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Abstract

We propose a new method, domain stabilization (DStab), to enhance the return predictive and forecasting ability of model-free option-implied moment estimators. Analyzing S&P 500 options data from January 2015 to December 2021, we show that DStab improves moment estimation consistency by stabilizing the integration domain, leading to better predictive and forecasting performance. When the options data characteristics are appropriately considered, DStab enhances both in-sample predictive and out-of-sample forecasting abilities of implied moments. DStab’s out-of-sample forecasting ability surpasses other treatment methods.

Since Breeden and Litzenberger (1978) demonstrate that the option-implied risk-neutral density (RND) can be inferred from the prices of synthesized Arrow–Debreu securities, the RND has been widely recognized as a valuable source of information and has consequently become the topic of research in numerous studies (Carr and Wu 2014; Mijatović and Schneider 2014). However, a limitation of the informational value of implied RND is its formulation as a function rather than a variable. Consequently, to capture RND characteristics, moments are employed as variables in empirical analysis. As these moments provide information about the shape of the RND and are easily interpretable, they have been utilized in numerous empirical studies investigating the options market (Conrad, Dittmar, and Hameed 2020; Yeap, Kwok, and Choy 2018; Žikeš and Baruník 2014). Particularly, non-parametric methods have gained popularity because they avoid model misspecification errors, as demonstrated by Britten-Jones and Neuberger (2000) and Bakshi, Kapadia, and Madan (2003) (hereafter, BKM).

While model-free implied moment estimators offer theoretical robustness by not relying on models, they face practical hurdles due to the limited availability of option prices. The non-parametric estimation of implied moments requires a set of option prices covering a continuum of strike prices across the positive real line. However, as highlighted by the study of Jiang and Tian (2005) (hereafter, JT), obtaining such a set of options is practically infeasible for two reasons. First, options market participants quote only a discrete set of strike prices. Second, reliable option quotes for a significant portion of the deep-out-of-the-money (DOTM) or deep-in-the-money (DITM) regions of the strike price domain remain entirely unobservable. Additionally, some available DOTM option quotes require filtration for research purposes owing to liquidity and market-microstructure concerns (Ballestra, D’Innocenzo, and Guizzardi 2024; Hu et al. 2024; Lee, Ryu, and Yang 2021).

The first issue of quoting a discrete set of strike prices can be easily addressed. Several interpolation methods are available to estimate Black–Scholes implied volatility values across a range of observed strike prices for a discrete set of strike prices. Contrarily, the second issue of approximating option prices for the DOTM and DITM regions of the strike price domain, where option prices are completely unobservable, poses a considerable challenge. Because there are only one-sided reference points beyond the minimum and maximum strike prices from which option prices can be extrapolated, it is essential to rely on stronger assumptions to extrapolate. JT terms the errors due to this complete unavailability “truncation errors” because it appears as if the option observation samples are “truncated.”

Prior research introduces two model-free approaches to mitigate truncation errors. First, JT suggests flat extrapolation, a method that derives DOTM option prices by extending the Black–Scholes implied volatility curve with flat lines, which are then converted into out-of-the-money (OTM) option prices. Second, Dennis and Mayhew (2002) propose domain symmetrization (DSym), a technique that excludes available option prices until the minimum and maximum strike prices of the integration domain are equidistant from the underlying price. Although these methods highlight the drawbacks of model-free implied moment estimation and are potentially effective in alleviating truncation errors, previous studies have reported counterevidence. Regarding extrapolation, Pan, Shiu, and Wu (2022) argue that truncation treatment methods involving extrapolation may distort the tail distribution and weaken the information content of implied higher moments. Lee and Ryu (2024) show that implied higher moment estimators remain sensitive to truncation, even after applying extrapolation. Pan, Shiu, and Wu (2024) demonstrate that extrapolation does not enhance the predictive power of volatility forecasting models. In the case of DSym, Shelton, Kane, and Favreau (2021) show that the method may not reduce truncation error but rather lead to biased and inconsistent estimation results. Given this counterevidence, further discussion may be needed on how to address truncation errors.

The consequences of incomplete truncation error reduction can be categorized into two factors: bias and inconsistency. Since the strike prices of option price observations do not cover the entire positive real line, the estimation tends to underestimate the magnitude of the implied moments. Additionally, the width of the integration domain varies with the availability of option price observations over time, and this inconsistency introduces noise. One approach to address both issues is to reconsider whether it is necessary to persist in estimating the true value of the moments of the implied RND. While it is inevitable to do so in the study of BKM, which estimates co-skewness based on implied moment estimation, it may not be necessary when the objective is merely to summarize option-implied information in a single figure. Therefore, it is expected that the information content in option prices can be extracted more efficiently by utilizing the implied moment estimators of BKM to estimate an alternative set of variables that successfully summarize the option-implied information and can be estimated more consistently than the true implied moments.

This study introduces a new concept, termed “truncated implied RND (TRND),” and estimates the moments of TRND using the BKM estimators and a set of OTM option price observations that are insufficient for estimating the moments of implied RND but sufficient for estimating the moments of TRND, thereby mitigating the issues of bias and inconsistency. We propose a method called domain stabilization (DStab), which aims to estimate the moments of TRND with BKM estimators. By processing the option price observations to ensure the moments of TRND are estimated using BKM estimators, we expect DStab to improve estimation consistency, thereby enriching the information content of implied moments.

Based on the recent findings that short-maturity options have gained popularity in recent years (Andersen, Fusari, and Todorov 2017; Ahn, Kang, and Ryu 2008; Bryzgalova, Pavlova, and Sikorskaya 2023; Ryu and Yang 2020; Song, Ryu, and Webb 2018), we estimate TRND moments using DStab for a one-week maturity and demonstrate that these moments effectively predict underlying index returns. When the characteristics of the options data are appropriately accounted for, BKM’s implied moment estimators show better in-sample predictive ability and out-of-sample forecasting ability of underlying returns with DStab. Furthermore, the out-of-sample test results reveal that the return predictive and forecasting ability of implied moments with DStab outperforms implied moment estimates for which other treatment methods are applied. Thus, it can be argued that DStab enables more efficient extraction of information from OTM option prices by improving estimation consistency.

This study introduces an empirical methodology, DStab, designed to improve the information content, return-predictive power, and forecasting ability of model-free, option-implied moments. With our methodology, future empirical research is expected to investigate the information dynamics in options and other related markets more effectively, capturing the information content in OTM option prices more accurately. By examining the dynamics of the short-term options market, we demonstrate that the empirical methodology presented in this study is applicable even when considering the recent tendency toward short-term maturity preferences in options markets.

The remainder of this article is organized as follows. Section 1 describes the theoretical background. Section 2 introduces DStab, and Section 3 describes the S&P 500 index options data used in this study. Section 4 presents the results of the empirical analysis, and Section 5 concludes the article.

