Local Estimation for Option Pricing: Improving Forecasts with Market State Information

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Local Estimation for Option Pricing: Improving Forecasts with Market State Information Hyung Joo Kim, Dong Hwan Oh 2025-076 Please cite this paper as: Kim, Hyung Joo, and Dong Hwan Oh (2025). “Local Estimation for Option Pricing: Improving Forecasts with Market State Information,” Finance and Economics Discussion Series 2025-076. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2025.076. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Local Estimation for Option Pricing: Improving Forecasts with * Market State Information Hyung Joo Kim(cid:132) Dong Hwan Oh(cid:133) July 30, 2025 Abstract We propose a novel estimation framework for option pricing models that incorporates local, state-dependent information to improve out-of-sample forecasting performance. Rather than modifyingtheunderlyingoptionpricingmodel,suchastheHeston-NandiGARCHortheHeston stochastic volatility framework, we introduce a local M-estimation approach that conditions on keystatevariablesincludingVIX,realizedvolatility,andtime. Ourmethodreweightshistorical observations based on their relevance to current market conditions, using kernel functions with bandwidths selected via a validation procedure. This adaptive estimation improves the model’s responsiveness to evolving dynamics while maintaining tractability. Empirically, we show that local estimators substantially outperform traditional non-local approaches in forecasting neartermoptionimpliedvolatilities. Theimprovementsareparticularlypronouncedinlow-volatility environments and across the cross-section of options. The local estimators also outperform the non-local estimators in explaining future option returns. Our findings suggest that local information, when properly incorporated into the estimation process, can enhance the accuracy and robustness of option pricing models. *Theanalysisandconclusionssetfortharethoseoftheauthorsanddonotindicateconcurrencebyothermembers of the research staff or the Board of Governors. (cid:132)Federal Reserve Board; hyungjoo.kim@frb.gov (cid:133)Federal Reserve Board; donghwan.oh@frb.gov

1 Introduction Accurately pricing options is essential for trading, hedging, risk management, and regulatory purposes. While the ability to forecast future option prices or their distributions is often more relevant for these applications, much of the literature on option valuation has focused on fitting historical or in-sampleoptionpricesmorecloselybydevelopingincreasinglycomplexmodels.1 Sincetheseminal work of Black and Scholes (1973), numerous models have been proposed to address the empirical limitations of the Black-Scholes (BS) framework—most notably its inability to capture volatility smilesandskewsduetotheassumptionofconstantvolatility. Modelssuchasthestochasticvolatility (SV) model of Heston (1993), the GARCH-based model of Heston and Nandi (2000), and their extended models have made significant strides in matching observed option prices, underlying return dynamics, and other option moments while offering tractable pricing formulas.2 Nonetheless, relatively little attention has been paid to their forecasting performance. Furthermore, the role of the estimation method itself has received limited attention. Most standardapproachesusefixed-parameterestimationtechniques, whichimplicitlytreatallhistorical observations as equally informative. This uniform weighting can be problematic in evolving or regime-shifting markets, where stale observations may dilute the influence of relevant information. Recent work by Dendramis, Kapetanios, and Marcellino (2020) and Oh and Patton (2024) shows that reweighting observations based on current market conditions can improve forecasting accuracy in macroeconomic and financial applications. These advances motivate our core research question: can option pricing models be improved by modifying the estimation method, without changing the model? In this paper, we develop a novel local M-estimation framework for option pricing models that conditions on market state variables using kernel-based weighting. Rather than altering the model structure, weadaptparameterestimatestoreflectthecurrentmarketenvironment. Ourframework 1Forexample,regulatoryapplicationsandcentralcounterparties(CCPs)oftenrequiremodelsthatcanforecastpotentialexposuresunderfuturemarketscenarios. Forecastedoptionvaluesareusedinmarginrequirementcalibration, stress testing, and systemic risk assessments. 2An extensive set of studies include, but not limited to, Bakshi, Cao, and Chen (1997), Bates (2000, 2019), Duffie,Pan,andSingleton(2000),Pan(2002),Eraker(2004),Christoffersen,Jacobs,Ornthanalai,andWang(2008), Christoffersen, Heston, and Jacobs (2009, 2013), Christoffersen, Jacobs, and Mimouni (2010), Andersen, Fusari, and Todorov (2015), Babaog˘lu, Christoffersen, Heston, and Jacobs (2018), Bardgett, Gourier, and Leippold (2019), Du and Luo (2019), A¨ıt-Sahalia, Karaman, and Mancini (2020), A¨ıt-Sahalia, Li, and Li (2021), Gruber, Tebaldi, and Trojani (2021), Heston, Jacobs, and Kim (2024). 1

allows for observation-specific weighting based on a range of informative variables, including the VIX, realized volatility, risk-neutral skewness and kurtosis, VIX volatility index (VVIX), and the varianceriskpremium(VRP),orweightingthatplacesgreateremphasisonmorerecentdatapoints. Bandwidths or tuning parameters governing the weighting scheme are selected using a split-sample validation procedure to avoid look-ahead bias. WeapplythisestimationtechniquetobothGARCHandSVoptionpricingmodelsandevaluate itsforecastingperformanceusingout-of-sample(OOS)optionimpliedvolatilitiesfrom2015to2023. Our empirical analysis demonstrates that the local estimation approach significantly outperforms traditional non-local methods in both time-series and cross-sectional dimensions. Our empirical findings demonstrate that local estimation substantially improves the accuracy of forecasting option prices or implied volatilities. Across multiple forecast periods from 2015 to 2023, local estimators consistently outperform their non-local counterparts in both time-series and cross-sectional environments. In the GARCH model, conditioning on VIX leads to the greatest improvements in implied volatility forecasting, while in the SV model, time-based local estimations perform best. These enhancements are especially pronounced in low-volatility periods, such as 2017, when market conditions evolve gradually. Although the outperformance of local estimation is less noticeable during high-volatility periods, it performs at least at par or slightly better relative to non-local estimation. In addition to reduced time-series forecast errors, local estimation leads to overall improved and more balanced pricing performance across different moneyness levels and maturities. We confirm that this improvement stems from the better alignment of model-implied risk-neutral distributions with nonparametric benchmarks. Moreover, local estimators yield expected option returns that match ex-post realized option returns better, indicating improvements in the joint modeling of physical and risk-neutral dynamics. These results not only support the theoretical value of statedependent estimation but also demonstrate its practical utility for traders, margin models, and short-horizon risk forecasts. This paper contributes to the literature in three key directions: First,ourpapercontributestothegrowingliteratureonforecastingoptionmoments. Onestrand ofthisliteraturefocusesonforecastingoption-impliedvolatilitysurfaces(e.g.,Almeida, Fan, Freire, 2

and Tang, 2023; Chen, Grith, and Lai, 2025; Dufays, Jacobs, and Rombouts, 2025). While these studies generate the entire volatility surface and then forecast it using various volatility forecasting models or their extensions, our approach emphasizes incorporating local state information into the estimation of dynamic option pricing models to better forecast implied volatilities. Instead of directly forecasting the volatility surface, we use a better-estimated option pricing model to generate the distribution of future underlying prices and variances, and then compute expected implied volatilities from those distributions. This approach allows us to forecast implied volatility, internally consistent with a pricing model. Another strand of the literature focuses on forecasting thecross-sectionofoptionreturns(e.g.,Zhan,Han,Cao,andTong,2022;Bali,Beckmeyer,Moerke, andWeigert,2023;Cao, Vasquez, Xiao, andZhan,2023). Thoughourgoalisnottofindforecasting models for option returns, we compute model-implied expected option returns and show that our local estimation model forecasts option returns better than a benchmark (non-local) model. Second, we improve option pricing performance by incorporating current information through a novel estimation method. Rather than extending or modifying existing models, we employ a local estimation method that naturally incorporates recent market information, which proves beneficial for pricing options or forecasting implied volatilities.3 This approach allows us to capture the most up-to-date market dynamics without increasing model complexity.4 By focusing on estimation rather than model structure, we maintain simplicity while enhancing adaptability to changing market conditions. This method is particularly effective in volatile or rapidly evolving markets where traditional models might struggle to adapt quickly. Furthermore, our approach can potentially reduce estimation time, parameter uncertainty, and model risk, which are common challenges in more complex option pricing models. Third, we bridge two strands of research—dynamic volatility modeling and local estimation methods in econometrics—and apply them to a forward-looking option pricing framework.5 Our 3Relatedly, Kim (2025) and Schreindorfer and Sichert (2025) exploit state information to better estimate the conditional pricing kernel implied by index returns and options, and Oh and Patton (2024) show that applying a local estimation method improves volatility forecasting. 4Asanexampleofcapturingstate-dependentmarketdynamicsinoptionpricingwithamodestincreaseinmodel complexity, Hansen and Tong (2025) improve the GARCH option pricing model fit by allowing a time-varying risk preference parameter. 5Thisperspectiveisparticularlyrelevantforpractitionersinvolvedinmarginsettingandshort-horizontrading. For example, clearinghouses often rely on forecasts of near-term option prices conditional on current market conditions to determine initial and variation margins. Similarly, trading desks require real-time pricing that reflects current volatility and tail risk exposure, which static parameter estimates may fail to capture. 3

approach aligns with how options are used in practice, such as in margin setting or short-horizon forecasting, where the objective is not to fit past prices but to forecast near-term values conditional on current information. By incorporating forward-looking local information into the estimation step, we improve the responsiveness of otherwise fixed-parameter models and enhance their forecasting performance without adding structural complexity. The remainder of the paper is organized as follows. Section 2 describes the benchmark option pricing models and the local M-estimation framework, along with implementation details including kernel choices, bandwidth selection, and validation procedures. Section 3 introduces the data, and Section 4 reports the empirical results. Section 5 concludes. 2 Model and Estimation This section outlines the option pricing models and the local estimation framework. We specify the physical and risk-neutral dynamics for the Heston-Nandi GARCH and Heston SV models, followed byalocalM-estimationapproachthatreweightsobservationsbasedonstatevariables. Thismethod preserves the model structure while adapting parameters to current market conditions. We also describekernelweighting, bandwidthselection, andthevalidationprocedureusedforout-of-sample evaluation. 2.1 Stock price dynamics and transition likelihoods We focus on two widely used option pricing models: the Heston and Nandi (2000) GARCH option pricingmodelandtheHeston(1993)SVoptionpricingmodel,aswellastheirunderlyingdynamics. The former is based on the discrete-time dynamic of the stock price, where its variance follows a GARCHprocessderivedfromhistoricalreturnsandpastvolatilities. Thelatterusesthecontinuoustime dynamic of the stock price in conjunction with its stochastic volatility, which follows a squareroot process. Both approaches are essential for pricing options and analyzing volatility dynamics because they incorporate the time-varying nature of volatility into the valuation process. They have also received ample attention in the literature, not only for their time-varying volatility but also for their quasi-closed-form option pricing formulas and their sufficient performance in terms of average option fits. 4

Forestimation,wespecifythephysicaldynamicsandmaximizethereturntransitionlikelihoods. It is worth noting that we do not use options data in model parameter estimations; instead, we use them for model validation and out-of-sample analysis. We discuss more details on how we convert the optimal physical dynamics to risk-neutral dynamics and how we price options in Section 2.2. 2.1.1 The GARCH dynamics WeemploythephysicalstockpriceandvariancedynamicsofHestonandNandi(2000)andChristoffersen, Heston, and Jacobs (2013) as our first benchmark model: (cid:18) (cid:18) (cid:19) (cid:19) 1 (cid:112) lnS(t+1) = lnS(t)+ r+ µ− v(t+1) + v(t+1)z(t+1), (1) 2 (cid:16) (cid:112) (cid:17)2 v(t+1) = ω+βv(t)+α z(t)−γ v(t) , where each time interval is one day, r is the risk-free rate, and z(t) follows a standard normal distribution. In this model, µ governs the equity risk premium. In the variance dynamic, α identifies the reactivity of variance to stock price, γ provides the leverage effect, and β partially captures the persistence of variance. Since the next-period daily variance v(t + 1) is determined once the daily return lnS(t) − lnS(t−1) or return shock z(t) is realized, it enables us to formulate the return transition likelihood recursively. At each point in time t+1, the log likelihood is given as 1 1 (cid:32) lnS(t+1)−lnS(t)− (cid:0) r+ (cid:0) µ− 1(cid:1) v(t+1) (cid:1)(cid:33)2 lnL(t+1) = − ln(2πv(t+1))− 2 , (2) (cid:112) 2 2 v(t+1) where the variance is given as (cid:32) lnS(t)−lnS(t−1)− (cid:0) r+ (cid:0) µ− 1(cid:1) v(t) (cid:1) (cid:112) (cid:33)2 v(t+1) = ω+βv(t)+α 2 −γ v(t) . (3) (cid:112) v(t) 5

