
(5)
2 ln 4 Pt
1
where the value 2 ln 4 corresponds to the variance of the distribution of the
high-low displacement difference, under the assumption of a stochastic Wiener
process (i.e. with normally distributed increments).
Garman et al. [12] further improves the efficiency of the estimator in Equation
(5) by including the information concerning nightly volatility.
" (o)
!#2
a Pt+1 1−a
σt3 = · ln (c)
+ · σ̂2 (t) (6)
f Pt 1−f
| {z } | {z }
Intraday volatility
Nightly volatility
Here, a is a weighting parameter, whose optimal value, according to the
authors is shown to be 0.17, regardless of the value of f .
Furthermore, the same study introduces a family of estimators based on the
normalization of the maximum, minimum and closing values by the opening
price of the considered day. We can then define:
! ! !
(h) (l) (c)
Pt Pt Pt
u = ln (o)
d = ln (o)
c = ln (o)
(7)
Pt Pt Pt
where u is the normalized high price, d is the normalized low price and c is
the normalized closing price.
We can derive Equation (8), by starting from a general, analytic form for the
estimator, and then deriving the optimal values of the coefficient by minimizing
the estimation variance.
σt4 = 0.511(u − d)2 − 0.019[c(u + d) − 2ud] − 0.383c2 (8)
The values of the coefficients are set assuming that the price dynamics fol-
lows a Brownian motion and enforcing scale invariance properties and price and
time symmetry conditions. For all the details concerning the proof, we refer the
interested reader to [12].
Equation (9) is derived from Equation (8) by eliminating the cross product
terms.
σt5 = 0.511(u − d)2 − (2 ln 2 − 1)c2 (9)
Last but not least, the best estimator in terms of estimation variance effi-
ciency is obtained by combining the overnight volatility measure with the optimal
estimator described in Equation (8).
Machine Learning for Multi-step Ahead Forecasting of Volatility Proxies 5
(o)
!2
a Pt+1 1−a
σt6 = · log (c)
+ · σ̂4 (t) (10)
f Pt 1−f
| {z } | {z }
Intraday volatility
Nightly volatility
GARCH-based volatility Even though the GARCH(p,q) [14] (Generalized
AutoRegressive Conditional Heteroskedasticity) family of models is generally
employed for volatility forecasting, we decided to consider it here as a filter,
that, given the original time series, returns its estimation of the series volatility.
All GARCH models assume that the return time series can be expressed as
the sum of two components: a deterministic trend µ and a stochastic time-
varying component εt . The stochastic component can be further decomposed
and expressed as the product between a sequence of independent and identically
distributed random variables Zt with null mean and unit variance and a time
varying scaling factor σtG .
rt = µ + εt (11)
εt = σtG zt zt ∼ N(0, 1) (12)
The core of the model is the variance equation, describing how the residuals εt
and the σtG past volatility affects the future volatility.
v
u X p Xq
G
u G
σt = ω +t 2
βj (σt−j ) + αi ε2t−i (13)
j=1 i=1
The coefficients ω, αi , βj are fitted according to the maximum likelihood es-
timated procedure proposed in [4]. In the case of our proxies, we consider the
estimation of the volatility made by a GARCH (p = 1,q = 1) model as suggested
in [13].
3 Multiple step ahead volatility forecasting
The Nonlinear Auto Regressive (NAR) formulation of a univariate time series
as an input-output mapping allows the use of supervised machine learning tech-
niques for time series one-step-ahead forecasting [6].
y = f (x) + ω (14)
y = [yt+1 ] (15)
x = [yt−d , · · · , yt−d−m+1 ] (16)
To be more precise, this model assumes an autoregressive dependence of the
future value of the time series on the past m (lag or embedding order) values,
with a given delay2 d and an additional null-mean noise term ω.
2
In the following of the paper we will assume d = 0 for the sake of simplicity.
6 Machine Learning for Multi-step Ahead Forecasting of Volatility Proxies
With this structure, the forecasting task can be reduced to a two-step process.
First the mapping f between the inputs x and the outputs y is learned through
a supervised learning task, and then such mapping is used to produce the one-
step-ahead forecast of the future values.
Extensions of this technique allows to perform multiple-step ahead forecast
(i.e. y = [yt+H , · · · , yt+1 ]). Such extensions can be summarised into two main
classes: single output (Direct and Recursive strategies) and multiple output
(MIMO) strategies. The former learns a multi-input single output dependency
while the latter learns a multi-input multiple output dependency. We invite the
interested reader to see [24], [6], [25] for more details.