1 Theoretical Background

1.1 BKM’s Model-Free Implied Moment Estimator

The estimators of the implied volatility, skewness, and kurtosis according to BKM are given by the following equations:
(1)
 
(2)
 
(3)
where Vt,τ⁠, Wt,τ⁠, and Xt,τ represent the fair values of volatility, cubic, and quartic contracts at time t for maturity τ⁠, respectively, and r represents the risk-free rate. The names of the three contracts are derived from their respective payoff functions, defined as Rt,τ2⁠, Rt,τ3⁠, and Rt,τ4⁠, where Rt,τ is the holding period log return of the underlying asset. μt,τ denotes the expected holding period log return under the risk-neutral measure. According to BKM, the fair values can be represented as a portfolio of OTM option prices, given by the following equations:
(4)
 
(5)
 
(6)
where S(t) denotes the underlying price at time t⁠, and Ct,τ;K and Pt,τ;K denote the OTM call and put prices for a strike price K and maturity τ at time t⁠, respectively. Using the fair values, BKM approximates μt,τ as:
(7)

1.2 Truncation Error and Proposed Remedies

In most options markets, OTM option price observations are available for only a limited range of strike prices, particularly when data filtration is applied. The restricted availability of observations prevents the integration of option prices across the positive real line with respect to the strike price. Consequently, integrations are typically performed over a subset of the original integration domain, and Vt,τ⁠, Wt,τ⁠, and Xt,τ are approximated as:
(8)
 
(9)
 
(10)
where Kmin(t,τ) and Kmin(t,τ) represent the minimum and maximum strike prices for which the corresponding OTM price observations are available. Truncation error refers to the estimation error resulting from the substitution of Vt,τ⁠, Wt,τ⁠, and Xt,τ with V^t,τ⁠, W^t,τ⁠, and X^t,τ⁠, respectively.

To alleviate truncation error, JT proposes flat extrapolation using the following steps. First, OTM option price observations are transformed into Black–Scholes implied volatility to establish an implied volatility curve with respect to the strike price K⁠. The endpoints of this curve are Kmin(t,τ) and Kmax(t,τ)⁠. Second, the curve is extended as needed through extrapolation, assuming that the Black–Scholes implied volatility remains constant at the level examined at Kmin(t,τ) for K∈(0,Kmin(t,τ)⁠) and the level examined at Kmax(t,τ) for K∈(0,Kmax(t,τ)⁠). Finally, the implied volatility curve is converted back to OTM option prices, which now encompass a wider strike price domain, making truncation trivial.

Another method for reducing truncation error, DSym, aims to minimize the truncation error by addressing a specific concern: as the fair value of Wt,τ⁠, defined in Equation (5), is determined by the difference between the weighted portfolio value of OTM calls and puts, the estimate may be biased when more call price observations are available than put price observations, or vice versa. Recognizing this potential issue, DSym seeks to reduce the error by further excluding OTM price observations to ensure that Kmin(t,τ) and Kmax(t,τ) are equidistant from S(t)⁠.

2 Domain Stabilization

2.1 Truncation and TRND

If OTM option prices are available only for a strike price domain [Kmin(t,τ),Kmax(t,τ)] at time t and maturity τ⁠, where 0≤Kmin(t,τ)≤S(t)≤Kmax(t,τ)<∞⁠, and integrations are conducted only over this domain for the fair value estimation of Vt,τ⁠, Wt,τ⁠, and Xt,τ as defined in Equations (4)–(6), this can be interpreted as assuming that the OTM option prices are zero for the strike price domains [0,Kmin(t,τ)) and (Kmax(t,τ),∞)⁠. This is equivalent to assuming that the implied RND is truncated and the risk-neutral probability is zero for the corresponding domains, as demonstrated in the following proposition: 

Proposition 1.
If truncation exists for the strike price domains [0,Kmin(t,τ)) and (Kmax(t,τ),∞)⁠, and the fair value of V⁠, W⁠, and X in BKM is estimated without considering the OTM option prices on the truncated domain, it is equivalent to assuming that, for a risk-neutral probability measure P*⁠,
(11)
where Pt*· is the conditional probability operator under P* with respect to the filtration Ft at time t⁠.
Proof. BKM specifies a payoff spanning function HS in Equation (2) of their paper as
 
(12)
where IC is an indicator function whose value equals one when condition C holds and zero otherwise. HS· (⁠HSS·⁠) is the first-order (second-order) derivatives of a twice-continuously differentiable payoff function HS with respect to S⁠. For any nonnegative real constants Kmin and Kmax satisfying 0≤Kmin(t,τ)≤S¯≤Kmax(t,τ)<∞ for a real constant S¯⁠, Equation (16) can be rearranged as
 
(13)
Following BKM’s Equation (3), Equation (17) leads to
(14)
where Et*· is the expectation operator under P* with respect to Ft at time t⁠. Ct,τ;K and Pt,τ;K represent the OTM call and put prices at time t for maturity τ and strike price K⁠, respectively.
If option prices are omitted for strike price domains [0,Kmin(t,τ)) and (Kmax(t,τ),∞)⁠, it is equivalent to assuming that Pt,τ;K=0 for the former, and Ct,τ;K=0 for the latter. This leads to
(15)
which is equivalent to
(16)
or
(17)

Given that HSSK⁠, K-S⁠, and S-K in Equation (17) are well-defined for all HS⁠, S⁠, and K⁠, the only way to make the assumption specified by Equation (17) is to assume that IS>Kmax and IS<Kmin are almost surely zero under P*⁠, which is equivalent to Equation (11). □

Proposition 1 suggests that estimating implied moments with truncation in the integration domain is equivalent to assuming that the implied RND is truncated. Hence, when estimating implied moments using multiple sets of option price observations with varying degrees of truncation, we are effectively calculating moments from randomly truncated RNDs. Consequently, BKM estimators would estimate moments of implied densities based on differing assumptions, which may be inappropriate when the objective is to estimate a single variable.

One way to address this issue is to impose consistent assumptions regarding the truncation on implied RND throughout the sample, ensuring that BKM estimators estimate a single variable. As highlighted in Proposition 1, there is a close relationship between truncation on the implied RND and the integration domain. By stabilizing truncation across the sample, we ensure that the implied RND is truncated consistently throughout the dataset. This, in turn, allows us to measure the moments of a density that can be consistently defined across the entire sample, even when varying levels of truncation are present. This is precisely the goal of DStab.

2.2 Methodology

To stabilize the effect of truncation on implied moment estimates across the entire sample period, DStab aims to ensure consistency in the degree of truncation. Drawing on the methodologies of flat extrapolation and DSym, DStab works to mitigate the variability of the integration domain’s width and asymmetry by stabilizing the domain’s endpoints. Specifically, we determine fixed endpoints for the integration domain, filter out observations that fall beyond these endpoints, and use flat extrapolation to fill gaps when observations are insufficient. This procedure allows us to stabilize the degree of truncation throughout the entire sample period without relying too heavily on the assumption of a flat implied volatility curve.

The detailed procedure is as follows. First, we calculate and collect the endpoint values of the daily integration domains over the entire sample period. Next, we select the percentile of these endpoint values to serve as the fixed endpoints, thereby determining the intensity level of DStab. Finally, we stabilize the width and asymmetry of daily integration domains by eliminating option price observations that fall beyond the endpoint values and applying flat extrapolation up to the endpoint values when the observations are insufficient to cover the integration domain between the two fixed endpoints. For each intensity level i⁠, the threshold values are the ith percentile value of the put-side endpoint locations and the (100-i)th percentile value of the call-side endpoint locations.

To determine the endpoints of the daily integration domains, it is crucial to accurately measure the location within the integration domain to effectively stabilize the degree of truncation using DStab. Since BKM estimators estimate the moments of the implied log-return density, and the moments are volatility-adjusted by definition, both underlying log-returns and their volatility must be considered when measuring location. Therefore, we adopt d1 from the Black–Scholes option pricing formula to measure endpoint locations. d1 can be expressed as:
(18)
where S(t) represents the underlying price, K denotes the strike price, σ(t,τ) represents the underlying return volatility, and r(t,τ) denotes the risk-free rate.