2.1.2 The SV dynamics As our second model, we adopt the continuous-time stochastic volatility model of Heston (1993). Its physical dynamics are given as (cid:18) (cid:18) (cid:19) (cid:19) 1 (cid:112) dlnS(t) = r+ µ− v(t) dt+ v(t)dz (t), (4) 1 2 (cid:112) dv(t) = κ(θ−v(t))dt+σ v(t)dz (t), 2 where z and z follow Brownian motions with correlation coefficient ρ. The square-root process or 1 2 the CIR process used for the variance dynamic fundamentally captures a mean-reverting property of variance, where κ and θ identify the mean-reverting speed and the long-run mean, respectively. Formodelestimation,thepresenceoflatentstochasticvariancemakesimplementationrelatively more complex than that of the GARCH dynamics, as it requires filtering the latent variable.6 To reducethecomputationalburden, weinsteadadoptthemethodofHeston, Jacobs, andKim(2023), which exploits the linear relationship between the squared VIX and the instantaneous stochastic variance implied by this SV model.7 Specifically, we assume v(t) = η +η VIX2(t), (5) 0 1 where η and η are free parameters to be estimated. 0 1 To characterize the transition likelihood function, we apply the Euler discretization to equation (4), resulting in: (cid:20) (cid:18) (cid:19) (cid:21) 1 lnR(t+∆) = r+ µ− v(t) ∆+ϵ (t+∆), (6) 1 2 v(t+∆)−v(t) = κ(θ−v(t))∆+ϵ (t+∆), 2 where R(t + ∆) = S(t + ∆)/S(t) denotes the gross return and ∆ = 1/252.8 The error terms 6To filter stochastic variance, the literature employs various approaches, including the Kalman filter, particle filter, and Markov Chain Monte Carlo (MCMC) techniques. For examples, see Ghysels and Jasiak (1994) and Ruiz (1994) for applications of the Kalman filter, Christoffersen, Jacobs, and Mimouni (2010) and Dufays, Jacobs, Liu, andRombouts(2023)fortheparticlefilter,andAndersen,Benzoni,andLund(2002),Eraker,Johannes,andPolson (2003), Jones (2003), Eraker (2004), and Bates (2006) for MCMC methods. 7Theexactlinearrelationshipinvolvesaspecificfunctionalformwithameasurementerror. However,toavoidthe additional filtering problem required to model this error, they further simplify the setup by relaxing the functional form and omitting the measurement error. 8NotethatlnR(t+∆)isthedailylogreturnbetweentandt+∆whilev(t)istheannualizedvarianceattimet. 6

ϵ(t+∆) = (ϵ (t+∆), ϵ (t+∆))′ follow a bivariate normal distribution with mean and covariance 1 2 matrix given as     0 v(t) σρv(t) 0 =  , Σ(t) =  ∆.     0 σρv(t) σ2v(t) Then, the joint log likelihood at t+∆ is expressed as: lnL(t+∆) = lnf (cid:0) logR(t+∆),VIX2(t+∆)|VIX2(t) (cid:1) = ln[f(logR(t+∆),v(t+∆)|v(t))×J(t+∆)] 1 1 = −ln(2π)− ln|Σ(t)|− ϵ(t+∆)′Σ−1(t)ϵ(t+∆)+lnη , (7) 1 2 2 where f(lnR(t+∆),v(t+∆)|v(t)) represents the conditional density of the discretized logR(t+∆) and v(t+∆), J(t+∆) is the Jacobian term relating VIX2(t+∆) to v(t+∆), given by η from 1 equation (5). Time t is measured in days. 2.2 Risk-neutral dynamics and option pricing We now turn to the discussion of how we price options under the two dynamics (GARCH and SV), respectively. It is noteworthy to mention that our physical dynamics in equations (1) and (4) alone are insufficient for option pricing. Either the pricing kernel or risk-neutral dynamics are necessary tocomputemodel-impliedoptionprices. Inthissection, wedescribeourchoiceofthepricingkernel and the internally consistent risk-neutral dynamics, followed by the option valuation procedure. To compute option prices, we first specify the pricing kernel and then derive the corresponding risk-neutral stock price dynamics. Following Heston and Nandi (2000), we assume that the representative agent is risk-averse to negative aggregate market returns. Specifically, we adopt the power utility function, which is parsimonious and widely used in the literature. The pricing kernel is then given by (cid:18) S(t) (cid:19)−ξ M(t) = M(0)e−δt , (8) S(0) where ξ represents the risk aversion parameter, and δ represents the time-preference parameter. Inferred from the given physical dynamics and the pricing kernel in equation (8), we obtain the In contrast, following the convention, v(t) in equation (1) represents the daily variance in the GARCH model. 7

following risk-neutral dynamics.9 For the GARCH model, (cid:18) (cid:19) 1 (cid:112) lnS(t+1) = lnS(t)+ r− v(t+1) + v(t+1)z∗(t+1), (9) 2 (cid:16) (cid:112) (cid:17)2 v(t+1) = ω+βv(t)+α z∗(t)−γ∗ v(t) , where (cid:112) z∗(t) = z(t)+ξ v(t), γ∗ = γ +ξ, ξ = µ. For the SV model, (cid:18) (cid:19) 1 (cid:112) dlnS(t) = r− v(t) dt+ v(t)dz∗(t), (10) 2 1 (cid:112) dv(t) = κ∗(θ∗−v(t))dt+σ v(t)dz∗(t), 2 where (cid:112) dz∗(t) = dz (t)+ξ v(t)dt, 1 1 (cid:112) dz∗(t) = dz (t)+ξρσ v(t)dt, 2 2 κ∗ = κ+ξρσ, θ∗ = κθ/κ∗, ξ = µ. Using the risk-neutral dynamics in equations (9) and (10), we follow the fast Fourier transformation of Carr and Madan (1999) for option valuation. The price of a call option with its strike 9Christoffersen,Heston,andJacobs(2013)andHeston,Jacobs,andKim(2023)providemoredetailedderivation oftheserelationshipsfortheGARCHandtheSVmodels,respectively. Theyalsodiscussamoregeneralizedpricing kernel that preserves the same functional form of the risk-neutral dynamics. 8

price K and maturity τ is given by a quasi-closed-form solution involving a numerical integration: e−ak (cid:90) ∞ (cid:104) (cid:105) C(S(t),v(t),t) = Re e−iukψ(u) du, (11) π 0 where i is the imaginary unit and k is the natural log of K. The function ψ(u) is the Fourier transform of a modified call price, which is the call price multiplied by eak for a > 0.10 We calculate ψ(u) as follows: e−rτfCH(u−i(a+1)|S(t),v(t)) ψ(u) = τ , (a+iu)(a+1+iu) where fCH(ϕ|S(t),v(t)) = E∗(cid:2) eiϕlogS(t+τ)(cid:3) represents the risk-neutral conditional characteristic τ t function of logS(t+τ). We derive closed-form expressions for fCH(ϕ|S(t),v(t)) following Heston τ and Nandi (2000) for the GARCH model and Heston (1993) for the SV model. The price of a put option with the same strike and maturity can be determined using put-call parity. Note that the pricing formula in equation (11) does not account for future dividends. In line with the existing literature, we adjust the current index level by discounting it with the dividend yield; that is, we use S(t)e−qτ, where q is the continuously compounded dividend yield at time t. Lastly, we convert each model-implied option price into the Black–Scholes implied volatility to facilitate consistent comparisons across the cross section and over time. 2.3 Local M-estimation Most stock price dynamics are typically estimated using M-estimation, a flexible framework for parameter estimation that minimizes or maximizes a specified objective function: T 1 (cid:88) θˆ = argmin L(Y ,g (θ)), (12) T t t−1 θ T t=1 whereθ ∈ Θ ⊆ Rp. Inthisframework, Lrepresents the lossfunction, Y denotesthe targetvariable, and g(θ) defines a parametric model of the target functional. M-estimation encompasses a wide range of methods, including maximum likelihood estimation (MLE) and least squares estimation. Undermildregularityconditions, M-estimatorsexhibitdesirablepropertiessuchasconsistencyand 10We use a=4, following Heston, Jacobs, and Kim (2023). 9

√ T-asymptotic normality; see Newey and McFadden (1994). However, the standard M-estimation framework assumes implicitly that all observations contributeequallytotheobjectivefunction,whichlimitsitsabilitytoincorporatetime-specificinformationthatcouldimproveparameterestimation. Inthecurrentpricingenvironment,certainhistorical data points may hold greater informational value for parameter estimation. As demonstrated by Oh and Patton (2024), maintaining the baseline model while optimally weighting observations can enhance forecasting performance. Following a similar approach, we preserve the standard option pricing model but assign greater weights to past data points that are more relevant to the specific day of pricing. Building on the approaches of Dendramis, Kapetanios, and Marcellino (2020) and Oh and Patton (2024), we define the estimator as: T 1 (cid:88) θ˜ (s) = argmin L(Y ,g (θ))K(s−S ;h ), fors ∈ Int(S), (13) h,T t t−1 t−1 T θ T t=1 where K represents the kernel function, h is a bandwidth parameter that decreases with sample T size, s is a predetermined value of the state variable, typically set to its current value at the time theforecastisgenerated, andInt(S)denotestheinteriorofthesupportofthestatevariableS. The weights are determined by a kernel function based on the distance between the current and past levels of state variables. Past values that deviate significantly from the current level are assigned smaller weights, while those closer to the current level receive larger weights. This approach allows the underlying dynamics to adapt effectively to the current environment, potentially improving option price forecasts. This local approach offers significant advantages over traditional estimation methods, particularly in the context of option pricing. There are two main reasons for its effectiveness: 1. Resolution of model misspecification: Standard option pricing models often suffer from misspecification, as they may not fully capture the complex dynamics of volatilities of underlying assets. Thelocalestimationapproachhelpsmitigatethisissuebyallowingthemodelparametersto varywiththestatevariable, thusadaptingtoprevailingmarketconditionswheretheoptionpricing isconducted. Thisflexibilityenablesthemodeltoadjusttodifferentmarketregimesandconditions, effectively reducing the impact of model misspecification. By giving more weight to observations 10

that are similar to the current market state, the method implicitly allows for nonlinearities and time-varying relationships that may not be captured by a fixed parametric model. 2. Informative state variables: The effectiveness of this approach relies on the choice of informative state variables that contain valuable information about potential model misspecification. These state variables serve as proxies for changing market conditions or regimes that affect option prices. For instance, variables such as option implied volatility, skewness and kurtosis, or macroeconomic indicators such as realized variance of S&P 500 returns can provide crucial information aboutthecurrentmarketenvironment. Byconditioningtheestimationonthesestatevariables, the local approach incorporates this additional information into the pricing model, leading to more accurate and adaptive option price estimates. The state variables essentially guide the model to focus onthemostrelevanthistoricaldataforthecurrentpricingscenario, improvingitspredictivepower. By addressing model misspecification and leveraging informative state variables, this local estimation approach can potentially lead to more accurate option pricing, especially in dynamic and evolving market conditions where traditional fixed-parameter models may struggle.11 The efficacy of this local estimation approach hinges critically on the judicious selection of state variables and the determination of appropriate bandwidth parameters. In the following section, we present a detailed exposition of our methodology for selecting these crucial elements. In our empirical analyses, we use the negative log likelihood (−lnL) in equation (2) or (7) as a loss function L, in conjunction with the realized stock return as a target variable Y and either the GARCH dynamics or the SV dynamics for a parametric model g(θ). The set of parameters θ is {µ, ω, β, α, γ} for the GARCH dynamics and {µ, κ, θ, σ, ρ, η , η } for the SV dynamics. 0 1 Furthermore, we impose a couple of restrictions. To relevantly capture the unconditional level of equity risk premium, we fix µ at the level of the time-series average equity premium between January 1996 and August 2023. To preserve variance positivity, we impose ω ≥ 0 and β ≥ 0 for the GARCH model and the Feller (1951) condition (2κθ > σ2) for the SV model. 11Foracomprehensivetheoreticalanalysisoftheimprovementsaffordedbythislocalestimationapproach,werefer readers to Section 2 of Oh and Patton (2024), which provides detailed discussions on the asymptotic properties of the estimator. 11