In what follows, we focus on two multi-step ahead single output learning
task, employing the Direct strategy [5], [23], [8]. In the first one (NAR), we will
focus on the multiple step ahead forecast of a primary volatility proxy σtP using
only its past values as input information, while in the second one (NARX), also
the past values of an additional volatility proxy σtX will be incorporated in the
model described in (14):
P P
yN AR = yN ARX = [σt+H , · · · , σt+1 ] (17)
P P
xN AR = [σt−d , · · · , σt−d−m+1 ] (18)
P P X X
xN ARX = [σt−d , · · · , σt−d−m+1 , σt−d , · · · , σt−d−m+1 ] (19)
We compare the two approaches for embedding orders m ∈ {2, 5}, several
forecasting horizons h ∈ {2, 5, 8, 10, 12} and for different estimators of the de-
pendency f . More precisely, as estimators of the dependency, we employ a naive
model, a GARCH(1,1), and three machine learning approaches: a feedforward
Artificial Neural Networks, a k-Nearest Neighbors approach and Support Vector
Machine based regression.
3.1 Naive
The Naive method is employed mainly as a benchmark for comparison for the
other models, simply consisting in taking the last available historical value:
P P
σ̂t+h = σt−1 (20)
3.2 GARCH(1,1)
The GARCH model corresponds to the one described in subsection 2, in equa-
tions (13) and (11), with p = 1 and q = 1.
3.3 Artificial Neural Networks
In machine learning and cognitive science, an artificial neural network (ANN) is a
network of interconnected processing elements, called neurons, which are used to
estimate or approximate functions that can depend on a large number of inputs
Machine Learning for Multi-step Ahead Forecasting of Volatility Proxies 7
that are generally unknown. For our task, we will focus on a specific family
of artificial neural networks, the multi-layer perceptron (MLP), with a single
hidden layer. Equations (21) and (22) describe the structure of the model for a
single forecasting horizon t + h in the context of the Direct strategy, respectively
for a NAR and a NARX model.
It should be noted that, as shown in Equation (21), the model can be de-
composed into a linear autoregressive component of order m and a nonlinear
component whose structure depends on the number of hidden nodes H (selected
through k-fold cross-validation). When an external regressor is added, its in-
fluence will affect both the linear and nonlinear component, as shown in (22).
In both cases the activity functions foh (·) and fh (·) are both logistic functions.
Finally, our implementation of the MLP models is based on the nnet package
for R [28].
!
m
X H
X m
X
P
σ̂t+h = foh bo + P
wio σt−i + wjo · fh P
wij σt−i + bj (21)
| i=1 j=1 i=1
{z } | {z }
Linear AR(m) Non-linear component
m
X
P X
+ wio σt−i + w(i+m)o σt−i
i=1
| {z }
P h Linear ARX(m)
σ̂t+h = fo bo H m
! (22)
X X P X
+
wjo · fh wij σt−i + w(i+m)j σt−i + bj
j=1 i=1
| {z }
Non-linear component
3.4 K Nearest Neighbors
The k-Nearest neighbors (kN N ) model is a local nonlinear model used for clas-
sification and regression. In the case of regression, the prediction for a given
input vector x∗ is obtained through local learning [3], a method that produces
predictions by fitting a simple local model in the neighborhood of the point to
be predicted. The neighborhood of a point is defined by taking the the k values
having the minimal values for a chosen distance metric defined on the space of
the input vector [1].
In this case, every data point is represented in the form (x, y) where x rep-
resents the vector of input values and y the corresponding output vector, as
described in Figure 1. Then the prediction for an unknown input vector x∗ is
computed as follows:
1 X
ŷ(x∗ ) = y(xi ) (23)
k
i∈kNN
where y(xi ) is the output vector of the ith nearest neighbor of the input
vector x in the dataset. The choice of the optimal number of neighbors k will
8 Machine Learning for Multi-step Ahead Forecasting of Volatility Proxies
be performed through automatic leave-one-out selection as described in [5]. Our
implementation of the kNN models is based on the R package gbcode [7].