3 Data

This study utilizes daily data on S&P 500 index options from January 2015 to December 2021. We retrieve the corresponding data from IvyDB OptionMetrics via Wharton Research Data Services, including closing bid and ask quotes, the underlying S&P 500 index level, and the continuously compounded zero-coupon interest rate and implied dividend rate curves. To approximate the closing prices, we calculate the midpoint of the quoted bid and ask prices at daily market closings. The risk-free and dividend rates for a specific maturity are determined by linearly interpolating the rates of the two nearest maturities. The zero curve provided by IvyDB OptionMetrics, derived from ICE LIBOR rates and CME Eurodollar futures settlement prices, is retrieved as the reference for risk-free rates. We use the implied dividend rate to calculate the discounted underlying prices employed in the implied moment estimation. The continuously compounded dividend rates are estimated by IvyDB based on put-call parity and constant dividend yield assumptions.

We apply a set of data filtration criteria to eliminate unreliable observations as follows. First, we remove options that are not OTM. Second, observations are discarded if the midpoint of the bid and ask quotes is below 0.375. Third, we remove observations in which the bid-ask spread is larger than the midpoint. Fourth, we include observations only if the daily trading volume is non-zero for the corresponding. Finally, we discard observations if any of the corresponding data entries are incomplete, the bid quote exceeds the ask quote or is zero, or if the no-arbitrage condition does not hold for the midpoint.

Table 1 provides the summary statistics for the filtered data, highlighting three notable findings. First, most observations have a time to maturity of no longer than two weeks, implying a recent tendency toward short-term maturity preference. Second, the implied volatility curve exhibits a skew or smirk, with implied volatility for puts being higher than that for calls. This pattern suggests that call observations are more affected by the minimum price filter. Third, the value of |d1| is more widespread for puts, which reveals that call observations are more severely truncated than puts, likely due to the minimum price filter. It should be noted that this asymmetric truncation is underestimated in Table 1 compared to its impact on the empirical analysis in this study. This underestimation arises from differences in the calculation procedure of d1⁠. For the summary statistics, we calculate d1 using the Black–Scholes implied volatility for each option to focus on the individual characteristics of observations. In contrast, when measuring the integration domain endpoints with |d1|⁠, we use the BKM model-free implied volatility to maintain consistency in the measurement. As a result, |d1| is underestimated for puts and overestimated for calls in Table 1.

Table 1
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Sample summary statistics

Panel A. OTM calls
Moneyness Maturity1st |d1| tertile group
2nd |d1| tertile group
3rd |d1| tertile group
Mean |d1|Mean σBS# of obs.Mean |d1|Mean σBS# of obs.Mean |d1|Mean σBS# of obs.
τ<70.3019.8314,0761.0217.3414,0771.7620.7514,077
7≤τ<140.3017.9340,7881.0815.1840,7891.8517.3940,789
τ≥140.2514.6011,3500.9912.4111,3501.7711.9411,351
Panel A. OTM calls
Moneyness Maturity1st |d1| tertile group
2nd |d1| tertile group
3rd |d1| tertile group
Mean |d1|Mean σBS# of obs.Mean |d1|Mean σBS# of obs.Mean |d1|Mean σBS# of obs.
τ<70.3019.8314,0761.0217.3414,0771.7620.7514,077
7≤τ<140.3017.9340,7881.0815.1840,7891.8517.3940,789
τ≥140.2514.6011,3500.9912.4111,3501.7711.9411,351
Panel B. OTM puts
Moneyness Maturity1st d1 tertile group
2nd d1 tertile group
3rd d1 tertile group
Mean d1Mean σBS# of obs.Mean d1Mean σBS# of obs.Mean d1Mean σBS# of obs.
τ<70.7023.1737,1461.6326.5437,1452.2038.3137,145
7≤τ<140.7621.97117,9791.7126.90117,9782.2938.69117,978
τ≥140.7719.2535,3721.7225.8135,3712.3135.0335,371
Panel B. OTM puts
Moneyness Maturity1st d1 tertile group
2nd d1 tertile group
3rd d1 tertile group
Mean d1Mean σBS# of obs.Mean d1Mean σBS# of obs.Mean d1Mean σBS# of obs.
τ<70.7023.1737,1461.6326.5437,1452.2038.3137,145
7≤τ<140.7621.97117,9791.7126.90117,9782.2938.69117,978
τ≥140.7719.2535,3721.7225.8135,3712.3135.0335,371

Notes: This table provides summary statistics for the data samples used in this study. After data filtration, the daily S&P 500 OTM index option price sample contains 770,132 observations, covering the period from January 2015 to December 2021. Panels A and B present the summary statistics for OTM calls and puts, respectively. We define moneyness using d1 from the Black–Scholes option pricing formula. τ represents the time to maturity in calendar days. σBS represents the Black–Scholes implied volatility, expressed as a percentage. We apply the following criteria to remove unreliable price observations: (1) the midpoint of bid and ask quotes is below 0.375; (2) the bid-ask spread is larger than the midpoint; (3) the daily trading volume is zero for the corresponding maturity; (4) any corresponding data entries are incomplete; (5) the bid quote exceeds the ask quote or is zero; and (6) the no-arbitrage condition does not hold for the midpoint price.

Table 1
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Sample summary statistics

Panel A. OTM calls
Moneyness Maturity1st |d1| tertile group
2nd |d1| tertile group
3rd |d1| tertile group
Mean |d1|Mean σBS# of obs.Mean |d1|Mean σBS# of obs.Mean |d1|Mean σBS# of obs.
τ<70.3019.8314,0761.0217.3414,0771.7620.7514,077
7≤τ<140.3017.9340,7881.0815.1840,7891.8517.3940,789
τ≥140.2514.6011,3500.9912.4111,3501.7711.9411,351
Panel A. OTM calls
Moneyness Maturity1st |d1| tertile group
2nd |d1| tertile group
3rd |d1| tertile group
Mean |d1|Mean σBS# of obs.Mean |d1|Mean σBS# of obs.Mean |d1|Mean σBS# of obs.
τ<70.3019.8314,0761.0217.3414,0771.7620.7514,077
7≤τ<140.3017.9340,7881.0815.1840,7891.8517.3940,789
τ≥140.2514.6011,3500.9912.4111,3501.7711.9411,351
Panel B. OTM puts
Moneyness Maturity1st d1 tertile group
2nd d1 tertile group
3rd d1 tertile group
Mean d1Mean σBS# of obs.Mean d1Mean σBS# of obs.Mean d1Mean σBS# of obs.
τ<70.7023.1737,1461.6326.5437,1452.2038.3137,145
7≤τ<140.7621.97117,9791.7126.90117,9782.2938.69117,978
τ≥140.7719.2535,3721.7225.8135,3712.3135.0335,371
Panel B. OTM puts
Moneyness Maturity1st d1 tertile group
2nd d1 tertile group
3rd d1 tertile group
Mean d1Mean σBS# of obs.Mean d1Mean σBS# of obs.Mean d1Mean σBS# of obs.
τ<70.7023.1737,1461.6326.5437,1452.2038.3137,145
7≤τ<140.7621.97117,9791.7126.90117,9782.2938.69117,978
τ≥140.7719.2535,3721.7225.8135,3712.3135.0335,371

Notes: This table provides summary statistics for the data samples used in this study. After data filtration, the daily S&P 500 OTM index option price sample contains 770,132 observations, covering the period from January 2015 to December 2021. Panels A and B present the summary statistics for OTM calls and puts, respectively. We define moneyness using d1 from the Black–Scholes option pricing formula. τ represents the time to maturity in calendar days. σBS represents the Black–Scholes implied volatility, expressed as a percentage. We apply the following criteria to remove unreliable price observations: (1) the midpoint of bid and ask quotes is below 0.375; (2) the bid-ask spread is larger than the midpoint; (3) the daily trading volume is zero for the corresponding maturity; (4) any corresponding data entries are incomplete; (5) the bid quote exceeds the ask quote or is zero; and (6) the no-arbitrage condition does not hold for the midpoint price.