2.4 Implementing Local Estimation and Model Evaluation: Methodological Details Toimplementthelocalestimationthroughequation(13), weneedtospecifytheweightingfunction andselectappropriatestatevariables. Webeginbydetailingthekernelfunctionsfordifferenttypes of state variables, then describe our process for bandwidth determination and model evaluation, and finally discuss the specific state variables used in our study. For stochastic state variables, we adopt a Gaussian kernel: (cid:26) x2 (cid:27) K (x;h) = exp − , for x ∈ R and h > 0 (14) G 2h2 Weexaminethebandwidthvalues,h,rangingfrom0.05σ to5σ ,whereσ representsthestandard S S S deviation of the state variable. Smaller bandwidths make the model parameters more ‘localized,’ but they reduce precision due to a smaller effective sample size. Conversely, as h increases, the local estimator converges to the non-local methods equivalent to the original estimation methods proposed by Heston and Nandi (2000) and Heston (1993). While h is a time-insensitive parameter, we implement a dynamic approach for σ to enhance precision. Specifically, we compute σ at each S S point in time as the rolling standard deviation of the state variable over the preceding three-year window. When employing time as the state variable, we use a one-sided exponential kernel defined by the bandwidth parameter λ and a window length of m: λj(1−λ) K (j;λ) = 1{j < m}, for j ∈ {0,1,2,...} and 0 < λ < 1 (15) E (1−λm) The values of λ explored range from 0.98 to 0.9999. Lower λ values result in reduced emphasis on older observations during estimation, leading to more temporally localized model parameters but increased estimation variability. As λ approaches 1, the weighting function flattens, and the local estimator converges to the non-local benchmark estimators described in Heston and Nandi (2000) and Heston (1993). Theprocessfordeterminingbandwidthparametersandevaluatingoptionpricingmodelsfollows 12

a structured approach. We divide our dataset into two distinct subsets: an estimation sample and an out-of-sample (OOS) set. The estimation sample is used to determine the optimal bandwidth parameter for each state variable. Subsequently, the selected bandwidth parameters are employed to evaluate option pricing errors during the OOS period, ensuring that bandwidth parameters are free from look-ahead bias.12 More specifically, our methodology follows a two-step process: first, identifying the optimal bandwidth, and second, evaluating the option pricing model in the OOS period. Step 1: Optimal Bandwidth Selection To determine the optimal bandwidth parameter for each state variable, we further subdivide the estimation sample into two parts: a ‘training sample’ (the first half) used to estimate model parameters across various bandwidths, and a ‘validation sample’ (the second half) used to identify the optimal bandwidth parameter. For each fixed bandwidth value13, we estimate the model using the training sample of underlying stock returns data through equation (13). We then calculate option pricing errors in the validation sample of option data using these estimates and pricing formulas. The bandwidth value yielding the lowest option pricing errors in the validation sample is selected as optimal. Step 2: OOS Model Evaluation Given the optimal bandwidth parameter for each state variable, we re-estimate the model using the entire estimation sample of underlying return data (i.e., combined training and validation samples) via equation (13). Given these parameter estimates with pricing formulas, we then calculate option pricing errors using option price data from the OOS period. To assess the performance of various models in our application, we compare their OOS average losses. It is important to note that these OOS losses are unweighted, which ensures that the local estimator does not have any inherent advantage in the evaluation process. This approach provides a fair and unbiased comparison between local and non-local estimation methods. 12Look-ahead bias occurs when a model incorporates information that would not have been available at the time of prediction. This could lead to unrealistically accurate estimates by inadvertently using future data in parameter estimation. 13Wetestbandwidthvaluesrangingfrom0.05σ to5σ with0.05incrementsforstochasticstatevariables,orfrom s s 0.98 to 0.9999 with 0.0005 increments for the time-based state variable. 13

For both validation and OOS evaluation, we first need to forecast option prices or implied volatilities. In this paper, we particularly rely on implied volatility. As a forecast measure for the i-day ahead implied volatility, we calculate the expected implied volatility by using parameter estimates at time t. To this end, we simulate the underlying price and variance for the next three days using their physical dynamics in equation (1) or (4), calculate option prices and implied volatilities over the joint distribution of the underlying price and variance, and then take their expectations for each time t+i. To evaluate pricing errors, we focus on forecasting performance for the next three-day option implied volatilities.14 Specifically, we evaluate option pricing performance by defining the the relative implied volatility error (IVR). Let IVR denote the mean squared relative errors of the t next three-day option implied volatilities:  (cid:104) (cid:105)2 1 (cid:88) 3 1 (cid:88) Nt IV t d + a i t , a j −E t IV t m +i o , d j el IVR t = 3 N  IVdata  , (16) i=1 t j=1 t+i,j (cid:104) (cid:105) whereIVdataistheBlack-Scholesimpliedvolatilityforoptionj attimet+iindata,andE IVmodel t+i,j t t+i,j isthetime-texpectedBlack-Scholesimpliedvolatilityobtainedfromthesimulationdiscussedabove. N represents the number of options included in the evaluation at time t. We then define IVR as t the time series average of IVR over an evaluation period, that is, t T 1 (cid:88) IVR = IVR . (17) t T t=1 We assess that a model is superior to another model when the former IVR is statistically smaller than the latter’s. Having established our methodology for local estimation and bandwidth determination, we now introduce the set of state variables considered to completely implement the local estimation. We consider six stochastic state variables that are well-established in the literature as informative and useful for pricing options or explaining option returns. These include two measures of volatility: 14We evaluate the performance every week. Excluding the evaluation date and a holiday if exists, we forecast the next three-day option implied volatilities. 14

the VIX and the 5-minute realized volatility (RV5) of the S&P 500 index. Furthermore, we use the VVIX, which captures volatility-of-volatility by measuring the expected volatility of the 30day forward price of the VIX.15 As tail measures, we incorporate model-free risk-neutral skewness and kurtosis based on Bakshi, Kapadia, and Madan (2003). We also include the variance risk premium (VRP), a measure of risk compensation that investors and policymakers use to assess sentimentregardinguncertainty. Inadditiontothesesixvariablesassociatedwithakernelfunction equation(14),weconsidertimeasastatevariableassociatedwithakernelfunctioninequation(15). This results in a total of seven possible state variables.16 3 Data and Empirical Setup In both the validation and out-of-sample (OOS) evaluation, we use daily S&P 500 call and put options with maturities between 14 and 183 days. The total sample period for the options data is from January 2011 to August 2023. We obtain the data from OptionMetrics and apply the following initial filters: discarding (i) options with implied volatility below 5% or above 150%, (ii) options with volume or open interest of fewer than ten contracts, (iii) options with mid price below $0.50 or a bid price below $0.375 to avoid low-valued options, and (iv) options with data errors, where the bid price exceeds the offer price or where a negative price is implied by put-call parity. Under our empirical setup, where we evaluate pricing performances for subsequent days of a week, we select one day per week to determine a weekly set of options. Following existing studies (see,e.g.,HestonandNandi,2000;Christoffersen,Heston,andJacobs,2013),wechooseWednesday because it is the day of the week least likely to be a holiday. It is also less likely than other days to be affected by day-of-the-week effects. Every Wednesday, we apply additional filters by retaining only out-of-the-money (OTM) call and put options and excluding those with moneyness below 0.75 or above 1.25. To enhance computational efficiency, instead of using all available data points, we select representative options that adequately span the implied volatility surface each Wednesday. We follow 15This forward price represents the price of a hypothetical VIX futures contract expiring in 30 days. 16The use of these state variables are supported by extant papers. For instance, Bali, Beckmeyer, Moerke, and Weigert (2023), Cao, Vasquez, Xiao, and Zhan (2023), and Horenstein, Vasquez, and Xiao (2025) highlight the importance ofvolatilitymeasures, includingimpliedvolatility, realizedvolatility, volatilityof volatility, andvariance risk premium; Kelly and Jiang (2014), tail risk. Oh and Patton (2024) also bolster the significance of volatility and time state variables in forecasting volatility. 15

the methodology of Dufays, Jacobs, Liu, and Rombouts (2023) to construct a balanced dataset. Specifically, we create eight buckets for moneyness (S/K) and five buckets for maturity.17 This results in 40 buckets, and we keep the most actively traded option in each bucket, although not all buckets are populated every week. Comparing the filtered set of Wednesday options with those available throughout the following week, we retain only the common options for each week from Wednesday. Thefinalfiltereddatasetcomprises64,970optioncontracts. PanelsAandBofTable1 provide descriptive statistics of the filtered option dataset. [Table 1 AROUND HERE] We obtain S&P 500 index returns from CRSP. The sample period of the return data is from January1996toAugust2023. Weusealongerreturnseriesthanoptionsdatatobettercapturethe time series dynamics of the underlying index’s variance. While extracting the underlying variance begins in 1996, our likelihood-based estimation sample starts in 2007, as all local estimation state variables are available from that year onward. We obtain the VIX and VVIX from Bloomberg. We calculate RV5 as the sum of open-to-close squared five-minute returns. The VRP is calculated as thedifferencebetweentheforward-lookingmarketvarianceimpliedbyoptionprices(VIX2)andthe annualized one-month forecast of realized variance over time (Bollerslev, Tauchen, and Zhou, 2009; Carr and Wu, 2009). We also compute model-free risk-neutral skewness (SKEW) and kurtosis (KURT) following Bakshi, Kapadia, and Madan (2003). Panel C of Table 1 reports descriptive statistics of these six state variables. The time series for the risk-free rate is proxied by the one-month Treasury Bill rates obtained from CRSP. Following the standard implementation in the literature, options are valued using a maturity-specific risk-free rate obtained from OptionMetrics. [Table 2 AROUND HERE] We conduct model estimation and evaluation across multiple periods using both expanding and rolling window approaches, as summarized in Table 2. In the expanding window approach, to determine an optimal bandwidth parameter, we initially estimate our models using data from 17In the moneyness dimension, we define buckets with moneyness levels spaced every 0.03 between 0.91 and 1.09, and assign two additional buckets for options with moneyness below 0.91 or above 1.09. In the maturity dimension, we define buckets with maturities of 30 days or less, 31–60 days, 61–90 days, 91–121 days, and more than 122 days. 16