3.5 Support Vector Regression
Support Vector Regression is a regression methodology, based on the Support
Vector Machine theoretical framework [9]. The key idea behind SVR is that the
regression model can be expressed using a subset of the input training examples,
called the support vectors. In more formal terms, the model (Equation (24)) is
a linear combination over all the n support vector of a bivariate kernel function
k(·, ·) taking as inputs the data point x whose forecast is required and the ith
support vector xi . The coefficients αi , αi∗ are determined through the minimiza-
tion of an empirical risk function (cf. [22]), solved as a continuous optimization
problem.
n
X
y= (αi − αi∗ )k(x, xi ) (24)
i=1
kx−xi k2
k(x, xi ) = e 2γ 2 (25)
Among the different available kernel functions we employ the radial basis one
(Equation(25)), for which the optimal value of the γ parameter is determined
through grid search. Here, the SVM implementation of the R package e1071
[18] is used for the experiments. As for ANN and kNN, we will be testing two
different dataset structures (cf. Figure 1), representing respectively the exclusion
(on the left) and the inclusion (on the right) of an external regressor.
Direct NAR Direct NARX
– A single model f h for each horizon h. – A single model f h for each horizon h.
– Forecast at h step is made using hth – Forecast at h step is made using hth
model. model.
x y x y
σ3P σ2P σ1P σ5P σ3P σ2P σ1P σ3X σ2X σ1X σ5P
σ4P σ3P σ2P σ6P σ4P σ3P σ2P σ4X σ3X σ2X σ6P
... ... ... ... ... ... ... ... ... ... ...
σTP −5 σTP −6 σTP −7 σTJ −2 σTP −5 σTP −6 σTP −7 σTX−5 σTX−6 σTX−7 σTP −2
Fig. 1: Comparison of the dataset structure and model identification procedure
for NAR and NARX forecasting strategies. The primary proxy is denoted with
σtP , while the secondary one is σtX . The example datasets are shown for a model
order m = 3 and a forecasting horizon h = 3.
Machine Learning for Multi-step Ahead Forecasting of Volatility Proxies 9
4 Experimental Results
4.1 Dataset description
The proxies have been computed on the 40 time series of the french stock market
index CAC40 from 05-01-2009 to 22-10-2014 (approximately 6 years) for a total
1489 OHLC (Opening, High, Low, Closing) samples for each time series. In
addition to the proxies, we include also the continuously compounded return
and the volume variable (representing the number of trades in given trading
day).
4.2 Statistical analysis
Figure 2 shows the aggregated correlation (over all the 40 time series) between all
the proxies under study, obtained by meta-analysis [11]. Black rectangles indicate
the results of an hierarchical clustering using [29] with k=3. As expected, we can
observe a correlation clustering phenomenon between proxies belonging to the
same family, i.e. σti and σtSD,n . The presence of σt0 in the σtSD,n cluster can be
explained by the fact that the former represents a degenerate case of the latter
when n = 1. Moreover, we find a correlation between the volume and the σti
family.
Volume
σ5SD
SD
SD
σ15
σ21
σG
σ1
σ6
σ4
σ5
σ2
σ3
σ0
rt
1
rt ?
Volume ? 0.8
σ1 ?
0.6
σ6 ?
0.4
σ4 ?
σ5 ?
0.2
σ2 ?
0
σ3 ?
−0.2
σ0 ?
σ5SD ?
−0.4
σ15
SD
?
−0.6
σ21
SD −0.8
?
σG ?
−1
Fig. 2: Summary of the correlations among the different volatility proxies for the
40 time series composing the CAC40 indexes. All the correlations shown in the
table (except those of rt ) are statistically significant (pv=0.05).
