Score-Driven Modeling with Jumps: An Application to S&P500 Returns and Options

Journal of Financial Econometrics, 2024, Vol. 22, No. 2, 375–406 https://doi.org/10.1093/jjfinec/nbad001 Advance Access Publication Date: 8 February 2023 Article Score-Driven Modeling with Jumps: An Application to S&P500 Returns and Options 2, *, and 1 University of Bologna, Italy and 2Vrije Universiteit Amsterdam, The Netherlands *Address correspondence to Enzo D’Innocenzo, Department of Econometrics and Data Science, Vrije Universiteit Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands, or e-mail: . Received November 16, 2021; revised December 30, 2022; editorial decision January 10, 2023; accepted January 11, 2023 Abstract We introduce a novel score-driven model with two sources of shock, allowing for both time-varying volatility and jumps. A theoretical investigation is performed which yields sufficient conditions to ensure stationarity and ergodicity. We extend the model to consider a time-varying jump intensity. Both an in-sample and an outof-sample analysis based on the S&P500 time series show that the proposed methodology provides excellent agreement with observed returns, outperforming more standard Generalized Autoregressive Conditional Heteroskedasticity (GARCH) specifications with jumps. Finally, we apply our models to option pricing via risk neutralization. Results show this novel approach produces reliable implied volatility surfaces. Supplementary Materials including proofs, the derivation of the conditional Fisher information, and two figures showing additional empirical results are available online. Key words: time-varying volatility, compound Poisson, observation-driven models, stationarity and ergodicity, option pricing, JEL Codes: C510, C530, C580 JEL classification: C510, C530, C580 Several empirical studies document that asset prices are affected by sharp and large discontinuities (jumps), due to unexpected news and events, see the recent works by Gürkaynak, Kisacikoglu, and Wright (2020), Engle et al. (2021) and Jeon, McCurdy, and Zhao (2021). To account for these discontinuities, nonstandard GARCH approaches that also allow for jumps in the return process have been proposed by Vlaar and Palm (1993), Chan and Maheu (2002), Maheu and McCurdy (2004), Duan, Ritchken, and Sun (2006), C The Author(s) 2023. Published by Oxford University Press. V This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs licence (https://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial reproduction and distribution of the work, in any medium, provided the original work is not altered or transformed in any way, and that the work is properly cited. For commercial re-use, please contact Luca Vincenzo Ballestra1, Enzo D’Innocenzo Andrea Guizzardi 1 376 Journal of Financial Econometrics Christoffersen, Jacobs, and Ornthanalai (2012), Guégan, Ielpo, and Lalaharison (2013), and Ornthanalai (2014). Importantly, GARCH models are not robust to misspecification or even small departures from the data-generating process (DGP) given that, in their original form, these models use squared-lagged innovations to update the value of the conditional variance. This makes them very sensitive to the presence of even a few outliers or extreme returns. A possible solution to this issue is provided by the so-called score-driven models, which were originally proposed by Creal, Koopman, and Lucas (2011) and Harvey (2013). The key feature of score-driven models is that the dynamic of the time-varying parameters is described by an autoregressive process driven by a scaled version of the score function, that is, the derivative of the (postulated) conditional log -density. Moreover, the score-driven approach guarantees that the Kullback–Leibler divergence between the probability density function of the DGP and the model implied probability distribution diminishes at least locally, see Blasques, Koopman, and Lucas (2015). In addition to guaranteeing theoretical optimality, this property is crucial in model misspecifications, when the chosen modeling framework is different from the true economic dynamic of the empirical data. Score-driven models also provide a general framework that fully exploits shape of the observation conditional density as its characteristics can be specifically incorporated as driving forces for the time-varying parameters. As a final advantage, score-driven models belong to the class of observation-driven models, see Cox et al. (1981), since the evolution of the unobserved dynamic parameters depends only on historical data. Therefore, exactly like GARCH models, they can be easily filtered and estimated by maximum likelihood, which makes them appealing for practical applications. Despite the strong advantages of this approach and the importance of accounting for jumps in asset price dynamics, literature on the score-driven approach with jumps is still lacking. To fill this gap, we develop a score-driven model with jumps (SDJ), where the conditional variance of the returns is assumed to follow an autoregressive process driven by the score of the predictive density, and the jumps are modeled by a compound Poisson process. This approach allows us to take into account the interaction between jumps and volatility, since the Poisson process used to specify the jumps is fully coupled with the dynamics of the variance. Moreover, from a theoretical standpoint, the proposed model offers the advantage of being strictly stationary and ergodic. In particular, based on the framework developed by Blasques, Koopman, and Lucas (2014), we establish one mild sufficient condition that ensures the ergodicity and strict stationarity of the return process. We also develop two extensions of the SDJ model in which the conditional variance of the returns and the conditional jump intensity follow a bivariate (coupled) score-driven autoregression. These two approaches, which we label SDSDJ-1 and SDSDJ-2, allow us to take into account the stochastic nature of the jump frequency, which depends on the contingent and continuously evolving macroeconomic conditions. We test the empirical performances of the proposed SDJ, SDSDJ-1, and SDSDJ-2 models, and we compare them against the GARCH models with jumps and both constant or time-varying intensity introduced by Christoffersen et al. (2008) and Christoffersen, Jacobs, and Ornthanalai (2012). We conduct both an in-sample and an out-of-sample exercise focusing on the S&P500 total return time series. The results reveal that the scoredriven approaches provide a very good fitting of empirical data as they significantly outperform their GARCH counterparts. In a Monte Carlo exercise, we show the superiority of the Ballestra et al. j Score-Driven Modeling with Jumps 377 1 Modeling Returns with Jumps Let St denotes the price of a risky asset and let us consider the log -return Rt ¼ ln  St þDt St1  , including the dividend Dt . Following Chris (...truncated)