January 2007 to December 2010 (training sample) and validate them using the subsequent period from January 2011 to December 2014 (validation sample). After identifying the optimal bandwidth parameter from this estimation sample, we conduct out-of-sample (OOS) evaluation for the period of January 2015 to December 2016. Our validation sample and OOS evaluation procedure employs an expanding window approach. We estimate the model at time t and evaluate it using calculated option prices from t+1 to t+3. We then expand the window to include t+1 data to estimate the model at t+1 and repeat the evaluation using t+2 to t+4 data. This method ensures our model is continuously updated with the most recent data. As illustrated in Table 2, we progressively expand our sample periods. For each subsequent period (P2, P3, and P4), we extend both the training and validation samples by one year and evaluate the next two years out-of-sample. This progressive expansion enables us to maintain the most appropriate bandwidth parameter that reflects recent marketconditions. Byregularlyupdatingourestimationwindow, wecanpotentiallyachievebetter results for both local and non-local estimations, as the model adapts to evolving market dynamics. It is important to note that for each subsequent period (P2, P3, and P4), we use a different bandwidthparameter,optimallychosenintherespectiveestimationsample. Thisapproachensures that our model remains responsive to changing market conditions across different time frames. In the rolling window approach, we fix the window size at 1,512 observations (approximately six years) for validation sample evaluation, and 2,016 observations (approximately eight years) for OOS evaluation. As the window advances one week at a time, we re-estimate the model using the most recent 1,512 or 2,016 data points, depending on the evaluation stage, and assess its performance over the subsequent three days. This procedure mirrors the evaluation horizon used in the expanding window setup but preserves a constant sample size throughout. By relying on a fixed-length, moving estimation sample, the rolling window approach offers a complementary perspective on model performance under evolving market conditions. 4 Empirical Results We report the out-of-sample (OOS) performance of the GARCH and SV models, specifically comparing several local estimation models with the non-local estimation approach. We then compare the best-performing local specification with the non-local model across various dimensions and 17

examine their associated economic implications. Finally, we assess the robustness of the local estimation model’s outperformance by examining alternative OOS performance metrics. 4.1 Out-of-sample option price forecasting performance Panel A of Table 3 provides a detailed evaluation of the pricing performance of Heston-Nandi GARCH option pricing models during the OOS period, spanning from January 2015 to August 2023, inferred from the expanding window estimation. These models are estimated using two distinct approaches: the non-local quasi-maximum likelihood (QML) method by Heston and Nandi (2000), denoted by ♦, and the local QML method described in Section 2.3. The table organizes the rows based on the average OOS loss, IVR, as defined in equation (17). Each local QML estimator employs a specific set of state variables and bandwidth parameters, which are outlined in the second and third columns, respectively. As described in Section 2.4, these bandwidth parameters are optimized based on the performance measured by the same loss function IVR in the validation sample. [Table 3 AROUND HERE] To further assess the relative performance of the models, the table reports Diebold-Mariano (DM) t-statistics based on Newey-West standard errors with five lags, following Diebold and Mariano (1995), which compare each model’s average OOS loss against that of the benchmark non-local QML method (denoted by ♦). Negative t-statistics indicate that a given model achieves a lower average loss compared to the benchmark. The final column provides an additional layer of evaluation by indicating, with a checkmark, whether a given model is included in the 95% model confidence set (MCS)—a procedure proposed by Hansen, Lunde, and Nason (2011).18 The benchmark method, which relies on non-local QML, performs the seventh among the eight estimation methods considered. Four local methods demonstrate significantly lower OOS losses compared to the benchmark, as evidenced by the DM test results, with their DM t-statistics falling below -2. The local method that achieves the best performance in the validation sample uses VIX as a state variable and also exhibits the lowest average loss during the OOS period. A direct 18To construct the MCS, we implement the stationary bootstrap, adopting an average block duration of ten observations. 18

comparison between the benchmark method and the best-performing local method selected from the validation sample yields a DM t-statistic of -5.85, providing strong evidence that the local approach outperforms the non-local benchmark. Among all estimation methods evaluated, the local methods utilizing VIX and time as state variables are included in the model confidence set. Panel B of Table 3 evaluates the pricing performance of SV option pricing models on the same dataset, inferred from the expanding window estimation. The benchmark method, which relies on non-local QML, ranks the seventh among the eight estimation methods. All local methods except the one with VVIX show significantly lower OOS losses compared to the benchmark, as indicated by the DM test results, with all DM t-statistics below -2. The local method that performs best in the validation sample incorporates time as a state variable and also exhibits the lowest average loss during the OOS period. Across the entire set of estimation methods, the local methods utilizing time, RV5, and VIX as state variables are included in the model confidence set. The two option pricing model classes, the GARCH and SV models, do not necessarily have to have the same state variable for the best-performing local estimation specification because one modelcancaptureacertaincharacteristicbetterthantheother,andviceversa. Asaresult,wefind that the optimal state variables for the GARCH and SV option pricing models are VIX and time, respectively. However, within the estimation specifications of the GARCH option pricing model, the time state variable is also included in the model confidence set, implying that the importance of the local time information is not statistically different from that of VIX. Similarly, within the SV model specifications, VIX is also included in the model confidence set. Additionally, Table 4 reports the OOS option pricing performance based on the rolling window estimation. Panel A presents results for the Heston-Nandi GARCH models, and Panel B for the Heston SV models. Interestingly, the MCS of each model class includes the same state variables as those identified using the expanding window approach. Although the most optimal state variable for the GARCH model under the rolling window is time, whereas VIX is the most optimal under the expanding window, both time and VIX are included in the MCS under both approaches. For the SV model, the state variables, time, RV5, and VIX, consistently constitute the MCS, with their rankings remaining unchanged. Similar to the expanding window approach, the non-local benchmark is ranked very low, namely seventh or eighth, with significantly higher OOS average losses compared to those from local estimation methods. Regardless of whether the expanding or 19

rolling window is used, the overall OOS pricing performances are highly consistent across the two approaches. [Table 4 AROUND HERE] 4.2 Model performance comparison and economic implications We further investigate when, where, and why a local estimation outperforms the non-local estimation. Specifically, we compare the time series and cross-sectional pricing errors of different specifications to see when and where performance improvement is pronounced. We also discuss how parameter estimates vary over time. Furthermore, simulated variance and return distributions help us better understand the differences between the local and non-local estimation performances. Lastly, since we estimate physical dynamics of the stock price and variance but evaluate the model using option prices, we are able to exploit the link between the physical distributions and the riskneutral distributions for an external validity test. Specifically, we employ expected option returns that measure the ratio between the physical expectation and the risk-neutral expectation of future payoffs. For brevity, in the following analyses, we focus on comparing the results between the non-local benchmark and the best-performing local estimation in either the GARCH or SV model. 4.2.1 Time series and cross sectional pricing errors Figure 1 presents the time series of out-of-sample (OOS) pricing errors. Panels A and C depict the errors under the GARCH models, based on the expanding window and rolling window approaches, respectively. Similarly, Panels B and D show the errors under the SV models, using the expanding window and rolling window approaches. As discussed in Tables 3 and 4, the local QML estimation with VIX as a state variable demonstrates the best performance for the expanding-window GARCH model in terms of the average loss (IVR) out of sample. For the other three models, the local estimation with time as a state variable performs best. The local estimation in all four panels generally outperforms the non-local benchmark during the pre-2019 period, while the difference is less pronounced during the post-2019 period, on average. Nevertheless, we still observe a statistically significant difference between the local and non-local approaches for the rolling-window GARCHmodelduringthepost-2019period. Acrossallpanels, thebiggestdifferenceoccursaround 20

2017, when the market was relatively calm. [Figure 1 AROUND HERE] Comparing the pricing errors between Panels A and C or between Panels B and D reveals that the time series dynamics of the non-local pricing errors are more dependent on the estimation data window than those of the local pricing errors. For instance, the non-local pricing errors during 2017 and 2018 decrease substantially when the estimation scheme changes from the expanding window to the rolling window, whereas the local pricing errors during the same period change little. This observation highlights a methodological advantage of local estimation. Unlike non-local estimation, which assigns equal weights to every data point, local estimation places more weight on states similar to current conditions, making it less dependent on the estimation sample size or period. In addition to comparing the time-series errors, we also examine the cross-sectional option pricing errors. Panels A and B of Table 5 present the OOS pricing errors from the expandingwindow GARCH model across moneyness and maturity buckets. Panel A shows the results from the local estimation using VIX, while Panel B displays the results from the non-local estimation. For deep OTM put options (S/K > 1.06), the non-local estimation error is slightly smaller, but for all other buckets, the local estimation error is significantly smaller. Furthermore, the non-local estimation exhibits larger variation in pricing errors across the moneyness dimension, whereas the local estimation shows much less variation overall. In other words, non-local estimation tends to focus more on reducing errors for OTM puts, while local estimation results in a more balanced allocation of importance across the cross-section. Ultimately, the use of local information incurs a slight performance cost for deep OTM puts but maintains small errors for those options while reducing overall error and improving performance. [Table 5 AROUND HERE] Panels C and D of Table 5 present the cross-sectional option pricing errors for the expandingwindow SV model. As shown in Table 3, the performance gap between the optimal local estimation and the non-local estimation is smaller in the SV model than in the GARCH model. The cross-sectional pricing errors follow the same pattern: the estimation using time-local information 21

statistically outperforms the non-local estimation overall, but the magnitude of improvement is smaller than in the GARCH model. Interestingly, unlike the GARCH model, the variation in errors under non-local estimation is more pronounced along the maturity dimension than the moneyness dimension for the SV model. When local estimation is used, the variation in errors across different maturities is significantly reduced, leading to more balanced results. Taken together, the SV option pricing model is already more improved compared to the GARCH model, and thus the marginal improvement from using local information is relatively smaller. Nevertheless, we still observe statistical improvement in the SV model as well. Table 6 repeats the cross-sectional pricing error analysis for the rolling-window GARCH and SV models. The rolling window approach produces highly consistent cross-sectional pricing error patterns with those from the expanding window approach, with one exception: the local GARCH model outperforms the non-local GARCH model for deep OTM put options. We again observe that local estimation leads to more balanced pricing errors across option maturity and moneyness compared to non-local estimation. Given the overall similarities between the results from the expanding window and rolling window approaches, we focus on further analyses based on the expanding window approach. [Table 6 AROUND HERE] 4.2.2 Time series of parameter estimates Panels A through D of Figure 2 depict the time series of the parameter estimates of ω, β, α, and γ, in order under the GARCH model. To get a better sense of economic intuition, we also report the variance persistence measured by β+αγ2 and the long-run variance measured by (ω+α)/(1− β −αγ2) in Panels E and F, respectively. The parameter estimates from the local estimation are highly time-varying, and they sometimes far deviate from the non-local estimates.19 For example, around 2017, which is a relatively calm period, the local estimate of the reactivity parameter α is approximately half of its non-local estimate, suggesting that variance is locally less reactive to the return shock during such a period. On the other hand, since the COVID-19 shock in 2020, we observe many data points where α is more than double of its non-local estimate, indicating that 19Notethatthenon-localparameterestimatesarealsoslightlytime-varyingbecauseweestimatethemeveryweek using the expanding window scheme. 22

variance is locally more reactive to the return shock. [Figure 2 AROUND HERE] Furthermore, variance persistence implied by the non-local estimation is set at between 0.95 and 0.98 over the course of the OOS period. In contrast, variance persistence implied by the local estimation ranges from 0.8 to 0.98, especially with low persistence during a calm period in 2017. This result indicates that variance tends to revert to its mean level relatively quickly during a period with low conditional variance, while it reverts to its mean level relatively slowly during a period with high conditional variance. Interestingly, the persistence implied by the local estimation rarely exceeds that implied by the non-local estimation. Figure 3 depicts the time series of parameter estimates under the SV model. As in the GARCH model, we also observe that the local parameter estimates are time-varying and highly volatile. First, κ represents the variance mean reversion speed, and 1−κ/252 corresponds to the variance’s daily persistence. In Panel A, we see that the local estimate of κ is almost always higher than the non-local estimate, which implies that the variance persistence implied by the local estimation is lower than that implied by the non-local estimation—consistent with the results from the GARCH modelestimation. Onedifferenceisthatwhilethepersistencewaslowestin2017undertheGARCH model, it is lowest during 2018–2019 under the SV model. Looking at θ, which corresponds to the long-run variance, the local estimate is consistently lower than the non-local estimate before the 2020 COVID-19 shock, but shows a sharp spike at the onset of COVID-19 and thereafter stays above or at a similar level to the non-local estimate. Compared to the GARCH model estimation results, one interesting point is that the locally estimated long-run variance in the GARCH model follows market volatility patterns like the VIX, while this pattern is relatively less pronounced in the SV model. [Figure 3 AROUND HERE] Most of the SV model’s local parameter estimates exhibit two jumps—one during the 2018 Volmageddon20 and another during the 2020 COVID-19 shock. Looking at the correlation parameter 20It refers to the sudden spike in volatility on February 5, 2018, which led to large losses in inverse volatility products. 23