After filtering the OTM options data, we construct daily implied volatility surfaces from which we extract the implied volatility curve for a single maturity of one week. This maturity is chosen to account for the recent trend of short-term maturity preference. We use the bilinear interpolation method, rather than the bicubic spline, to approximate surfaces, minimizing fluctuations and allowing for surface approximation based on observations for two maturities. Under this specification, we require observations for at least one maturity no longer than one week, and at least one maturity no shorter than one week. The required number of maturities can be reduced to one if observations for exactly a one-week maturity are available. We derive the implied volatility curve for the one-week maturity from daily surfaces, convert the implied volatility values to a set of OTM option prices, and trim the endpoints using a minimum option price filter of 0.375, which is equivalent to the filter applied to the S&P 500 index options data. The strike price gap is set to 0.25 to address estimation errors caused by strike price discreteness.

4 Empirical Analysis

4.1 Implied Moment Estimates and Truncation

Table 2 presents the summary statistics for the daily model-free implied moment estimates, which are calculated using BKM estimators without any truncation error treatment. The table shows that the implied RND exhibits negative skewness and leptokurtosis, which is consistent with the findings of several previous studies (Diavatopoulos et al. 2012; Duan and Zhang 2014). The underlying log-return distribution is also negatively skewed and leptokurtic, in line with the implied RND and prior research (Bali, Cakici, and Chabi-Yo 2015; Cheng et al. 2023). Given these characteristics, we can conclude that truncation errors are not significant enough to distort the fundamental features of the implied RND, although they may introduce noise and reduce the accuracy of tracking implied moment dynamics.

Table 2
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Daily underlying log-prices and model-free implied moment estimates

Panel A. Level
Panel B. First-order difference
ln(S)VOLSKEWKURTΔln(S)·102ΔVOLΔSKEWΔKURT
Mean7.9470.160−2.01712.8150.0560.0000.0000.006
Std. dev.0.2360.1010.9018.8361.2250.0390.5195.420
5th pct.7.6250.075−3.7954.581−1.743−0.047−0.835−7.630
25th pct.7.7690.102−2.4796.476−0.314−0.014−0.250−1.730
Median7.9280.135−1.7999.9000.079−0.001−0.002−0.043
75th pct.8.0870.186−1.33116.1310.5740.0110.2581.718
95th pct.8.3980.316−0.95232.0891.5140.0480.8088.061
Skewness0.4554.012−1.1661.841−1.2452.302−0.2310.245
Kurtosis2.40228.5774.4517.12424.38634.1086.88012.232
# of obs.1,5281,5281,5281,5281,5271,5271,5271,527
Panel A. Level
Panel B. First-order difference
ln(S)VOLSKEWKURTΔln(S)·102ΔVOLΔSKEWΔKURT
Mean7.9470.160−2.01712.8150.0560.0000.0000.006
Std. dev.0.2360.1010.9018.8361.2250.0390.5195.420
5th pct.7.6250.075−3.7954.581−1.743−0.047−0.835−7.630
25th pct.7.7690.102−2.4796.476−0.314−0.014−0.250−1.730
Median7.9280.135−1.7999.9000.079−0.001−0.002−0.043
75th pct.8.0870.186−1.33116.1310.5740.0110.2581.718
95th pct.8.3980.316−0.95232.0891.5140.0480.8088.061
Skewness0.4554.012−1.1661.841−1.2452.302−0.2310.245
Kurtosis2.40228.5774.4517.12424.38634.1086.88012.232
# of obs.1,5281,5281,5281,5281,5271,5271,5271,527

Notes: This table provides summary statistics for the daily model-free implied moment estimates derived using the BKM estimators, without applying any truncation error treatment. S, VOL, SKEW, and KURT denote the underlying price, implied volatility, skewness, and kurtosis, respectively. The option prices are approximated for a time to maturity of seven calendar days from daily implied volatility surfaces. Panels A and B present the summary statistics for the levels and first-order differences, respectively.

Table 2
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Daily underlying log-prices and model-free implied moment estimates

Panel A. Level
Panel B. First-order difference
ln(S)VOLSKEWKURTΔln(S)·102ΔVOLΔSKEWΔKURT
Mean7.9470.160−2.01712.8150.0560.0000.0000.006
Std. dev.0.2360.1010.9018.8361.2250.0390.5195.420
5th pct.7.6250.075−3.7954.581−1.743−0.047−0.835−7.630
25th pct.7.7690.102−2.4796.476−0.314−0.014−0.250−1.730
Median7.9280.135−1.7999.9000.079−0.001−0.002−0.043
75th pct.8.0870.186−1.33116.1310.5740.0110.2581.718
95th pct.8.3980.316−0.95232.0891.5140.0480.8088.061
Skewness0.4554.012−1.1661.841−1.2452.302−0.2310.245
Kurtosis2.40228.5774.4517.12424.38634.1086.88012.232
# of obs.1,5281,5281,5281,5281,5271,5271,5271,527
Panel A. Level
Panel B. First-order difference
ln(S)VOLSKEWKURTΔln(S)·102ΔVOLΔSKEWΔKURT
Mean7.9470.160−2.01712.8150.0560.0000.0000.006
Std. dev.0.2360.1010.9018.8361.2250.0390.5195.420
5th pct.7.6250.075−3.7954.581−1.743−0.047−0.835−7.630
25th pct.7.7690.102−2.4796.476−0.314−0.014−0.250−1.730
Median7.9280.135−1.7999.9000.079−0.001−0.002−0.043
75th pct.8.0870.186−1.33116.1310.5740.0110.2581.718
95th pct.8.3980.316−0.95232.0891.5140.0480.8088.061
Skewness0.4554.012−1.1661.841−1.2452.302−0.2310.245
Kurtosis2.40228.5774.4517.12424.38634.1086.88012.232
# of obs.1,5281,5281,5281,5281,5271,5271,5271,527

Notes: This table provides summary statistics for the daily model-free implied moment estimates derived using the BKM estimators, without applying any truncation error treatment. S, VOL, SKEW, and KURT denote the underlying price, implied volatility, skewness, and kurtosis, respectively. The option prices are approximated for a time to maturity of seven calendar days from daily implied volatility surfaces. Panels A and B present the summary statistics for the levels and first-order differences, respectively.

Figure 1 shows the time series dynamics of the implied moment estimates. Panel A depicts the dynamics of the implied volatility estimate. The panel clearly shows the time trend of the implied volatility level without being significantly affected by noise, suggesting that the implied volatility estimator is not greatly influenced by truncation in our dataset. In contrast, Panels B and C, which illustrate the dynamics of the implied skewness and kurtosis estimates, respectively, show that these dynamics are evidently driven by abrupt and heavy noise. Notably, the higher moment estimates are more unstable in 2017, 2018, and 2021, even compared to the COVID-19 outbreak in early 2020. If the integration domain exhibits a similar pattern, it may suggest that the noise in implied higher moment estimates is related to time-varying truncation.

Graphical representation of the time-series dynamics for three daily model-free implied moment estimates—volatility, skewness, and kurtosis—illustrating the daily levels of each estimate over time.
Figure 1

Time-series dynamics of daily model-free implied moment estimates.