4.3 Forecasts
Table 1 shows the performances of the machine learning techniques under study
both with (NARX) and without (NAR) external regressor on proxy σ 4 , averaged
across the 40 times series of CAC40. For each combination of time series, method,
10 Machine Learning for Multi-step Ahead Forecasting of Volatility Proxies
Horizon - H
X
σ Family m 2 5 8 10 12
Average ∅ - - 2.1409 1.5064 1.3456 1.2928 1.2716
GARCH(1,1) ∅ - - 4.7169 3.2481 2.8989 2.7878 2.7560
∅ - 2 0.7914 0.8401 0.8488 0.8469 0.8527
∅ - 5 0.6140 0.7180 0.7707 0.7702 0.7767
σ6 = 2 0.5786 0.5772 0.5877+ 0.5924+ 0.6063+
V olume = 2 0.5492 0.5738 0.5945+ 0.6022+ 0.6126+
ANN-Dir SD,5
σ 6= 2 0.5856 0.5835 0.5958+ 0.6016+ 0.6168+
SD,15
σ 6= 2 0.5856 0.5835 0.5958+ 0.6016+ 0.6168+
SD,21
σ 6= 2 0.5890 0.5715 0.5710 0.5649 0.5756
∅ - 2 0.6375 0.7684 0.7924 0.7905 0.7933
∅ - 5 0.6147 0.7132 0.7425 0.7381 0.7436
σ6 = 2 0.5030 0.5665+ 0.5822+ 0.5825+ 0.5877+
V olume = 2 0.5901 0.5618+ 0.5773+ 0.5794+ 0.5842+
kNN-Dir
σ SD,5 6= 2 0.5596 0.5753+ 0.5835+ 0.5849+ 0.5892+
σ SD,15 6= 2 0.6306 0.5760+ 0.5817+ 0.5770+ 0.5780+
σ SD,21 6= 2 0.5396 0.5790+ 0.5744+ 0.5670+ 0.5678+
∅ - 2 0.4261 0.5965 0.6442 0.6463 0.6503
∅ - 5 0.4070 0.5718 0.6158 0.6145 0.6208
σ6 = 2 0.7265 0.5577+ 0.5442+ 0.5339+ 0.5329+
V olume = 2 0.5901 0.5618+ 0.5773+ 0.5794+ 0.5842+
SVR-Dir SD,5
σ 6= 2 0.6605 0.5691+ 0.5489+ 0.5408+ 0.5407+
SD,15
σ 6= 2 0.8313 0.5764+ 0.5489+ 0.5336+ 0.5307+
SD,21
σ 6= 2 0.5890 0.5715+ 0.5710+ 0.5649+ 0.5756+
Table 1: σ 4 - Naive Normalized MASE - 40 TS. All the ML methods perform
significantly better (pv=0.05) than the GARCH(1,1) method across all the hori-
zons. The superscript + denotes a significant improvement (pv=0.05) with re-
spect to the corresponding method with no additional regressor. The bold nota-
tion is used to identify the best method for each horizon.
forecasting horizon and model order we performed a number of training and test
tasks by following a rolling origin strategy [26]. The size of the training set is
2N
3 and the procedure is repeated for 50 testing sets of length H. The regressor
combinations have been selected in order to test whether the belonging (=) or
not to the same proxy family (6=) impacts the forecasting performance. The
employed error measure is the Mean Absolute Scaled Error [15], normalized at
each forecasting horizon by the the MASE of the Naive method.
PT P P
t=1 σt − σ̂t
MASE = T P T
(26)
P P
T −1 t=2 σt − σt−1
We include in our analysis a GARCH(1,1) method [13] as a baseline reference
method. While employing an additional regressor, model orders higher than 2
have not been tested due to the excessive computational time required by the
Machine Learning for Multi-step Ahead Forecasting of Volatility Proxies 11
corresponding technique for the given task or due to numerical convergence
problems. A first observation from the table is that all the ML methods, both
in the single input and the multiple input configuration, are able to outperform
the reference GARCH method. Moreover, both the increase of the model or-
der m and the introduction of an additional regressor are able to improve the
methods’ performances. However, only the addition of an external regressor, for
horizons greater than 8 steps ahead is shown to bring a statistically significant
improvement (paired t-test, pv=0.05). Even though no model appear to clearly
outperform all the others on every horizons, we can observe that the SVR model
family is generally able to produce smaller forecast errors than those based on
ANN and k-NN.
5 Conclusion and Future work
After having shown the benefits of including an additional proxies in our models,
our main aim is to investigate how the forecasting quality of volatility could be
improved, mainly by tuning three parameters in our methods: the choice of the
additional proxy, the employed machine learning technique and the size of the
training window. In order to further advance our research, we also plan to study
how the current approach could be generalized, in order to include an arbitrary
number of volatility proxies.
Acknowledgments. Jacopo De Stefani acknowledges the support of the ULB-
WORLDLINE agreement. Gianluca Bontempi acknowledges the funding of the
Brufence project (Scalable machine learning for automating defense system) sup-
ported by INNOVIRIS (Brussels Institute for the encouragement of scientific
research and innovation).
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