ρ in Panel D, we see that the negative correlation between return and variance shocks changes significantly at these two events, with the magnitude of the correlation decreasing.21 Although not as large as in the SV model results, in fact, Panel D of Figure 2 shows that the leverage parameter γ in the GARCH model also exhibits a negative jump during the same periods, which likewise suggests a reduction in the size of the negative correlation. 4.2.3 Simulated return and variance distributions We have confirmed that both the GARCH and SV models improve their OOS pricing performances in both time series and cross section when local information is incorporated in estimation. Then, why does model performance improve when local information is used? To investigate this, we simulate returns and variances using each model’s estimates and examine their forward risk-neutral distributions. We select two representative days from our sample period—one with low conditional volatility and one with high. The year 2017 is a low-volatility period, and during this time, the differencebetweenlocalandnon-localestimationerrorsappearsconsistently,makingitareasonable period for this analysis. Additionally, the peak of the COVID-19 crisis is undoubtedly a highvolatility period. As a result, we select July 12, 2017, and March 18, 2020, as the respective lowand high-volatility days for analysis. For each model and each of the selected dates, we take the parameter estimates and run simulations to obtain the τ-horizon forward risk-neutral distribution.22 To assess how well these model-implied distributions match the actual distribution, we compare them against the nonparametric risk-neutral distribution of Breeden and Litzenberger (1978). Since we evaluate models out-of-sample by checking whether the model estimated today can price options accurately over the next three days, we compare the model-implied distributions with the data-driven nonparametric risk-neutral return distribution corresponding to the day after each of the two representative dates. Figures4and5showtheone-monthforwardrisk-neutraldistributionsfromtheGARCHandSV models, respectively.23 Panels A and B describe the return and variance distributions for the low 21This observation is in line with Pyun (2019). 22We run 50,000 simulation draws at a daily frequency. 23The overall implications remain consistent for other option maturities or horizons; we report the three-month distributions in Figures A.1 and A.2 in Appendix. 24

conditional volatility case, while Panels C and D correspond to the high conditional volatility case. In all panels, we show both the optimal local and non-local estimation results, and for the return distributions, we also display the nonparametric distribution for comparison. Since our analysis focuses only on OTM options, the portion of the distribution with negative returns (Return < 1) corresponds to the state prices used to compute OTM put option prices, while the positive return region (Return > 1) corresponds to that used for OTM call options. [Figure 4 AROUND HERE] Looking at Panels A and C of Figure 4, we can see that in the positive return region, the return distribution derived from the local estimation using VIX is much closer to the nonparametric distribution than that from the non-local estimation. This is consistent with the results from Table 5, where the difference in OTM call pricing errors between local and non-local estimations is substantial. Furthermore, in Panel C, the return distribution inferred from the local estimation is also closer to the nonparametric distribution in the negative return region, while in Panel A, the results are more mixed or similar. This partially aligns with the earlier observation that local estimation incurs some performance cost for deep OTM puts in the cross-sectional error analysis. [Figure 5 AROUND HERE] Moreover, Figure 5 shows that in the SV model, the results are similar under low conditional volatility, but under high volatility, the differences between the distributions inferred from the local and non-local estimations become much smaller. Indeed, Figure 3 shows that during the COVID-19 period, the difference in errors is quite minimal. This suggests that the SV model itself is already potentially more beneficial for capturing return and volatility dynamics during high-volatility periods. Taking a step further, why do the return distributions differ between local and non-local estimation? We can find a clue by looking at the variance distributions in Panels B and D. In the low conditional volatility case, the variance distribution from the local estimation is narrower and more concentrated around lower variances. As a result, the return distribution from the local estimation also becomes narrower and closer to the nonparametric distribution than that from the non-local estimation. In contrast, in the high conditional volatility case, the variance distribution from the 25

local estimation is much wider and more concentrated around higher variances. This makes the return distribution from the local estimation also wider and closer to the nonparametric distribution compared to the non-local estimation. The reason why the variance distributions change in this way is that as seen in Figure 2, the parameter estimates move appropriately in response to local information. For example, during low-volatility periods, both the long-run variance and persistence estimated by the local method are lower than those from the non-local estimation, resulting in patterns like Panel B. During high-volatility periods, the opposite pattern appears. 4.2.4 Expected option returns So far, we have explored various ways in which local estimation outperforms non-local estimation, focusing on how well each approach explains out-of-sample (OOS) option prices. A noteworthy aspect of our estimation framework is that the model parameters are estimated solely using return data to capture the physical dynamics, while the implied risk-neutral dynamics are derived under a powerutilityassumptiontocomputeoptionprices. Intheend,ourvalidationandOOSperformance assessments have concentrated on how well the risk-neutral dynamics align with actual outcomes or how accurately they explain future option prices. However, we have not yet examined the implications for the physical dynamics used in the estimation process. Therefore, we now aim to investigate whether both the physical and risk-neutral dynamics jointly contribute to performance improvement. To do this, we evaluate how well the local and non-local estimation methods explain option returns. Specifically, we compare the holding-toexpiration realized option returns with the model-implied expected option returns. Since we focus exclusively on index options, which are European-style, all payoffs are determined at maturity without early exercise. The option payoff is driven by the comparison between the index spot price at maturity and the strike price, and thus is determined by the physical dynamics. In contrast, the current option price is determined under risk-neutral dynamics. Consequently, the option return reflects information from both the physical and risk-neutral dynamics as of the current date. We compute the realized option return as follows: (S −K)+ (K −S )+ Retcall = T , and Retput = T , (18) t,T E∗ (cid:2) e−r(T−t)(S −K)+ (cid:3) t,T E∗ (cid:2) e−r(T−t)(K −S )+ (cid:3) t T t T 26

and the expected option return is calculated as: E [(S −K)+] E [(K −S )+] ERetcall = t T , and ERetput = t T . (19) t,T E∗ (cid:2) e−r(T−t)(S −K)+ (cid:3) t,T E∗ (cid:2) e−r(T−t)(K −S )+ (cid:3) t T t T In the model-implied expected option return, the numerator is the expectation under the physical measure. This can be derived in closed form using the same characteristic function employed for option pricing but using the physical dynamics instead of the risk-neutral dynamics. To ensure a more consistent comparison – given that characteristics such as maturity and moneynessvarydailyinthemarketdata–weutilizetheVolatilitySurfacedatafromOptionMetrics. This dataset provides interpolated market option prices on a uniform grid of Black-Scholes Deltas and maturities. We conduct our analysis across 1-month and 3-month horizons. The sample period matches our OOS window, and we consistently use option data from Wednesdays only to compute both realized and expected option returns. We then compare their time-series averages. Figure 6 presents the results. [Figure 6 AROUND HERE] We find that the expected option returns computed from the local estimation consistently align more closely with the realized option returns than those from the non-local estimation. This pattern holds across all panels, regardless of the underlying option pricing model (GARCH or SV) or maturity. When comparing local- and non-local-implied option returns, we observe that in-themoney options (roughly |Delta| > 50), whether calls (positive Delta) or puts (negative Delta), show relatively similar results between the two methods. However, as we move toward out-of-the-money options (|Delta| < 50), the divergence between the two expected returns grows in both calls and puts, with the local-implied return aligning more closely with the realized return. This pattern becomes more pronounced as the maturity increases. This finding connects benefits of the optimal physical dynamics obtained through the local estimation approach with the improved OOS option pricing performance. Taken together, they demonstrate that the local estimation provides superior economic implications in terms of option returns compared to the non-local estimation. 27

4.3 Robustness to choice of performance measures In this section, we investigate whether local estimation continues to robustly outperform non-local estimation even when alternative performance evaluation measures are used. 4.3.1 Performance measure: IVR given future stock prices and variances First, we assess the robustness of our main results by calculating future option prices and the corresponding implied volatilities under the assumption that future stock prices and variances are known and given. In the main analysis, we computed the expected value of implied volatility without access to any future information at the time of evaluation. However, forecasting option prices inherently involves two components: one related to predicting the option pricing model or its parameters, and the other related to the future evolution of the underlying prices and variances. Since we forecast implied volatilities directly without explicitly forecasting the underlying variables better, it is difficult to clearly disentangle the sources of uncertainty associated with each component. To address this concern, we conduct an additional analysis in which future option values are computed based on ex-post realized future stock prices and variances, and examine whether the results remain robust. Table 7 presents the results. Overall, the findings are highly consistent with our main out-ofsample performance results reported in Table 3. For the GARCH and SV models, local estimation using the VIX and time as the state variable, respectively, continues to deliver the best performance in both the validation and out-of-sample periods. One notable difference is that, while the MCS set under the SV model includes time, RV5, and VIX in Table 3, only time and RV5 are included in Table 7. Nevertheless, VIX still ranks third and yields highly negative DM statistics. Looking at average losses, we find that the GARCH models show a reduction of approximately 0.001 to 0.005 across different specifications compared to the main results. For the SV models, the reduction ranges from about 0.002 to 0.005. While these improvements reflect the benefit of knowing future underlying prices, the relative performance of different estimation methods remains largely unchanged. This suggests that the ranking of estimation methods is robust to this assumption—that is, whether or not future stock prices and variances are known, the local estimation method continues to outperform the non-local approach. 28

[Table 7 AROUND HERE] 4.3.2 Performance measure: variance of delta-hedged portfolio return Second,wetesttherobustnessofourapproachusinganalternativeperformancemeasureinplaceof the relative implied volatility error, which we used in the main analysis. As an alternative measure, we consider the variance of returns from a delta-hedged portfolio.24 Delta hedging is commonly used in option trading to reduce or neutralize directional risk. However, as emphasized by Hull and White (2017), reducing the variance of the portfolio’s return may be more important than simply achieving zero Delta, as this accounts for risk from both price changes and expected changes in volatility. Furthermore, while in the Black and Scholes (1973) model the delta-hedged portfolio has zero Delta, this is no longer the case under models such as the Heston-Nandi GARCH or the Heston SV model. Therefore, rather than simply analyzing the returns of the hedged portfolio, we use the variance of those returns as our performance measure to test the effectiveness of the estimation approach. Table 8 presents the results. Looking at Panel A, which shows the performance of the Heston- Nandi GARCH model, we find that using time as the local information variable yields the best performance in both the validation and OOS samples under this new performance measure. However, the results from using VIX as the local variable are also very close to those of the time specification. VIX is included in the MCS set and significantly outperforms the non-local estimation in a statistically meaningful way. Compared to the main results in Panel A of Table 3, although the rankings have shifted slightly, it is strongly consistent that VIX and time remain the two best-performing state variables despite the change in performance measure. Panel B shows the results for the Heston SV model using the new performance measure. Here, time performs best in the validation sample, whereas in the OOS sample, VIX leads, followed by RV5 and time, in that order. Nevertheless, the differences in OOS average loss are small, all three state variables are included in the MCS set, and they statistically outperform the non-local estimation. ComparingthiswithPanelBofTable3,whereIVRisusedastheperformancemeasure, 24The delta-hedged call return calculation follows Almeida, Freire, and Hizmeri (2025): max(S(t+τ)−K,0)−Call(t,τ,K)−∆(S(t+τ)−S(t))−(Call(t,τ,K)−∆S(t))rτ . S(t) 29

we again find that the three state variables consist of the MCS set. In sum, we evaluate 1- to 3-day-ahead option forecast performance using two different option pricing models and two different performance measures. Across all cases, a local estimation consistently and statistically outperforms the non-local estimation, demonstrating the effectiveness of the local approach for option forecasting. It is also encouraging to see that, despite using different performance metrics, the best-performing state variables or those included in the MCS set remain virtually unchanged. [Table 8 AROUND HERE] 5 Conclusion ThispaperdevelopsandempiricallyevaluatesalocalM-estimationframeworkthatenhancesout-ofsample option pricing accuracy by incorporating state-dependent information into standard pricing models. Rather than modifying the model structure, the framework reweights historical observations based on their relevance to current market conditions using kernel-based techniques. We implement this approach for both GARCH and stochastic volatility (SV) models, conditioning on a range of variables including VIX, realized volatility (RV5), variance risk premium (VRP), VVIX, skewness, kurtosis, and calendar time. Ourempiricalanalysis,basedondatafrom2015to2023,showsthatthelocalestimationmethod consistently outperforms traditional non-local approaches in forecasting near-term option prices. These improvements are statistically significant in both time-series and cross-sectional dimensions. For the GARCH model, VIX- or time-based local estimation achieves the lowest implied volatility forecast errors and is included in the model confidence set. For the SV model, time-based local estimation performs best, with RV5 and VIX also yielding strong results. The gains are especially pronounced in calm periods around 2017, where local parameter estimates differ substantially from their non-local counterparts. In addition, local estimation reduces pricing error dispersion across moneyness and maturities, resulting in more balanced cross-sectional performance. Furthermore, local estimation produces risk-neutral return distributions more closely aligned with nonparametric benchmarks, particularly in low-volatility environments. Model-implied expected option returns under local estimation more accurately match realized returns, reflecting 30

improved alignment between physical and risk-neutral dynamics. Lastly, robustness checks using alternative evaluation metrics—including forecasts given ex-post realized underlyings and deltahedged return variances—corroborate the consistency and superiority of the local approach. Overall,ourfindingsunderscorethepracticalvalueofadaptingestimationmethodstoprevailing market conditions. Without increasing model complexity, the local estimation framework yields substantial gains in predictive accuracy, making it particularly well-suited for applications such as risk management, margin setting, and regulatory forecasting. Future research may extend this framework to incorporate jumps, alternative pricing kernels, or machine learning techniques for automated state variable selection and bandwidth tuning. 31

