Revisiting the puzzle of jumps in volatility forecasting: The new insights of high‐frequency jump intensity

• Published date: 01 February 2024
• Author: Hui Qu,Tianyang Wang,Peng Shangguan,Mengying He
• Date: 01 February 2024
• DOI: http://doi.org/10.1002/fut.22468

Received: 6 June 2022

|

Accepted: 16 October 2023

DOI: 10.1002/fut.22468

RESEARCH ARTICLE

Revisiting the puzzle of jumps in volatility forecasting:

The new insights of high‐frequency jump intensity

Hui Qu

1

|Tianyang Wang

2

|Peng Shangguan

1

|Mengying He

3

1

School of Management and Engineering,

Nanjing University, Nanjing, China

2

Finance and Real Estate Department,

Colorado State University, Fort Collins,

Colorado, USA

3

Department of Systems Engineering and

Engineering Management, The Chinese

University of Hong Kong, Hong Kong,

China

Correspondence

Hui Qu, School of Management and

Engineering, Nanjing University, Nanjing

210093, China.

Email: linda59qu@nju.edu.cn

Funding information

National Natural Science Foundation of

China, Grant/Award Number: 72171110

Abstract

Motivated by the puzzling null impact of high‐frequency‐based jumps on

future volatility, this paper exploits the rich information content in high‐

frequency jump intensity with a mark structure under the heterogeneous

autoregressive framework. Our proposed model shows that harnessing jump

intensity information from the marked Hawkes process leads to significantly

superior in‐sample fit and out‐of‐sample forecasting accuracy. In addition to

statistical significance evidence, we also illustrate the economic significance in

terms of trading efficiency. Our findings hold for a variety of competing

models and under different market conditions, underlying the robustness of

our results.

KEYWORDS

HAR model, high‐frequency data, jump intensity, marked Hawkes process, volatility

forecasting

JEL CLASSIFICATION

C5, G1

1|INTRODUCTION

Financial assets often exhibit volatility clustering and jumps caused by significant new information, such as financial

earnings and news announcements. Advances in high‐frequency econometrics offer a means of separating jumps from

continuous asset price movements (Barndorff‐Nielsen & Shephard, 2006), enabling the explicit inclusion of jumps in

volatility modeling. Despite the importance of jumps in the dispersion of beliefs and heterogeneous information, which

both impact volatility (Giot & Laurent, 2007; Shalen, 1993), the effect of high‐frequency‐based jumps on volatility

forecasting has been puzzling in the literature and contradicts the evidence from lower frequency‐based generalized

autoregressive conditional heteroskedasticity (GARCH) and related models (Jorion, 1988; Maheu & McCurdy, 2004).

This study aims to address the gap in the literature regarding this puzzle.

Many research efforts have attempted to solve this puzzle, but the accumulation of negative evidence against the

predictive value of high‐frequency‐based jump dynamics has continued to persist (c.f. Andersen, Bollerslev, &

Diebold, 2007; Barndorff‐Nielsen & Shephard, 2006; Busch et al., 2011; Forsberg & Ghysels, 2007; Giot & Laurent, 2007;

Prokopczuk et al., 2016). For example, Andersen, Bollerslev, & Diebold (2007) found that the jump size in equity, fixed

income, and foreign exchange markets is neither persistent nor predictable and has no significant impact on volatility

forecasting. Corsi et al. (2010) shed some light on this apparent puzzle, suggesting that it may partially be due to small

sample bias of bipower variation (Barndorff‐Nielsen & Shephard, 2004) and market microstructure effects. However,

this puzzle remains unresolved. Prokopczuk et al. (2016) concluded that explicitly modeling jumps does not

J Futures Markets. 2024;44:218–251.wileyonlinelibrary.com/journal/fut218

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© 2023 Wiley Periodicals LLC.

significantly improve forecasting accuracy when examining the predictive ability of high‐frequency‐based jump size in

major energy markets.

