DeepVol Volatility Forecasting from High-Frequency Data with Dilated Causal Convolutions Fernando Moreno-Pinoaband Stefan Zohrenac

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DeepVol: Volatility Forecasting from High-Frequency Data with

Dilated Causal Convolutions

Fernando Moreno-Pinoa,b and Stefan Zohrena,c

aOxford-Man Institute of Quantitative Finance, University of Oxford

bSignal Processing and Learning Group, Universidad Carlos III de Madrid

cMachine Learning Research Group, University of Oxford

ABSTRACT

Volatility forecasts play a central role among equity risk measures. Besides tradi-

tional statistical models, modern forecasting techniques based on machine learning

can be employed when treating volatility as a univariate, daily time-series. More-

over, econometric studies have shown that increasing the number of daily obser-

vations with high-frequency intraday data helps to improve volatility predictions.

In this work, we propose DeepVol, a model based on Dilated Causal Convolutions

that uses high-frequency data to forecast day-ahead volatility. Our empirical ﬁnd-

ings demonstrate that dilated convolutional ﬁlters are highly eﬀective at extracting

relevant information from intraday ﬁnancial time-series, proving that this architec-

ture can eﬀectively leverage predictive information present in high-frequency data

that would otherwise be lost if realised measures were precomputed. Simultane-

ously, dilated convolutional ﬁlters trained with intraday high-frequency data help

us avoid the limitations of models that use daily data, such as model misspeciﬁca-

tion or manually designed handcrafted features, whose devise involves optimising the

trade-oﬀ between accuracy and computational eﬃciency and makes models prone

to lack of adaptation into changing circumstances. In our analysis, we use two years

of intraday data from NASDAQ-100 to evaluate the performance of DeepVol. Our

empirical results suggest that the proposed deep learning-based approach eﬀectively

learns global features from high-frequency data, resulting in more accurate pre-

dictions compared to traditional methodologies and producing more accurate risk

measures.

KEYWORDS

Volatility forecasting; Realised volatility; High-frequency data; Deep learning;

Dilated causal convolutions

1. Introduction

In recent years, the assessment of portfolios’ risk through volatility measures has

garnered signiﬁcant attention (Brownlees and Gallo 2010), leading to the growing

adoption of volatility conditional portfolios (Harvey et al. 2018). Various studies have

reported overall gains in their Sharpe ratios (Moreira and Muir 2017) and reductions

in the likelihood of observing extreme heavy-tailed returns when using them (Harvey

et al. 2018). Consequently, there has been widespread interest in the development of

volatility forecasting models.

In volatility forecasting, as in many other forecasting problems in economics and

ﬁnance, the variable of interest is latent. This is exempliﬁed in Figure 1, where we

Corresponding author: fernando.moreno-pino@eng.ox.ac.uk

arXiv:2210.04797v3 [q-fin.RM] 8 Aug 2024

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Daily

Close (USD)

−10

Daily

Returns

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2020-02-12

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2021-06-09

2021-08-02

100

Volatility

Figure 1. Apple’s daily data. The top row shows the price trend, while the second

row depicts the associated daily returns. In the bottom row, an estimation of the

unobserved latent volatility process is calculated using a 5-day moving window over

the daily returns. Notice that this method of estimating the volatility serves just as

an approximation to the underlying latent volatility process, providing insights into

the dynamic nature of market volatility.

employ a rolling window to estimate the unobserved latent volatility process from

the observed returns. To address this complexity, an unbiased estimator of this

latent variable must be chosen as a proxy measure. Within the scope of volatility

forecasting, the squared returns of an asset over a speciﬁc period of time is one of the

most obvious realised volatility measures that can be interpreted as a conditionally

unbiased estimator of the true unobserved conditional variance of the asset over that

same time span (Patton 2011).

In a context where volatility forecasting methods oﬀer considerable advantages, with

an extensive literature also pointing out that intraday volatility forecasts are impor-

tant for improving the understanding of the risk involved in trading strategies (Bates

2019), pricing derivatives and the development of quantitative strategies (Engle and

Sokalska 2012), as well as for risk management and trading applications (Stroud and

Johannes 2014), among others, numerous studies have explored the challenges of in-

traday and daily volatility prediction. Delving into the latter, practitioners commonly

rely on classic methodologies, with a heavy reliance on models that provide forecasts

of daily volatility from daily returns. The Generalized Autoregressive Conditional Het-

eroskedasticity (GARCH) model (Bollerslev 1986) and traditional stochastic volatility

models (Andersen et al. 2006; Calvet, Fisher, and Thompson 2006) are predominant

in this regard. These models leverage volatility spillover eﬀects and past volatility

along with daily squared returns as the driving variables for predicting day-ahead

volatility. However, their eﬀectiveness is constrained by their inability to eﬀectively

leverage high-frequency data (Andersen et al. 2003, 2006; Engle and Patton 2007)

and their ineﬃcacy to analysing simultaneously multiple assets in high-dimensionality.

Recent advancements have addressed some of these challenges by incorporating

realised measures as predictors for realised volatility, thereby enhancing the prediction

accuracy of classic models (Hansen, Huang, and Shek 2012; Patton and Zhang 2022).