References A¨ıt-Sahalia, Yacine, Mustafa Karaman, and Loriano Mancini, 2020, The Term Structure of Equity and Variance Risk Premia, Journal of Econometrics. A¨ıt-Sahalia, Yacine, Chenxu Li, and Chen Xu Li, 2021, Closed-form implied volatility surfaces for stochastic volatility models with jumps, Journal of Econometrics 222, 364–392. Almeida, Caio, Jianqing Fan, Gustavo Freire, and Francesca Tang, 2023, Can a machine correct option pricing models?, Journal of Business & Economic Statistics 41, 995–1009. Almeida, Caio, Gustavo Freire, and Rodrigo Hizmeri, 2025, 0DTE asset pricing, Working paper, Princeton University. Andersen, Torben G, Luca Benzoni, and Jesper Lund, 2002, An empirical investigation of continuous-time equity return models, The Journal of Finance 57, 1239–1284. Andersen, Torben G., Nicola Fusari, and Viktor Todorov, 2015, Parametric Inference and Dynamic State Recovery from Option Panels, Econometrica 83, 1081–1145. Babaog˘lu, Kadir, Peter Christoffersen, Steven Heston, and Kris Jacobs, 2018, Option valuation with volatility components, fat tails, and nonmonotonic pricing kernels, The Review of Asset Pricing Studies 8, 183–231. Bakshi, Gurdip, CharlesCao, andZhiwuChen, 1997, EmpiricalPerformanceofAlternativeOption Pricing Models, Journal of Finance 52, 2003–2049. Bakshi, Gurdip, Nikunj Kapadia, and Dilip Madan, 2003, Stock return characteristics, skew laws, and the differential pricing of individual equity options, Review of Financial Studies 16, 101–143. Bali, Turan G, Heiner Beckmeyer, Mathis Moerke, and Florian Weigert, 2023, Option return predictability with machine learning and big data, Review of Financial Studies 36, 3548–3602. Bardgett, Chris, Elise Gourier, and Markus Leippold, 2019, Inferring Volatility Dynamics and Risk Premia from the S&P 500 and VIX Markets, Journal of Financial Economics 131, 593–618. Bates, David S., 2000, Post-’87 Crash Fears in the S&P 500 Futures Option Market, Journal of Econometrics 94, 181–238. Bates, David S., 2006, Maximum Likelihood Estimation of Latent Affine Processes, Review of Financial Studies 19, 909–965. Bates, David S., 2019, How Crashes Develop: Intradaily Volatility and Crash Evolution, Journal of Finance 74, 193–238. Black, Fischer, andMyronScholes, 1973, ThePricingofOptionsandCorporateLiabilities, Journal of Political Economy 81, 637–654. Bollerslev, Tim, George Tauchen, and Hao Zhou, 2009, Expected Stock Returns and Variance Risk Premia, Review of Financial Studies 22, 4463–4492. Breeden, DouglasT., andRobertH.Litzenberger, 1978, PricesofState-ContingentClaimsImplicit in Option Prices, Journal of Business 51, 621–651. 32

Cao, Jie Jay, Aurelio Vasquez, Xiao Xiao, and Xintong Eunice Zhan, 2023, Why does volatility uncertainty predict equity option returns?, Quarterly Journal of Finance 13, 2350005. Carr, Peter, and Dilip Madan, 1999, Option Valuation Using the Fast Fourier Transform, Journal of Computational Finance 2, 61–73. Carr, Peter, and Liuren Wu, 2009, Variance risk premiums, Review of Financial Studies 22, 1311– 1341. Chen, Ying, Maria Grith, and Hannah LH Lai, 2025, Neural tangent kernel in implied volatility forecasting: a nonlinear functional autoregression approach, Journal of Business & Economic Statistics pp. 1–15. Christoffersen, Peter, Steven Heston, and Kris Jacobs, 2009, The Shape and Term Structure of the Index Option Smirk: Why Multifactor Stochastic Volatility Models Work So Well, Management Science 55, 1914–1932. Christoffersen, Peter, Steven Heston, and Kris Jacobs, 2013, Capturing Option Anomalies with a Variance-Dependent Pricing Kernel, Review of Financial Studies 26, 1963–2006. Christoffersen, Peter, Kris Jacobs, and Karim Mimouni, 2010, Volatility Dynamics forthe S&P500: EvidencefromRealizedVolatility,DailyReturns,andOptionPrices,Review of Financial Studies 23, 3141–3189. Christoffersen, Peter, Kris Jacobs, Chayawat Ornthanalai, and Yintian Wang, 2008, Option valuation with long-run and short-run volatility components, Journal of Financial Economics 90, 272–297. Dendramis, Y., G. Kapetanios, and M. Marcellino, 2020, A Similarity-Based Approach for Macroeconomic Forecasting, Journal of the Royal Statistical Society Series A: Statistics in Society 183, 801–827. Diebold, Francis X., and Roberto S. Mariano, 1995, Comparing Predictive Accuracy, Journal of Business & Economic Statistics 13, 253–263. Du, Du, and Dan Luo, 2019, The pricing of jump propagation: Evidence from spot and options markets, Management Science 65, 2360–2387. Dufays, Arnaud, Kris Jacobs, Yuguo Liu, and Jeroen Rombouts, 2023, Fast Filtering with Large Option Panels: Implications for Asset Pricing, Journal of Financial and Quantitative Analysis pp. 1–56. Dufays, Arnaud, Kris Jacobs, and Jeroen Rombouts, 2025, A Framework for Real-Time Modeling and Forecasting of Large Unbalanced Option Implied Volatility Surfaces, Working paper, University of Houston. Duffie, Darrell, Jun Pan, and Kenneth Singleton, 2000, Transform Analysis and Asset Pricing for Affine Jump-Diffusions, Econometrica 68, 1343–1376. Eraker, Bjørn, 2004, Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices, Journal of Finance 59, 1367–1404. Eraker, Bjørn, Michael Johannes, and Nicholas Polson, 2003, The Impact of Jumps in Volatility and Returns, Journal of Finance 58, 1269–1300. 33

Feller, William, 1951, Two Singular Diffusion Problems, The Annals of Mathematics 54, 173–182. Ghysels, Eric, and Joanna Jasiak, 1994, Stochastic volatility and time deformation: an application oftradingvolumeandleverageeffects,Cahier de recherche, Universit´e de Montr´eal. D´epartement de sciences ´economiques. Gruber, Peter H, Claudio Tebaldi, and Fabio Trojani, 2021, The price of the smile and variance risk premia, Management Science 67, 4056–4074. Hansen, Peter R., Asger Lunde, and James M. Nason, 2011, The Model Confidence Set, Econometrica 79, 453–497. Hansen, Peter Reinhard, and Chen Tong, 2025, Option Pricing with Time-Varying Volatility Risk Aversion, Review of Financial Studies, Forthcoming. Heston, Steven, Kris Jacobs, and Hyung Joo Kim, 2023, The Pricing Kernel in Options, Finance and Economics Discussion Series, Board of Governors of the Federal Reserve System 2023-053. Heston, Steven, Kris Jacobs, and Hyung Joo Kim, 2024, A new closed-form discrete-time option pricing model with stochastic volatility, Working paper, University of Maryland. Heston, Steven L., 1993, A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, Review of Financial Studies 6, 327–343. Heston, Steven L., and Saikat Nandi, 2000, A Closed-Form GARCH Option Valuation Model, Review of Financial Studies 13, 585–625. Horenstein, Alex R, Aurelio Vasquez, and Xiao Xiao, 2025, Common factors in equity option returns, Working paper, University of Cambridge. Hull,John,andAlanWhite,2017,Optimaldeltahedgingforoptions,JournalofBanking&Finance 82, 180–190. Jones, Christopher S., 2003, The Dynamics of Stochastic Volatility: Evidence from Underlying and Options Markets, Journal of Econometrics 116, 181–224. Kelly, Bryan, and Hao Jiang, 2014, Tail risk and asset prices, The Review of Financial Studies 27, 2841–2871. Kim, Hyung Joo, 2025, Characterizing the conditional pricing kernel: A new approach, Working paper. Newey, WhitneyK., andDanielMcFadden, 1994, Chapter36Largesampleestimationandhypothesis testing, in (Elsevier, ). Oh, Dong Hwan, and Andrew J. Patton, 2024, Better the devil you know: Improved forecasts from imperfect models, Journal of Econometrics 242, 105767. Pan, Jun, 2002, The Jump-Risk Premia Implicit in Options: Evidence from an Integrated Time- Series Study, Journal of Financial Economics 63, 3–50. Pyun,Sungjune,2019,Varianceriskinaggregatestockreturnsandtime-varyingreturnpredictability, Journal of Financial Economics 132, 150–174. 34

Ruiz, Esther, 1994, Quasi-maximum likelihood estimation of stochastic volatility models, Journal of econometrics 63, 289–306. Schreindorfer, David, and Tobias Sichert, 2025, Conditional risk and the pricing kernel, Journal of Financial Economics, Forthcoming. Zhan, Xintong, Bing Han, Jie Cao, and Qing Tong, 2022, Option return predictability, The Review of Financial Studies 35, 1394–1442. 35

Table 1: Summary statistics Panel A: Option data by moneyness Number of contracts Average IV (%) Average price Average spread S/K ≤0.91 2,472 18.015 11.799 0.678 0.91<S/K ≤0.94 4,857 15.011 17.993 0.818 0.94<S/K ≤0.97 8,562 13.857 25.958 0.923 0.97<S/K ≤1.00 10,045 14.913 54.112 1.267 1.00<S/K ≤1.03 10,138 17.054 62.736 1.415 1.03<S/K ≤1.06 10,084 19.489 42.598 1.109 1.06<S/K ≤1.09 9,129 21.280 30.880 0.966 S/K >1.09 9,683 25.867 21.640 0.836 All 64,970 18.471 37.546 1.058 Panel B: Option data by maturity Number of contracts Average IV (%) Average price Average spread DTM≤30 10,720 18.977 16.246 0.630 30<DTM≤61 19,200 18.373 26.190 0.883 61<DTM≤91 15,575 18.207 38.477 1.140 91<DTM≤122 10,420 18.565 52.693 1.304 DTM>122 9,055 18.422 67.814 1.510 Panel C: State variables Mean Standard deviation Skewness Kurtosis VIX (%) 20.357 8.241 2.081 10.743 RV5 0.023 0.053 13.284 312.590 VRP 0.012 0.022 5.276 78.072 VVIX (%) 93.280 16.429 1.177 5.847 SKEW -2.544 1.295 -1.411 4.817 KURT 21.091 20.733 2.272 8.361 Notes: Panels A and B present descriptive statistics for closing OTM option contracts for the January 2011 to August 2023 period. Panel C reports on the six state variables for the January 2007 to August 2023 period. VIX and VVIX are in percentage terms. Mean and standard deviation of the first four state variables (VIX, RV5, VRP, and VVIX) are annualized. Source: Bloomberg Finance LP; OptionMetrics, Ivy DB US, via WRDS; and authors’ calculations. 36