Motivated by the puzzle of the null impact of high‐frequency‐based jumps on future volatility, we present a new

framework for volatility forecasting that leverages the information contained in high‐frequency jump intensity to

reconcile this puzzle. Our framework addresses the puzzle from multiple angles. First, we utilize the rich information

contained in the high‐frequency jump intensity to improve the accuracy of volatility forecasting. Previous research has

primarily focused on the size of jumps, but jump intensity has received limited attention despite its demonstrated

persistence and predictability, particularly during periods of high jump activity (Aït‐Sahalia et al., 2015;

Andersen, Bollerslev, & Diebold, 2007). Given that the likelihood and magnitude of future jumps may increase with

jump occurrence (Luciano & Schoutens, 2006; Ma et al., 2020), jump intensity (probability) may play a crucial role in

predicting future volatility. By incorporating jump intensity into the heterogeneous autoregressive (Heterogeneous

Autoregressive with Continuous volatility and Jumps model [HAR‐CJ]) framework (Andersen, Bollerslev, &

Diebold, 2007), our study enhances forecasting accuracy by capitalizing on the information contained in jump

intensity.

Second, we explore the clustering feature of jump intensity, which has received limited attention in volatility

forecasting despite its significance during periods of high clustered jump activity in the market (Maheu &

McCurdy, 2004). Intuitively, a higher likelihood of clustered jump movements indicates higher volatility in

the next period due to market instability. Daal et al. (2007) found that incorporating jump intensities

and volatility feedback in the jump component in GARCH‐type models leads to a better fit for the dynamics of

equity returns. Hainaut and Moraux (2018) illustrate the improved efficiency of hedging strategies for stock

options by considering the presence of jump clustering. Our study incorporates a dynamic clustering structure,

along with high‐frequency‐based jump intensity information, within the heterogeneous autoregressive (HAR)

framework.

Third, we introduce the event characteristics associated with the jump intensity into the volatility forecasting

framework. Jump intensity, which is dynamic in nature, has distinct features for each event such as jump size,

asymmetrical positive and negative jumps (c.f. Ma et al., 2019; Patton & Sheppard, 2015), and concurrent market

conditions. These characteristics have not been previously explored in the literature but could have a significant impact

on future volatility. Our study compares the simple Hawkes process and marked Hawkes process, using median

realized volatility (RV) as a mark to characterize market changes. We enhance these models by introducing the

asymmetric Hawkes process and asymmetric marked Hawkes process for volatility forecasting and find that these

models, incorporating both jump intensity and a mark structure, have improved in‐sample fit and out‐of‐sample

accuracy compared to benchmark models.

In addition, we incorporate the leverage effect into our framework. As demonstrated by Choi and Richardson

(2016), the leverage effect corresponds to a negative correlation between past returns and future volatility, and financial

leverage has a significant impact on equity volatility. Additionally, leverage and asset volatility have both permanent

and transitory effects on equity volatility, which help explain the short‐and long‐term dynamics of equity volatility.

Carr and Wu (2017) also highlights how index volatility increases with the market's aggregate financial leverage, and

positive shocks to systematic risk increase the cost of capital and decrease the valuation of future cash flows, resulting

in a negative correlation between the index return and its volatility, regardless of financial leverage. Moreover, large

negative market disruptions are observed to have self‐exciting behaviors. This study employs the use of realized positive

and negative semivariances (Barndorff‐Nielsen et al., 2010), as established in the literature on the leverage effect (Gong

& Lin, 2018; Ma, Wei, et al., 2018; Patton & Sheppard, 2015; Sévi, 2014; Wen et al., 2016), and investigates the

usefulness of realized semivariances under varying jump intensities. The results of our study demonstrate that the

model that includes the leverage effect, high‐frequency‐based jump intensity, and the (asymmetric) marked Hawkes

process is the best‐performing model. The robustness of our results is proven through statistical significance and their

practicality in different market conditions.

We also contribute to the growing literature on extracting information from index futures due to their dominant

role in the market. Although recent studies have focused on predicting index futures returns (c.f. Jondeau et al., 2020;

Cao et al., 2020,2023), this study emphasizes on examining the information extracted from the S&P 500 index futures.