Notes: This figure presents the time-series dynamics throughout the entire sample period from January 2015 to December 2021 of daily model-free implied moment estimates, derived using the BKM estimators, without applying any truncation error treatment. Panels A, B, and C illustrate the time-series dynamics of implied volatility, skewness, and kurtosis estimates, respectively.

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Figure 2 illustrates the time series dynamics of the integration domain measured in various terms. When the integration domain is measured in nominal price, as shown in Panels A and B, it appears relatively unstable in 2018, 2020, and 2021. Although it seems somewhat related, this pattern differs slightly from that observed in Figure 1, instead aligning more closely with the period during which the implied volatility estimate is less stable. In contrast, when measured in terms of d1⁠, the dynamics of the integration domain endpoints, especially those of the put-side endpoint, show a more evident tendency to become noisier when the implied higher moment estimates show instability. Given this similarity, the noise in the implied higher moments can be related to the noise in the width and asymmetry of the integration domain, particularly when the domain endpoint location is measured in terms of d1⁠.

Graphical representation of the time-series dynamics for the endpoints of the daily integration domain following the data filtration procedure, showing the daily minimum and maximum endpoints alongside the underlying price level.
Figure 2

Time-series dynamics of integration domain endpoints.

Notes: This figure illustrates the time-series dynamics of the endpoints of the daily integration domain after the data filtration procedure. The minimum and maximum endpoints are measured as Kmin and Kmax in Panel A, Kmin-S and Kmax-S in Panel B, and as -d1(Kmin) and -d1(Kmax) in Panel C, where S is the underlying price, Kmin and Kmax are the minimum and maximum strike prices of the integration domain, and d1 is derived from the Black–Scholes option pricing formula.

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To investigate the relationship between the dynamics of implied moment estimates and the integration domain endpoints in more detail, we conduct a set of regression analyses, employing the first-order differences of implied moments as the dependent variables, and the first-order differences of integration domain width and asymmetry, measured in terms of strike price and d1⁠, as the independent variables. We define domain width as the gap between the two endpoints, and domain asymmetry as the log-ratio of put-side width to call-side width. Table 3 reports the regression results, which support the visual implications of Figure 2. The results demonstrate that the dynamics of implied higher moments are closely related to domain width and asymmetry, especially when the domain endpoints are measured in terms of d1⁠. The fact that only the higher moment estimates, which exhibit noise, are closely related to the properties of the integration domain in terms of d1⁠, suggests a possibility to reduce the noise in the estimates, at least to some degree, by stabilizing these d1-denominated integration domain properties and consistently estimate the moments of TRND.

Table 3
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Relationship between the dynamics of implied moment estimates and integration domain

ΔVOL
ΔSKEW
ΔKURT
Strike priced1Strike priced1Strike priced1
Δ(Domain width)0.026***−0.008*−0.106***−0.229***1.252***3.807***
(7.67)(−1.83)(−8.51)(−11.63)(8.31)(14.91)
Δ(Domain asymmetry)−0.047***0.043***−1.779***−1.357***14.802***7.239
(−5.49)(3.84)(−28.04)(−21.33)(17.77)(9.50)
Intercept0.0000.0000.0000.000−0.0010.057*
(−0.14)(−0.05)(0.04)(0.07)(−0.01)(1.76)
# of obs.1,5271,5271,5271,5271,5271,526
R20.40950.03260.66490.72370.48630.6519
ΔVOL
ΔSKEW
ΔKURT
Strike priced1Strike priced1Strike priced1
Δ(Domain width)0.026***−0.008*−0.106***−0.229***1.252***3.807***
(7.67)(−1.83)(−8.51)(−11.63)(8.31)(14.91)
Δ(Domain asymmetry)−0.047***0.043***−1.779***−1.357***14.802***7.239
(−5.49)(3.84)(−28.04)(−21.33)(17.77)(9.50)
Intercept0.0000.0000.0000.000−0.0010.057*
(−0.14)(−0.05)(0.04)(0.07)(−0.01)(1.76)
# of obs.1,5271,5271,5271,5271,5271,526
R20.40950.03260.66490.72370.48630.6519

Notes: This table presents the OLS regression results of the first-order differences of implied moment estimates on the first-order differences of width and asymmetry of the daily integration domain, without any truncation error treatment method applied. VOL, SKEW, and KURT represent the implied volatility, skewness, and kurtosis, respectively, and Δ denotes the first-order difference operator. The integration domain width for puts and calls is measured as S-Kmin and Kmax-S when the domain endpoints are measured in terms of strike price, and as d1(Kmin) and -d1(Kmax) when the domain endpoints are measured in terms of d1 derived from the Black–Scholes option pricing formula. S is the underlying price, Kmin and Kmax are the minimum and maximum strike prices of the integration domain. The domain width is defined as Kmax-Kmin in terms of strike price and as d1Kmin-d1(Kmax) in terms of d1⁠. The domain asymmetry is defined as ln[(S-Kmin)/(Kmax-S)|] in terms of strike price and ln[d1(Kmin)/-d1(Kmax)] in terms of d1⁠. *** and * indicate statistical significance at the 1% and 10% levels, respectively.

Table 3
Open in new tab

Relationship between the dynamics of implied moment estimates and integration domain

ΔVOL
ΔSKEW
ΔKURT
Strike priced1Strike priced1Strike priced1
Δ(Domain width)0.026***−0.008*−0.106***−0.229***1.252***3.807***
(7.67)(−1.83)(−8.51)(−11.63)(8.31)(14.91)
Δ(Domain asymmetry)−0.047***0.043***−1.779***−1.357***14.802***7.239
(−5.49)(3.84)(−28.04)(−21.33)(17.77)(9.50)
Intercept0.0000.0000.0000.000−0.0010.057*
(−0.14)(−0.05)(0.04)(0.07)(−0.01)(1.76)
# of obs.1,5271,5271,5271,5271,5271,526
R20.40950.03260.66490.72370.48630.6519
ΔVOL
ΔSKEW
ΔKURT
Strike priced1Strike priced1Strike priced1
Δ(Domain width)0.026***−0.008*−0.106***−0.229***1.252***3.807***
(7.67)(−1.83)(−8.51)(−11.63)(8.31)(14.91)
Δ(Domain asymmetry)−0.047***0.043***−1.779***−1.357***14.802***7.239
(−5.49)(3.84)(−28.04)(−21.33)(17.77)(9.50)
Intercept0.0000.0000.0000.000−0.0010.057*
(−0.14)(−0.05)(0.04)(0.07)(−0.01)(1.76)
# of obs.1,5271,5271,5271,5271,5271,526
R20.40950.03260.66490.72370.48630.6519

Notes: This table presents the OLS regression results of the first-order differences of implied moment estimates on the first-order differences of width and asymmetry of the daily integration domain, without any truncation error treatment method applied. VOL, SKEW, and KURT represent the implied volatility, skewness, and kurtosis, respectively, and Δ denotes the first-order difference operator. The integration domain width for puts and calls is measured as S-Kmin and Kmax-S when the domain endpoints are measured in terms of strike price, and as d1(Kmin) and -d1(Kmax) when the domain endpoints are measured in terms of d1 derived from the Black–Scholes option pricing formula. S is the underlying price, Kmin and Kmax are the minimum and maximum strike prices of the integration domain. The domain width is defined as Kmax-Kmin in terms of strike price and as d1Kmin-d1(Kmax) in terms of d1⁠. The domain asymmetry is defined as ln[(S-Kmin)/(Kmax-S)|] in terms of strike price and ln[d1(Kmin)/-d1(Kmax)] in terms of d1⁠. *** and * indicate statistical significance at the 1% and 10% levels, respectively.