Table 2: Sample periods for estimation and out-of-sample evaluation Panel A: Expanding window Estimation sample Out-of-sample (OOS) Training sample Validation sample Start End Start End Start End P1 Jan 2007 Dec 2010 Jan 2011 Dec 2014 Jan 2015 Dec 2016 P2 Jan 2007 Dec 2011 Jan 2012 Dec 2016 Jan 2017 Dec 2018 P3 Jan 2007 Dec 2012 Jan 2013 Dec 2018 Jan 2019 Dec 2020 P4 Jan 2007 Dec 2013 Jan 2014 Dec 2020 Jan 2021 Aug 2023 Panel B: Rolling window Estimation sample Out-of-sample (OOS) Training sample Validation sample Start End Start End Start End P1 Jan 2007 Dec 2012 Jan 2013 Dec 2014 Jan 2015 Dec 2016 P2 Jan 2009 Dec 2014 Jan 2015 Dec 2016 Jan 2017 Dec 2018 P3 Jan 2011 Dec 2016 Jan 2017 Dec 2018 Jan 2019 Dec 2020 P4 Jan 2013 Dec 2018 Jan 2019 Dec 2020 Jan 2021 Aug 2023 Notes: This table outlines the sequential sample periods used for model estimation and out-of-sample (OOS) evaluation with an expanding window (Panel A) and a rolling window (Panel B). Each row in each panel corresponds to a forecasting period (P1 through P4), in which the model is first trained on historical data (training sample), then validated on a holdout window (validation sample) to select optimal bandwidths, and finally evaluated using an OOS test. The estimation sample includes both the training and validation periods. The validation and OOS periods involve weekly forecasts of three-day-ahead option prices and are strictly forward-looking, ensuring no look-ahead bias in model assessment. 37

Table 3: Out-of-sample option pricing performance: expanding window approach Panel A: Heston-Nandi GARCH model Rank State variable Optimal bandwidth parameters Out-of-sample performance P1 P2 P3 P4 Avg. loss DM stat MCS 1∗ VIX 1.51 1.31 1.16 1.16 0.0315 -5.85 ✓ 2 time 0.9945 0.9945 0.9945 0.9945 0.0315 -5.32 ✓ 3 VRP 0.50 0.50 0.50 0.65 0.0448 -3.70 4 RV5 1.01 1.01 0.96 1.01 0.0473 -3.21 5 BKMskew 0.65 0.55 0.60 0.60 0.0575 -1.00 6 BKMkurt 0.70 0.45 0.50 0.70 0.0630 -0.09 7 ♦ 0.0637 8 VVIX 1.50 1.35 0.50 1.20 0.0659 0.38 Panel B: Heston SV model Rank State variable Optimal bandwidth parameters Out-of-sample performance P1 P2 P3 P4 Avg. loss DM stat MCS 1∗ time 0.9980 0.9975 0.9975 0.9975 0.0181 -4.82 ✓ 2 RV5 2.71 2.31 2.01 2.01 0.0192 -5.34 ✓ 3 VIX 4.51 3.36 2.86 2.16 0.0207 -5.22 ✓ 4 VRP 3.15 1.95 1.70 1.30 0.0226 -4.37 5 BKMskew 1.45 1.30 0.95 0.85 0.0228 -4.77 6 BKMkurt 1.85 1.60 0.80 0.85 0.0269 -2.78 7 ♦ 0.0310 8 VVIX 2.00 1.90 1.95 1.90 0.0315 0.36 Notes: This table presents metrics evaluating pricing performance during the out-of-sample (OOS) period, January 2015 to August 2023, for Heston-Nandi GARCH option pricing models (Panel A) and Heston SV option pricing models (Panel B). These models are estimated using either the (non-local) quasi-maximum likelihood (QML) approach, indicated by ♦, or the local QML approach. In each panel, the rows are ordered by the average OOS loss, calculated as the relative implied volatility error in equation (17). The local method that demonstrates the best performance in the validation sample (the latter half of the estimation sample) is highlighted with an asterisk (*) in the first column. Local estimators utilize state variables listed in the second column and bandwidth parameters specified in the next four columns, which are optimized using the validation sample. All models are estimated using an expanding window. The penultimate column reports Diebold-Mariano t-statistics for each model compared to the benchmark non-local method (identified as ♦), where negative values suggest lower average loss. Finally, the last column marks models included in the 95% model confidence set with a checkmark. 38

Table 4: Out-of-sample option pricing performance: rolling window approach Panel A: Heston-Nandi GARCH model Rank State variable Optimal bandwidth parameters Out-of-sample performance P1 P2 P3 P4 Avg. loss DM stat MCS 1∗ time 0.9940 0.9970 0.9965 0.9940 0.0329 -6.57 ✓ 2 VIX 1.36 1.56 1.61 1.26 0.0384 -6.50 ✓ 3 RV5 0.96 1.56 1.06 1.11 0.0541 -3.79 4 BKMkurt 0.30 0.65 1.05 2.25 0.0550 -3.99 5 BKMskew 0.15 0.30 0.90 2.25 0.0557 -3.62 6 VRP 0.40 0.60 0.80 1.95 0.0595 -2.03 7 ♦ 0.0726 8 VVIX 0.25 1.85 0.65 1.40 0.0797 0.83 Panel B: Heston SV model Rank State variable Optimal bandwidth parameters Out-of-sample performance P1 P2 P3 P4 Avg. loss DM stat MCS 1∗ time 0.9960 0.9985 0.9985 0.9970 0.0187 -4.81 ✓ 2 RV5 2.06 4.71 2.91 1.81 0.0194 -5.84 ✓ 3 VIX 2.66 4.96 4.96 1.41 0.0199 -5.48 ✓ 4 VRP 2.00 5.00 3.25 1.85 0.0217 -3.58 5 BKMskew 1.35 5.00 1.85 2.20 0.0221 -4.24 6 BKMkurt 1.50 5.00 1.65 1.90 0.0229 -3.94 7 VVIX 1.45 3.65 3.00 2.60 0.0233 -1.47 8 ♦ 0.0243 Notes: This table presents metrics evaluating pricing performance during the out-of-sample (OOS) period, January 2015 to August 2023, for Heston-Nandi GARCH option pricing models (Panel A) and Heston SV option pricing models (Panel B). These models are estimated using either the (non-local) quasi-maximum likelihood (QML) approach, indicated by ♦, or the local QML approach. In each panel, the rows are ordered by the average OOS loss, calculated as the relative implied volatility error in equation (17). The local method that demonstrates the best performance in the validation sample (the latter half of the estimation sample) is highlighted with an asterisk (*) in the first column. Local estimators utilize state variables listed in the second column and bandwidth parameters specified in the next four columns, which are optimized using the validation sample. All models are estimated using a rolling window. The penultimate column reports Diebold-Mariano t-statistics for each model compared to the benchmark non-local method (identified as ♦), where negative values suggest lower average loss. Finally, the last column marks models included in the 95% model confidence set with a checkmark. 39

Table 5: Out-of-sample option pricing error by maturity and moneyness: expanding window approach Heston-Nandi GARCH model Panel A: Local - VIX S/K ≤0.94 0.94<S/K ≤1.00 1.00<S/K ≤1.06 S/K >1.06 τ ≤30 0.0456 0.0558 0.0204 0.0429 30<τ ≤61 0.0330 0.0435 0.0163 0.0337 61<τ ≤91 0.0330 0.0322 0.0149 0.0342 91<τ ≤122 0.0387 0.0237 0.0158 0.0368 τ >122 0.0361 0.0194 0.0160 0.0405 Panel B: Nonlocal S/K ≤0.94 0.94<S/K ≤1.00 1.00<S/K ≤1.06 S/K >1.06 τ ≤30 0.0727 0.1206 0.0410 0.0314 30<τ ≤61 0.0507 0.1192 0.0435 0.0243 61<τ ≤91 0.0639 0.1217 0.0445 0.0227 91<τ ≤122 0.0828 0.1077 0.0402 0.0240 τ >122 0.0896 0.0880 0.0395 0.0253 Heston SV model Panel C: Local - time S/K ≤0.94 0.94<S/K ≤1.00 1.00<S/K ≤1.06 S/K >1.06 τ ≤30 0.0359 0.0310 0.0138 0.0359 30<τ ≤61 0.0251 0.0209 0.0096 0.0212 61<τ ≤91 0.0207 0.0184 0.0089 0.0169 91<τ ≤122 0.0222 0.0167 0.0097 0.0179 τ >122 0.0240 0.0165 0.0110 0.0189 Panel D: Nonlocal S/K ≤0.94 0.94<S/K ≤1.00 1.00<S/K ≤1.06 S/K >1.06 τ ≤30 0.0304 0.0445 0.0113 0.0252 30<τ ≤61 0.0272 0.0470 0.0138 0.0120 61<τ ≤91 0.0364 0.0615 0.0183 0.0085 91<τ ≤122 0.0543 0.0661 0.0199 0.0099 τ >122 0.0693 0.0690 0.0264 0.0118 Notes: This table presents the cross-sectional out-of-sample (OOS) pricing errors from Heston-Nandi GARCH option pricing models (Panels A and B) and Heston SV option pricing models (Panels C and D) across moneyness and maturity buckets. Panels A and C show the results from the local estimation using the VIX and time as a state variable, respectively, while Panels B and D show the result from the non-local estimation. All models are estimated using an expanding window. 40

Table 6: Out-of-sample option pricing error by maturity and moneyness: rolling window approach Heston-Nandi GARCH model Panel A: Local - time S/K ≤0.94 0.94<S/K ≤1.00 1.00<S/K ≤1.06 S/K >1.06 τ ≤30 0.0912 0.0608 0.0262 0.0463 30<τ ≤61 0.0492 0.0439 0.0210 0.0347 61<τ ≤91 0.0347 0.0335 0.0192 0.0335 91<τ ≤122 0.0412 0.0265 0.0193 0.0364 τ >122 0.0244 0.0172 0.0196 0.0355 Panel B: Nonlocal S/K ≤0.94 0.94<S/K ≤1.00 1.00<S/K ≤1.06 S/K >1.06 τ ≤30 0.1238 0.1233 0.0503 0.0461 30<τ ≤61 0.0726 0.1213 0.0530 0.0434 61<τ ≤91 0.0784 0.1223 0.0529 0.0450 91<τ ≤122 0.1009 0.1029 0.0474 0.0463 τ >122 0.0878 0.0791 0.0453 0.0509 Heston SV model Panel C: Local - time S/K ≤0.94 0.94<S/K ≤1.00 1.00<S/K ≤1.06 S/K >1.06 τ ≤30 0.0347 0.0288 0.0139 0.0368 30<τ ≤61 0.0240 0.0199 0.0103 0.0224 61<τ ≤91 0.0185 0.0187 0.0106 0.0186 91<τ ≤122 0.0207 0.0184 0.0120 0.0200 τ >122 0.0232 0.0183 0.0137 0.0212 Panel D: Nonlocal S/K ≤0.94 0.94<S/K ≤1.00 1.00<S/K ≤1.06 S/K >1.06 τ ≤30 0.0305 0.0340 0.0123 0.0340 30<τ ≤61 0.0252 0.0281 0.0102 0.0198 61<τ ≤91 0.0325 0.0341 0.0129 0.0164 91<τ ≤122 0.0455 0.0374 0.0157 0.0181 τ >122 0.0510 0.0405 0.0196 0.0207 Notes: This table presents the cross-sectional out-of-sample (OOS) pricing errors from Heston-Nandi GARCH option pricing models (Panels A and B) and Heston SV option pricing models (Panels C and D) across moneyness and maturity buckets. Panels A and C show the results from the local estimation using time as a state variable, while Panels B and D show the result from the non-local estimation. All models are estimated using a rolling window. 41