By focusing on the high‐frequency (5‐and 10‐min) prices of the S&P 500 index and S&P 500 index futures, we

demonstrate significant improvements in volatility forecasting accuracy of S&P 500 index futures by combining jump

intensity with a mark structure under the HAR framework and utilizing the rich information of event characteristics

related to jump intensity.

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Moreover, we go beyond statistical measures. While most studies primarily explore the statistical gains in prediction

tasks related to financial markets, statistical significance does not always translate to economic significance (Della

Corte et al., 2009; Leitch & Tanner, 1991). Therefore, we also investigate the economic significance of modeling and

forecasting financial asset volatility. Specifically, we evaluate the trading efficiency of a strategy for VIX futures using

the tradable VIX exchange‐traded products. We find that our proposed framework is both statistically and economically

significant, underscoring its practical benefit and the potential economic gains it can bring to financial decision‐

making.

The rest of the paper is organized as follows. Section 2introduces the models and methods. Section 3describes the

data and provides the in‐sample and out‐of‐sample analysis. Section 4presents the robustness check results. Section 5

evaluates the economic significance. Section 6concludes.

2|JUMP INTENSITY AND VOLATILITY FORECASTING MODELS

The accuracy of volatility estimation is crucial in various financial applications such as investment, portfolio

management, risk management, and derivatives pricing. Traditional volatility models, such as GARCH models, only

model the conditional variance of daily returns and do not account for the daily volatility. This weak signal of the

current volatility level is unable to capture rapid changes in volatility. In contrast, the introduction of RV by Andersen

and Bollerslev (1998) solves this issue. RV is calculated as the sum of high‐frequency intraday squared returns, making

volatility directly observable for the first time.

Recent research has focused on using RV to model financial volatility. It has been found that volatility can be

separated into continuous and jump components. Methods have been proposed to estimate continuous volatility and

identify significant jumps in asset prices, respectively (Andersen, Bollerslev, & Dobrev, 2007; Andersen et al., 2012;

Barndorff‐Nielsen & Shephard, 2006). The study of RV as a linear function of lagged realized volatilities over different

time horizons using the HAR model has been proposed and shown to have good forecasting performance (Corsi, 2009).

The use of the HAR‐CJ model, which differentiates between contributions from the continuous and jump components,

has been widely adopted and has resulted in further improvements in forecasting accuracy (Andersen, Bollerslev, &

Diebold, 2007). This paper further explores the use of high‐frequency data to explicitly model jump intensity in

volatility forecasting, capturing the self‐exciting, clustering, and asymmetric properties of jump occurrence that cannot

be captured by jump size alone.

2.1 |Jump intensity models

Let pti,denote the logarithmic price of the financial asset observed at the end of the

i

t

h

interval on trading day

t

, where

iM

=1,2,…,

and

M

is the numberof sampling intervalsper day. Let rti,be the

i

t

h

intraday return of day

t

(magnified

100 times). Following Andersen and Bollerslev (1998), the RV on day

t

is defined as:

RV r=

ti

Mti

=1 ,

2

.

High‐frequency returns may suddenly exhibit wide variations in intraday continuous time, which are referred to as

jumps. The ABD test proposed by Andersen, Bollerslev, & Dobrev (2007) is used to estimate the jump component in RV

in this study. It employs a unified judgment rule to simultaneously identify the presence of multiple significant jumps

within 1 day. Specifically, the ABD test detects a jump in the

i

t

h

interval on day

t

when:

⋅⋅



r BPV>ΦΔ

,

ti βt,1−2(1)

where

Φ

is the critical value of the standard normal distribution,

β

αα=1−(1 −),

M

1represents the significance level,

Δ

=

M

1

, and

⋅BPVr r=

tπM

Mi

Mti ti

2−1=2 ,,−1

is the realized bipower variation (BPV; Barndorff‐Nielsen & Shephard, 2004),

which estimates the integrated variance of day

t

. The jump component of volatility on day

t

is thus the sum of the

squared intraday returns that satisfy Equation (1).

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