4.2 Applying Domain Stabilization

Figure 3 illustrates the changes in the integration domain when DStab is applied at various intensity levels. Panel A demonstrates the case for 0% DStab, which does not further discard OTM price observations but conducts flat extrapolation up to the sample maximum of |d1| for each of the puts and calls. Although no further sample exclusion leads to no loss of information, the strong assumption of a flat implied volatility curve significantly affects extrapolation-generated DOTM option prices, which are heavily weighted when estimating higher moments. When 50% DStab is employed, as shown in Panel B, the impact of this assumption decreases significantly, at the expense of the information contained in additionally filtered DOTM option price observations. Furthermore, although applied less extensively, the flat extrapolation is still in use, and the estimates are affected by the constant implied volatility assumption to some degree. The impact of the assumption is completely removed only when 100% DStab is applied, and observations are discarded when their |d1| exceeds the sample minimum for each of the puts of calls, as demonstrated in Panel C, so that there are no gaps to fill with extrapolation. It is noteworthy that the filtering is considerably extensive in this case, as illustrated by the narrow integration domain width in Panel C.

Graphical representation of the time-series dynamics for the integration domain in terms of Black–Scholes d1 after applying DStab at varying intensity levels—0%, 50%, and 100%—demonstrating progressively aggressive discarding of option price observations as the intensity level increases.
Figure 3

Integration domain after DStab.

Notes: This figure illustrates the time-series dynamics of the integration domain in terms of Black–Scholes d1 after applying DStab at various intensity levels. An n percent stabilization is implemented by further discarding OTM option price observations whose d1 is lower than the nth percentile of the minimum d1 or higher than the (100-n)th percentile of the maximum d1 for each daily integration domain. The areas filled with dark (light) colors indicate the integration domain covered by observed (extrapolated) OTM option prices. Panels A, B, and C present the time-series dynamics for 0%, 50%, and 100% stabilizations, respectively.

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Figure 4 demonstrates the changes in implied moment estimates when DStab is employed at varying intensity levels, revealing two interesting features. First, the implied volatility estimate is not significantly affected by DStab, even at the 100% intensity level. This lack of impact may be due to the construction of the BKM implied volatility estimator, where DOTM option prices—either discarded or extrapolated by DStab—are not heavily weighted. Second, the higher moment estimators are significantly affected by DStab. Both implied skewness and estimates show a decrease in both the time trend and the noisy component as the intensity level increases. The findings in Figure 4 suggest that DStab can mitigate the noisy dynamics of higher moment estimates, though the informative part of the dynamics might also be weakened. The summary statistics of implied moments for various DStab intensity levels, which are summarized in Table 4, also provide similar implications.

Graphical representation of the time-series dynamics for the implied moment estimates at varying intensity levels—0%, 50%, and 100%—demonstrating that the estimates exhibit reduced volatility as the intensity level increases.
Figure 4

Time-series dynamics of implied moment estimates after DStab.

Notes: This figure illustrates the time-series dynamics of the implied volatility (VOL), skewness (SKEW), and kurtosis (KURT) estimates after applying DStab at various intensity levels. Panels A, B, and C present the time-series dynamics for 0%, 50%, and 100% stabilizations, respectively.

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Table 4
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Implied moment estimates after applying DStab

Panel A. Implied volatility estimates
Level
First-order difference
Intensity0%25%50%75%100%0%25%50%75%100%
Mean0.1610.1600.1590.1570.1470.0000.0000.0000.0000.000
Std. dev.0.1020.1010.1010.1010.0960.0390.0400.0390.0390.037
5th pct.0.0760.0750.0740.0730.066−0.047−0.047−0.047−0.045−0.042
25th pct.0.1030.1020.1010.1000.092−0.014−0.014−0.013−0.013−0.012
Median0.1350.1350.1330.1320.1230.000−0.001−0.001−0.001−0.001
75th pct.0.1870.1850.1840.1820.1700.0110.0100.0100.0100.009
95th pct.0.3170.3130.3120.3080.2880.0480.0460.0450.0440.042
Skewness4.0484.0684.0944.1254.2082.4322.4252.4022.3542.256
Kurtosis29.02429.26629.61730.07231.26835.38237.09237.82437.85438.232
# of obs.1,5281,5281,5281,5281,5281,5271,5271,5271,5271,527
Panel A. Implied volatility estimates
Level
First-order difference
Intensity0%25%50%75%100%0%25%50%75%100%
Mean0.1610.1600.1590.1570.1470.0000.0000.0000.0000.000
Std. dev.0.1020.1010.1010.1010.0960.0390.0400.0390.0390.037
5th pct.0.0760.0750.0740.0730.066−0.047−0.047−0.047−0.045−0.042
25th pct.0.1030.1020.1010.1000.092−0.014−0.014−0.013−0.013−0.012
Median0.1350.1350.1330.1320.1230.000−0.001−0.001−0.001−0.001
75th pct.0.1870.1850.1840.1820.1700.0110.0100.0100.0100.009
95th pct.0.3170.3130.3120.3080.2880.0480.0460.0450.0440.042
Skewness4.0484.0684.0944.1254.2082.4322.4252.4022.3542.256
Kurtosis29.02429.26629.61730.07231.26835.38237.09237.82437.85438.232
# of obs.1,5281,5281,5281,5281,5281,5271,5271,5271,5271,527
Panel B. Implied skewness estimates
Level
First-order difference
Intensity0%25%50%75%100%0%25%50%75%100%
Mean−2.127−1.992−1.839−1.658−1.2920.0000.0000.0000.0000.000
Std. dev.0.9540.7220.5620.4300.1980.5420.3910.3090.2460.124
5th pct.−4.020−3.342−2.862−2.425−1.633−0.878−0.638−0.493−0.392−0.201
25th pct.−2.633−2.455−2.198−1.921−1.415−0.282−0.232−0.182−0.145−0.072
Median−1.902−1.895−1.781−1.617−1.2770.0100.000−0.004−0.005−0.001
75th pct.−1.393−1.414−1.406−1.335−1.1500.2700.2250.1800.1470.076
95th pct.−1.000−1.041−1.057−1.043−1.0040.9010.6560.5100.3940.199
Skewness−1.094−0.737−0.661−0.676−0.638−0.215−0.140−0.082−0.035−0.055
Kurtosis4.1013.2953.3803.6593.8606.6694.7654.5924.8825.392
# of obs.1,5281,5281,5281,5281,5281,5271,5271,5271,5271,527
Panel B. Implied skewness estimates
Level
First-order difference
Intensity0%25%50%75%100%0%25%50%75%100%
Mean−2.127−1.992−1.839−1.658−1.2920.0000.0000.0000.0000.000
Std. dev.0.9540.7220.5620.4300.1980.5420.3910.3090.2460.124
5th pct.−4.020−3.342−2.862−2.425−1.633−0.878−0.638−0.493−0.392−0.201
25th pct.−2.633−2.455−2.198−1.921−1.415−0.282−0.232−0.182−0.145−0.072
Median−1.902−1.895−1.781−1.617−1.2770.0100.000−0.004−0.005−0.001
75th pct.−1.393−1.414−1.406−1.335−1.1500.2700.2250.1800.1470.076
95th pct.−1.000−1.041−1.057−1.043−1.0040.9010.6560.5100.3940.199
Skewness−1.094−0.737−0.661−0.676−0.638−0.215−0.140−0.082−0.035−0.055
Kurtosis4.1013.2953.3803.6593.8606.6694.7654.5924.8825.392
# of obs.1,5281,5281,5281,5281,5281,5271,5271,5271,5271,527
Panel C. Implied kurtosis estimates
Level
First-order difference
Intensity0%25%50%75%100%0%25%50%75%100%
Mean14.82812.20610.0978.0734.1870.0070.0040.0020.0000.000
Std. dev.9.9665.8463.7932.3820.7116.0263.1191.9761.2710.394
5th pct.5.1615.1145.0724.8463.233−9.006−5.255−3.118−2.043−0.622
25th pct.7.4357.3767.0776.2593.680−2.165−1.618−1.130−0.716−0.218
Median11.54911.0829.5447.7064.069−0.128−0.108−0.046−0.022−0.004
75th pct.18.92015.65812.4349.4664.6072.1531.5771.1200.7350.221
95th pct.36.23423.35517.01712.4475.5149.0395.1653.1751.9490.593
Skewness1.6050.9090.7970.8570.9620.1700.2150.1690.0990.027
Kurtosis5.7923.5233.5524.0044.52010.6395.8295.6276.2356.889
# of obs.1,5281,5281,5281,5281,5281,5271,5271,5271,5271,527
Panel C. Implied kurtosis estimates
Level
First-order difference
Intensity0%25%50%75%100%0%25%50%75%100%
Mean14.82812.20610.0978.0734.1870.0070.0040.0020.0000.000
Std. dev.9.9665.8463.7932.3820.7116.0263.1191.9761.2710.394
5th pct.5.1615.1145.0724.8463.233−9.006−5.255−3.118−2.043−0.622
25th pct.7.4357.3767.0776.2593.680−2.165−1.618−1.130−0.716−0.218
Median11.54911.0829.5447.7064.069−0.128−0.108−0.046−0.022−0.004
75th pct.18.92015.65812.4349.4664.6072.1531.5771.1200.7350.221
95th pct.36.23423.35517.01712.4475.5149.0395.1653.1751.9490.593
Skewness1.6050.9090.7970.8570.9620.1700.2150.1690.0990.027
Kurtosis5.7923.5233.5524.0044.52010.6395.8295.6276.2356.889
# of obs.1,5281,5281,5281,5281,5281,5271,5271,5271,5271,527