Realised measures, non-parametric estimators of an asset price variation over a

time interval, extract and summarize information embedded in high-frequency data

(Andersen, Bollerslev, and Diebold 2010). Moreover, ongoing research continues to

introduce novel methodologies for computing realised measures. E.g., Pascalau and

Poirier (2023) proposes a new approach to estimate and forecast realised volatility

that contrast to the usual calendar approach.

However, methodologies that take advantage of realised measures require pre-

processing steps to use them, as they cannot directly model the complex relations

exhibited by intraday ﬁnancial data. One concrete example of this pre-processing is

the procedure followed to obtain the realised measures themselves, which summarises

vectors of daily intraday high-frequency data into single scalars to avoid managing

the microstructure noise associated with the former. In contrast, our approach

uses raw high-frequency data as input to the model, eliminating the need for data

preprocessing and mitigating associated drawbacks, such as information loss during

the aggregation of intraday data into daily realised measures. Our proposed model is

capable of directly handling the microstructure noise linked to higher intraday data’s

sampling frequencies.

Among the methodologies employing realised measures, the HEAVY model

(Shephard and Sheppard 2010) is of special appeal among industry practitioners

(Karanasos, Yfanti, and Hunter 2022; Papantonis, Rompolis, and Tzavalis 2022;

Yuan, Li, and Wang 2022). HEAVY is based on insights from the ARCH architecture,

with superior performance over other classical benchmarks, as shown in Section

5. Nevertheless, the inability of realised measures-based models, such as HEAVY,

Realised GARCH (Hansen, Huang, and Shek 2012), or HAR (Corsi 2009) to use

unprocessed raw high-frequency data directly as input, exposes them to several

disadvantages. Firstly, the dependence on the realised measures for day-ahead

volatility forecasting artiﬁcially limits the amount of information these architectures

use, which is not the case when using raw intraday data, as proven in Section 5.

Furthermore, some of the most used realised measures of volatility lack robustness to

microstructure noise (Baars 2014), implying that the trained models may be based on

biased data. Finally, methodologies based on realised measures often rely on manually

designed handcrafted features, as the realised measures design themselves, formulated

to optimise the trade-oﬀ between accuracy and increasing computational costs.

These issues, coupled with common model misspeciﬁcation in classical model-based

approaches, undermines their reported performances.

Here, we use Deep Neural Networks (DNN) (LeCun, Bengio, and Hinton 2015)

to take advantage of the abundance of high-frequency data without prejudice,

preventing the constraints of models based on realised measures in the context of

day-ahead volatility forecasting. Despite the success of these DNN architectures in

diﬀerent areas, such as healthcare (Shamshirband et al. 2021), image recognition

(Kaur and Gandhi 2020; Chen et al. 2021), time-series modelling and forecasting

(Li et al. 2019; Jim´enez Rama et al. 2023; Moreno-Pino et al. 2024b), and text

analytics (Conneau et al. 2016), they have not been widely adopted for the problem

of volatility forecasting, leading to a large gap between modern machine learning

models and those applied in the volatility framework. Among DNN-based models,

Recurrent Neural Networks (RNN) (Rumelhart, Hinton, and Williams 1985) and

Long Short-Term Memory (LSTM) (Hochreiter and Schmidhuber 1997) are the most

popular approaches with regard to time-series forecasting (Lim and Zohren 2021).

Furthermore, the addition of the attention mechanism (Bahdanau, Cho, and Bengio

2014) into these base architectures allowed them to focus on the most relevant input

data while producing predictions, making them especially prominent in ﬁelds such as

Natural Language Processing (NLP). These advances also lead to the appearance of

Transformer models (Vaswani et al. 2017), which were initially introduced for NLP,

and later used for the problem of time-series forecasting (Li et al. 2019; Moreno-Pino,

Olmos, and Art´es-Rodr´ıguez 2023). These models are also applied in the context

of ﬁnancial time-series through diverse variations (Su 2021; Scalzo et al. 2021; Lin

et al. 2022; Arroyo et al. 2022; Christensen, Siggaard, and Veliyev 2023; Zhu et al.

2023; Song et al. 2023; Arroyo et al. 2024; Souto and Moradi 2024; Ghosh et al.

2024). More speciﬁcally, regarding volatility forecasting, various studies have delved

into the application of deep-learning architectures, such as LSTM (Yu and Li 2018),

Convolutional Neural Networks (CNN) (Vidal and Kristjanpoller 2020), Graph

Neural Networks (GNN) (Chen and Robert 2021; Zhang et al. 2022), Transformer

models (Ramos-P´erez, Alonso-Gonz´alez, and N´u˜nez-Vel´azquez 2021), and NLP-based

word embedding techniques (Rahimikia and Poon 2020; Rahimikia, Zohren, and Poon

2021). Furthermore, models combining traditional volatility forecasting methods with

deep-learning techniques can be found in the literature (Kim and Won 2018; Mademlis

and Dritsakis 2021), as well as other approaches using DNN as calibration methods

for implying volatility surfaces (Horvath, Muguruza, and Tomas 2019), proving how

neural network-based approaches work as complex pricing function approximators.