Table 7: Out-of-sample pricing performance given future stock prices and variances Panel A: Heston-Nandi GARCH model Rank State variable Optimal bandwidth parameters Out-of-sample performance P1 P2 P3 P4 Avg. loss DM stat MCS 1∗ VIX 1.51 1.36 1.21 1.16 0.0300 -5.78 ✓ 2 time 0.9950 0.9950 0.9955 0.9955 0.0313 -4.92 ✓ 3 VRP 0.60 0.50 0.50 0.65 0.0427 -3.27 4 RV5 1.16 1.06 0.96 1.06 0.0445 -3.12 5 BKMskew 1.05 0.55 0.65 0.65 0.0553 -0.62 6 ♦ 0.0586 7 BKMkurt 0.70 0.45 0.70 0.70 0.0589 0.05 8 VVIX 1.90 1.90 0.50 1.20 0.0616 0.51 Panel B: Heston SV model Rank State variable Optimal bandwidth parameters Out-of-sample performance P1 P2 P3 P4 Avg. loss DM stat MCS 1∗ time 0.9980 0.9980 0.9975 0.9980 0.0158 -5.07 ✓ 2 RV5 2.81 2.46 2.16 2.16 0.0171 -5.42 ✓ 3 VIX 4.61 4.16 3.36 2.56 0.0186 -4.99 4 BKMskew 1.45 1.25 1.00 0.90 0.0190 -5.16 5 VRP 3.00 1.95 1.65 1.25 0.0206 -3.61 6 BKMkurt 1.80 1.55 0.80 0.85 0.0228 -2.82 7 ♦ 0.0265 8 VVIX 2.00 1.90 2.00 1.95 0.0269 0.37 Notes: This table presents metrics evaluating pricing performance during the out-of-sample (OOS) period, January 2015 to August 2023, for Heston-Nandi GARCH option pricing models (Panel A) and Heston SV option pricing models (Panel B). These models are estimated using either the (non-local) quasi-maximum likelihood (QML) approach, indicated by ♦, or the local QML approach. In each panel, the rows are ordered by the average OOS loss, calculated as the relative implied volatility error under the assumption that future stock prices and variances are known and given. The local method that demonstrates the best performance in the validation sample (the latter half of the estimation sample) is highlighted with an asterisk (*) in the first column. Local estimators utilize state variables listed in the second column and bandwidth parameters specified in the next four columns, which are optimized using the validation sample. All models are estimated using an expanding window. The penultimate column reports Diebold-Mariano t-statistics for each model compared to the benchmark non-local method (identified as ♦), where negative values suggest lower average loss. Finally, the last column marks models included in the 95% model confidence set with a checkmark. 42

Table 8: Out-of-sample delta hedging performance Panel A: Heston-Nandi GARCH model Rank State variable Optimal bandwidth parameters Out-of-sample performance P1 P2 P3 P4 Avg. loss DM stat MCS 1∗ time 0.9960 0.9935 0.9935 0.9905 8.22E-06 -3.99 ✓ 2 VIX 1.96 2.11 1.86 1.46 9.72E-06 -3.35 ✓ 3 RV5 1.56 1.61 1.41 1.41 1.08E-05 -2.07 4 ♦ 1.20E-05 5 BKMkurt 0.70 0.55 0.70 1.30 1.29E-05 0.94 6 BKMskew 1.05 0.50 0.30 1.35 1.33E-05 1.35 7 VRP 0.55 0.50 0.50 1.15 1.38E-05 0.69 8 VVIX 0.60 0.35 0.35 1.25 1.91E-05 0.95 Panel B: Heston SV model Rank State variable Optimal bandwidth parameters Out-of-sample performance P1 P2 P3 P4 Avg. loss DM stat MCS 1 VIX 1.56 1.71 1.61 1.36 6.43E-06 -7.68 ✓ 2 RV5 2.06 1.91 1.76 1.56 6.50E-06 -8.23 ✓ 3∗ time 0.997 0.9975 0.9975 0.9965 7.04E-06 -3.55 ✓ 4 VRP 4.75 1.95 1.75 1.35 7.67E-06 -3.59 5 BKMskew 2.10 1.35 1.10 0.90 7.70E-06 -3.68 6 BKMkurt 2.70 1.95 0.80 0.95 8.49E-06 -0.69 7 ♦ 8.67E-06 8 VVIX 1.95 1.90 2.00 1.60 8.93E-06 0.95 Notes: Notes: This table presents metrics evaluating delta hedging performance during the out-of-sample (OOS) period, January 2015 to August 2023, for Heston-Nandi GARCH option pricing models (Panel A) and Heston SV option pricing models (Panel B). These models are estimated using either the (non-local) quasi-maximum likelihood (QML) approach, indicated by ♦, or the local QML approach. In each panel, the rows are ordered by the average OOS loss, calculated as the variance of returns from a delta-hedged portfolio. The local method that demonstrates the best performance in the validation sample (the latter half of the estimation sample) is highlighted with an asterisk (*) in the first column. Local estimators utilize state variables listed in the second column and bandwidth parameters specified in the next four columns, which are optimized using the validation sample. All models are estimated using an expanding window. The penultimate column reports Diebold-Mariano t-statistics for each model compared to the benchmark non-local method (identified as ♦), where negative values suggest lower average loss. Finally, the last column marks models included in the 95% model confidence set with a checkmark. 43

Figure 1: Time-series out-of-sample option pricing error Panel A: GARCH, Expanding window Panel B: SV, Expanding window Panel C: GARCH, Rolling window Panel D: SV, Rolling window Notes: This figure shows the time-series out-of-sample (OOS) option pricing errors measured by the relative IV error in equation (16) for the Heston-Nandi GARCH model (Panels A and C) and for the Heston SV model (Panels B and D). Panels A and B present the results from the expanding window estimation, while Panels C and D present the results from the rolling window estimation. The solid blue lines represent the results from the local estimation using the VIX as the state variable, and the dashed red lines represent the results from the non-local estimation. 44

Figure 2: Time-series out-of-sample GARCH parameter estimates Panel A: ω Panel B: β Panel C: α Panel D: γ Panel E: Variance persistence Panel F: Long-run variance Notes: This figure shows the time-series out-of-sample (OOS) GARCH model parameter estimates. Panels A, B, C, and D plot the estimates of ω, β, α, and γ, respectively. Panels E and F plot the model-implied variance persistence and long-run variance, measured by β+αγ2 and (ω+α)/(1−β−αγ2), respectively. The solid blue lines represent the results from the local estimation using the VIX as the state variable, and the dashed red lines represent the results from the non-local estimation. 45

Figure 3: Time-series out-of-sample SV parameter estimates Panel A: κ Panel B: θ Panel C: σ Panel D: ρ Notes: This figure shows the time-series out-of-sample (OOS) SV model parameter estimates. Panels A, B, C, and D plot the estimates of κ, θ, σ, and ρ, respectively. The solid blue lines represent the results from the local estimation using time as the state variable, and the dashed red lines represent the results from the non-local estimation. 46

Figure 4: GARCH-simulated one-month forward variance and return distributions Low conditional volatility, 07/12/2017 Panel A: Return Panel B: Variance High conditional volatility, 03/18/2020 Panel C: Return Panel D: Variance Notes: This figure shows the simulated risk-neutral distributions of the one-month forward return and annualized daily variance for Heston-Nandi GARCH option pricing models. Panels A and B display these distributions conditional on a low-volatility environment, using parameter estimates from July 12, 2017. Assuming the underlying stock price is observed the following day, the distributions of one-month forward returns and variances are simulated for that day. Panels C and D show the corresponding distributions under a high-volatility environment, using parameter estimates from March 18, 2020, with an identical simulation procedure to the low-volatility case. The solid blue lines represent the results from the local estimation using the VIX as the state variable, the dashed red lines represent the results from the non-local estimation, and the dotted black lines represent the nonparametric return distribution from data by Breeden and Litzenberger (1978). Source: OptionMetrics, Ivy DB US, via WRDS; and authors’ calculations. 47

Figure 5: SV-simulated one-month forward variance and return distributions Low conditional volatility, 07/12/2017 Panel A: Return Panel B: Variance High conditional volatility, 03/18/2020 Panel C: Return Panel D: Variance Notes: This figure shows the simulated risk-neutral distributions of the one-month forward return and annualized daily variance for Heston SV option pricing models. Panels A and B display these distributions conditional on a low-volatility environment, using parameter estimates from July 12, 2017. Assuming the underlying stock price is observed the following day, the distributions of one-month forward returns and variances are simulated for that day. Panels C and D show the corresponding distributions under a high-volatility environment, using parameter estimates from March 18, 2020, with an identical simulation procedure to the low-volatility case. The solid blue lines represent the results from the local estimation using time as the state variable, the dashed red lines represent the results from the non-local estimation, and the dotted black lines represent the nonparametric return distribution from data by Breeden and Litzenberger (1978). Source: OptionMetrics, Ivy DB US, via WRDS; and authors’ calculations. 48

Figure 6: Expected option returns Panel A: 1-month return, GARCH Panel B: 3-month return, GARCH Panel C: 1-month return, SV Panel D: 3-month return, SV Notes: This figure shows the comparison between the model-implied holding-to-expiration expected option returns with the realized option returns. Panels A and B present the one-month and three-month option returns under Heston-Nandi GARCH option pricing models across the Black-Scholes Delta dimension (x-axis). Panels C and D present the corresponding results for Heston SV option pricing models. The solid blue lines represent the result from the optimal local estimation, the dashed red lines represent the result from the non-local estimation, and the dotted black lines represent the data. Source: OptionMetrics, Ivy DB US, via WRDS; Center for Research in Security Prices, CRSP 1925 US Indices Database and CRSP US Treasury Database - Daily/Monthly (updated monthly), via WRDS; and authors’ calculations. 49

Appendix Figure A.1: GARCH-simulated three-month forward variance and return distributions Low conditional volatility, 07/12/2017 Panel A: Return Panel B: Variance High conditional volatility, 03/18/2020 Panel C: Return Panel D: Variance Notes: This figure shows the simulated risk-neutral distributions of the three-month forward return and annualized daily variance for Heston-Nandi GARCH option pricing models. Panels A and B display these distributions conditional on a low-volatility environment, using parameter estimates from July 12, 2017. Assuming the underlying stock price is observed the following day, the distributions of three-month forward returns and variances are simulated for that day. Panels C and D show the corresponding distributions under a high-volatility environment, using parameter estimates from March 18, 2020, with an identical simulation procedure to the low-volatility case. The solid blue lines represent the results from the local estimation using the VIX as the state variable, the dashed red lines represent the results from the non-local estimation, and the dotted black lines represent the nonparametric return distribution from data by Breeden and Litzenberger (1978). Source: OptionMetrics, Ivy DB US, via WRDS; and authors’ calculations. 50

Figure A.2: SV-simulated three-month forward variance and return distributions Low conditional volatility, 07/12/2017 Panel A: Return Panel B: Variance High conditional volatility, 03/18/2020 Panel C: Return Panel D: Variance Notes: This figure shows the simulated risk-neutral distributions of the three-month forward return and annualized daily variance for Heston SV option pricing models. Panels A and B display these distributions conditional on a low-volatility environment, using parameter estimates from July 12, 2017. Assuming the underlying stock price is observed the following day, the distributions of three-month forward returns and variances are simulated for that day. Panels C and D show the corresponding distributions under a high-volatility environment, using parameter estimates from March 18, 2020, with an identical simulation procedure to the low-volatility case. The solid blue lines represent the results from the local estimation using time as the state variable, the dashed red lines represent the results from the non-local estimation, and the dotted black lines represent the nonparametric return distribution from data by Breeden and Litzenberger (1978). Source: OptionMetrics, Ivy DB US, via WRDS; and authors’ calculations. 51