Notes: This table provides summary statistics for the levels and first-order differences of implied moment estimates after applying DStab at various intensity levels. Panels A, B, and C present the summary statistics for implied volatility, skewness, and kurtosis estimates, respectively. The table reports statistics for intensity levels of 0%, 25%, 50%, 75%, and 100%.

Table 4
Open in new tab

Implied moment estimates after applying DStab

Panel A. Implied volatility estimates
Level
First-order difference
Intensity0%25%50%75%100%0%25%50%75%100%
Mean0.1610.1600.1590.1570.1470.0000.0000.0000.0000.000
Std. dev.0.1020.1010.1010.1010.0960.0390.0400.0390.0390.037
5th pct.0.0760.0750.0740.0730.066−0.047−0.047−0.047−0.045−0.042
25th pct.0.1030.1020.1010.1000.092−0.014−0.014−0.013−0.013−0.012
Median0.1350.1350.1330.1320.1230.000−0.001−0.001−0.001−0.001
75th pct.0.1870.1850.1840.1820.1700.0110.0100.0100.0100.009
95th pct.0.3170.3130.3120.3080.2880.0480.0460.0450.0440.042
Skewness4.0484.0684.0944.1254.2082.4322.4252.4022.3542.256
Kurtosis29.02429.26629.61730.07231.26835.38237.09237.82437.85438.232
# of obs.1,5281,5281,5281,5281,5281,5271,5271,5271,5271,527
Panel A. Implied volatility estimates
Level
First-order difference
Intensity0%25%50%75%100%0%25%50%75%100%
Mean0.1610.1600.1590.1570.1470.0000.0000.0000.0000.000
Std. dev.0.1020.1010.1010.1010.0960.0390.0400.0390.0390.037
5th pct.0.0760.0750.0740.0730.066−0.047−0.047−0.047−0.045−0.042
25th pct.0.1030.1020.1010.1000.092−0.014−0.014−0.013−0.013−0.012
Median0.1350.1350.1330.1320.1230.000−0.001−0.001−0.001−0.001
75th pct.0.1870.1850.1840.1820.1700.0110.0100.0100.0100.009
95th pct.0.3170.3130.3120.3080.2880.0480.0460.0450.0440.042
Skewness4.0484.0684.0944.1254.2082.4322.4252.4022.3542.256
Kurtosis29.02429.26629.61730.07231.26835.38237.09237.82437.85438.232
# of obs.1,5281,5281,5281,5281,5281,5271,5271,5271,5271,527
Panel B. Implied skewness estimates
Level
First-order difference
Intensity0%25%50%75%100%0%25%50%75%100%
Mean−2.127−1.992−1.839−1.658−1.2920.0000.0000.0000.0000.000
Std. dev.0.9540.7220.5620.4300.1980.5420.3910.3090.2460.124
5th pct.−4.020−3.342−2.862−2.425−1.633−0.878−0.638−0.493−0.392−0.201
25th pct.−2.633−2.455−2.198−1.921−1.415−0.282−0.232−0.182−0.145−0.072
Median−1.902−1.895−1.781−1.617−1.2770.0100.000−0.004−0.005−0.001
75th pct.−1.393−1.414−1.406−1.335−1.1500.2700.2250.1800.1470.076
95th pct.−1.000−1.041−1.057−1.043−1.0040.9010.6560.5100.3940.199
Skewness−1.094−0.737−0.661−0.676−0.638−0.215−0.140−0.082−0.035−0.055
Kurtosis4.1013.2953.3803.6593.8606.6694.7654.5924.8825.392
# of obs.1,5281,5281,5281,5281,5281,5271,5271,5271,5271,527
Panel B. Implied skewness estimates
Level
First-order difference
Intensity0%25%50%75%100%0%25%50%75%100%
Mean−2.127−1.992−1.839−1.658−1.2920.0000.0000.0000.0000.000
Std. dev.0.9540.7220.5620.4300.1980.5420.3910.3090.2460.124
5th pct.−4.020−3.342−2.862−2.425−1.633−0.878−0.638−0.493−0.392−0.201
25th pct.−2.633−2.455−2.198−1.921−1.415−0.282−0.232−0.182−0.145−0.072
Median−1.902−1.895−1.781−1.617−1.2770.0100.000−0.004−0.005−0.001
75th pct.−1.393−1.414−1.406−1.335−1.1500.2700.2250.1800.1470.076
95th pct.−1.000−1.041−1.057−1.043−1.0040.9010.6560.5100.3940.199
Skewness−1.094−0.737−0.661−0.676−0.638−0.215−0.140−0.082−0.035−0.055
Kurtosis4.1013.2953.3803.6593.8606.6694.7654.5924.8825.392
# of obs.1,5281,5281,5281,5281,5281,5271,5271,5271,5271,527
Panel C. Implied kurtosis estimates
Level
First-order difference
Intensity0%25%50%75%100%0%25%50%75%100%
Mean14.82812.20610.0978.0734.1870.0070.0040.0020.0000.000
Std. dev.9.9665.8463.7932.3820.7116.0263.1191.9761.2710.394
5th pct.5.1615.1145.0724.8463.233−9.006−5.255−3.118−2.043−0.622
25th pct.7.4357.3767.0776.2593.680−2.165−1.618−1.130−0.716−0.218
Median11.54911.0829.5447.7064.069−0.128−0.108−0.046−0.022−0.004
75th pct.18.92015.65812.4349.4664.6072.1531.5771.1200.7350.221
95th pct.36.23423.35517.01712.4475.5149.0395.1653.1751.9490.593
Skewness1.6050.9090.7970.8570.9620.1700.2150.1690.0990.027
Kurtosis5.7923.5233.5524.0044.52010.6395.8295.6276.2356.889
# of obs.1,5281,5281,5281,5281,5281,5271,5271,5271,5271,527
Panel C. Implied kurtosis estimates
Level
First-order difference
Intensity0%25%50%75%100%0%25%50%75%100%
Mean14.82812.20610.0978.0734.1870.0070.0040.0020.0000.000
Std. dev.9.9665.8463.7932.3820.7116.0263.1191.9761.2710.394
5th pct.5.1615.1145.0724.8463.233−9.006−5.255−3.118−2.043−0.622
25th pct.7.4357.3767.0776.2593.680−2.165−1.618−1.130−0.716−0.218
Median11.54911.0829.5447.7064.069−0.128−0.108−0.046−0.022−0.004
75th pct.18.92015.65812.4349.4664.6072.1531.5771.1200.7350.221
95th pct.36.23423.35517.01712.4475.5149.0395.1653.1751.9490.593
Skewness1.6050.9090.7970.8570.9620.1700.2150.1690.0990.027
Kurtosis5.7923.5233.5524.0044.52010.6395.8295.6276.2356.889
# of obs.1,5281,5281,5281,5281,5281,5271,5271,5271,5271,527