We capitalise on the increased availability of high-frequency data. In this work, we

employ a Dilated Causal Convolutions (DCC)-based model. This architecture, initially

proposed as a fully probabilistic model for audio generation (Oord et al. 2016), with

equivalents for image-related problems (Van Oord, Kalchbrenner, and Kavukcuoglu

2016), can handle lengthy sequences of data eﬀectively without a signiﬁcant increase

in the number of parameters thanks to the use of dilated connections. In the

literature, other works use DCC in the context of realised volatility forecasting. More

speciﬁcally, Reisenhofer, Bayer, and Hautsch (2022) propose a model based on dilated

convolutions, strongly inspired by the well-known Heterogeneous Autoregressive

(HAR) model (Corsi 2009). However, their approach does not use unprocessed raw

intraday high-frequency data as input. Conversely, it still bases its predictions on

the pre-computed daily realised variance, therefore requiring pre-processing steps

to obtain the indispensable realised measures for forecasting the one-step-ahead

volatility. This, in our judgment, does not fully explore the capabilities of DCC-based

methodologies of exploiting a more dynamic representation of the intraday data.

Hence, models adopting DCC-based approaches that operate from daily data still

succumb to the limitations enumerated previously.

Motivated by the improved performance of classical methods that employ realised

measures (Hansen, Huang, and Shek 2012; Shephard and Sheppard 2010), we propose

using DCCs to bypass the estimation of these non-parametric estimators of assets’

variance, aiming to tackle the volatility forecasting problem from a data-driven

perspective. The proposed model, DeepVol, oﬀers several advantages for volatility

forecasting. Primarily, it does not require any pre-processing steps, as the model

directly uses raw high-frequency data as input. Furthermore, DeepVol is not bounded

to static realised measures whose use may be counter-productive, i.e. the optimal

realised measure to use may vary depending on the traded assets’ liquidity. Instead,

through the attention mechanism and internal non-linearities, DeepVol intelligently

performs the required transformations over the input data to maximise the accuracy

of the predictions, combining relevant intraday datapoints and merging them for

each day’s volatility forecast, dinamically adapting to diﬀerent scenarios. Moreover,

through the use of dilated convolutions, DeepVol’s large receptive ﬁeld easily processes

long sequences of high-frequency data, enabling the model to exponentially increase its

input window without triggering an unrestrained increase in the model’s complexity

while performing the predictions. While DeepVol’s operation does not directly produce

any type of realised measure —meaning the model does not explicitly construct an

ex-post estimate of the returns variation— its aggregation of high-frequency data

resembles how realised measures condense intraday data into daily statistics. Unlike

the computation of these realised measures, DeepVol undertakes this aggregation

in a data-driven manner, diverging from traditional approaches used for computing

realised measures of volatility. DeepVol’s ability to dynamically select and weight the

most relevant time-steps from the conditioning range at each time instant stands

out as its primary distinction from traditional methods computing realised measures,

allowing DeepVol to hierarchically integrate the most relevant high-frequency data

into the predictions. We perform extensive experiments to show the eﬀectiveness of

the proposed architecture, which consistently outperforms the base models used by

practitioners.

This paper presents three main contributions. Firstly, it empirically demonstrates

the advantages of DCCs for forecasting realised volatility using high-frequency data,

providing a data-driven solution that consistently outperforms classical methodologies.

Further, the proposed model overcomes the limitations of classical methods, such as

model misspeciﬁcation or their inability to directly use intraday data to perform the

forecast. Secondly, we oﬀer an analysis showing how DeepVol optimizes the balance

between extracting signals from high-frequency data and minimizing the microstruc-

ture noise inherent to higher sampling frequencies. The reported results are consistent

with studies validating this trade-oﬀ for constructing realised measures. Thirdly, the

proposed volatility forecasting model generates appropriate risk measures through

its predictions in an out-of-sample forecasting task, both in low and high volatility

regimes. Moreover, we evaluate the proposed model’s generalisation capabilities on

out-of-distribution stocks, demonstrating DeepVol’s capabilities to transfer learning

as it performs accurate predictions into data distributions not observed during the

training phase.

The structure of the paper is as follows. Section 2 details the dataset used, while

Section 3 contains a brief overview on volatility forecasting, describing the baselines

used for benchmarking purposes and the metrics that will be utilised for model

comparison. Section 4 presents the proposed model, which is empirically evaluated in

Section 5. Finally, Section 6 summarises the ﬁndings and provides concluding remarks.

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DeepVol:VolatilityForecastingfromHigh-FrequencyDatawithDilatedCausalConvolutionsFernandoMoreno-Pinoa,bandStefanZohrena,caOxford-ManInstituteofQuantitativeFinance,UniversityofOxfordbSignalProcessingandLearningGroup,UniversidadCarlosIIIdeMadridcMachineLearningResearchGroup,UniversityofOxfordABSTRACTVo...

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DeepVol Volatility Forecasting from High-Frequency Data with Dilated Causal Convolutions Fernando Moreno-Pinoaband Stefan Zohrenac

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