Notes: This table provides summary statistics for the levels and first-order differences of implied moment estimates after applying DStab at various intensity levels. Panels A, B, and C present the summary statistics for implied volatility, skewness, and kurtosis estimates, respectively. The table reports statistics for intensity levels of 0%, 25%, 50%, 75%, and 100%.

Table 5 reports the correlations among the levels and first-order differences of the underlying log-price and implied moment estimates for various DStab intensity levels, revealing three noteworthy findings. First, the correlation between the level of the underlying log-price and implied moments decreases as the intensity level increases. This may be because the underlying price level is a primary determinant of the integration domain endpoints, even when we measure the endpoint location in terms of d1⁠. If d1⁠, r⁠, τ⁠, and σ in Equation (22) are fixed, as well as the continuous dividend rate for the underlying asset, OTM option price increases as the underlying price increases, according to the definition of the Black–Scholes formula, regardless of whether the option is a call or a put. As a result, OTM price observations become less affected by the minimum price filter, and the integration domain widens. Since the endpoint location of the integration domain in terms of d1 is related to all three implied moment estimates, particularly the higher ones, the integration domain may function as a conduit, making the underlying price and implied moment estimates more correlated. By contrast, when DStab is employed, the method may mitigate these spurious correlations by making domain endpoint locations as constant as possible.

Table 5
Open in new tab

Correlations among underlying log-prices and implied moments

Panel A. No stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000−−−Δln(S)1.000−−−
VOL0.0371.000−−ΔVOL−0.7671.000−−
SKEW−0.3550.1291.000−ΔSKEW−0.017−0.1261.000−
KURT0.415−0.116−0.9661.000ΔKURT0.0210.074−0.9421.000
Panel A. No stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000−−−Δln(S)1.000−−−
VOL0.0371.000−−ΔVOL−0.7671.000−−
SKEW−0.3550.1291.000−ΔSKEW−0.017−0.1261.000−
KURT0.415−0.116−0.9661.000ΔKURT0.0210.074−0.9421.000
Panel B. 0% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0351.000––ΔVOL−0.7671.000––
SKEW−0.3400.1421.000–ΔSKEW−0.015−0.1231.000–
KURT0.383−0.146−0.9711.000ΔKURT0.0280.060−0.9451.000
Panel B. 0% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0351.000––ΔVOL−0.7671.000––
SKEW−0.3400.1421.000–ΔSKEW−0.015−0.1231.000–
KURT0.383−0.146−0.9711.000ΔKURT0.0280.060−0.9451.000
Panel C. 25% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0291.000––ΔVOL−0.7691.000––
SKEW−0.2780.1881.000–ΔSKEW−0.017−0.1421.000–
KURT0.309−0.221−0.9761.000ΔKURT0.0180.094−0.9541.000
Panel C. 25% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0291.000––ΔVOL−0.7691.000––
SKEW−0.2780.1881.000–ΔSKEW−0.017−0.1421.000–
KURT0.309−0.221−0.9761.000ΔKURT0.0180.094−0.9541.000
Panel D. 50% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0251.000––ΔVOL−0.7711.000––
SKEW−0.2350.2141.000–ΔSKEW−0.030−0.1461.000–
KURT0.256−0.269−0.9701.000ΔKURT0.0260.107−0.9491.000
Panel D. 50% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0251.000––ΔVOL−0.7711.000––
SKEW−0.2350.2141.000–ΔSKEW−0.030−0.1461.000–
KURT0.256−0.269−0.9701.000ΔKURT0.0260.107−0.9491.000
Panel E. 75% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0211.000––ΔVOL−0.7741.000––
SKEW−0.1980.2251.000–ΔSKEW−0.056−0.1271.000–
KURT0.211−0.302−0.9611.000ΔKURT0.0550.091−0.9401.000
Panel E. 75% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0211.000––ΔVOL−0.7741.000––
SKEW−0.1980.2251.000–ΔSKEW−0.056−0.1271.000–
KURT0.211−0.302−0.9611.000ΔKURT0.0550.091−0.9401.000
Panel F. 100% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0171.000––ΔVOL−0.7831.000––
SKEW−0.1770.2771.000–ΔSKEW−0.1790.0111.000–
KURT0.210−0.335−0.9531.000ΔKURT0.151−0.004−0.9201.000
Panel F. 100% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0171.000––ΔVOL−0.7831.000––
SKEW−0.1770.2771.000–ΔSKEW−0.1790.0111.000–
KURT0.210−0.335−0.9531.000ΔKURT0.151−0.004−0.9201.000

Notes: This table presents the correlation coefficient estimates between the levels and first-order differences of the daily underlying log price and the implied moment estimates. S, VOL, SKEW, and KURT represent the underlying price and the implied volatility, skewness, and kurtosis estimates, respectively. Δ denotes the first-order difference operator. Panel A provides the correlation coefficient estimates for the no stabilization case, while Panels B–F present the estimates for DStab at intensity levels of 0%, 25%, 50%, 75%, and 100%, respectively.

Table 5
Open in new tab

Correlations among underlying log-prices and implied moments

Panel A. No stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000−−−Δln(S)1.000−−−
VOL0.0371.000−−ΔVOL−0.7671.000−−
SKEW−0.3550.1291.000−ΔSKEW−0.017−0.1261.000−
KURT0.415−0.116−0.9661.000ΔKURT0.0210.074−0.9421.000
Panel A. No stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000−−−Δln(S)1.000−−−
VOL0.0371.000−−ΔVOL−0.7671.000−−
SKEW−0.3550.1291.000−ΔSKEW−0.017−0.1261.000−
KURT0.415−0.116−0.9661.000ΔKURT0.0210.074−0.9421.000
Panel B. 0% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0351.000––ΔVOL−0.7671.000––
SKEW−0.3400.1421.000–ΔSKEW−0.015−0.1231.000–
KURT0.383−0.146−0.9711.000ΔKURT0.0280.060−0.9451.000
Panel B. 0% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0351.000––ΔVOL−0.7671.000––
SKEW−0.3400.1421.000–ΔSKEW−0.015−0.1231.000–
KURT0.383−0.146−0.9711.000ΔKURT0.0280.060−0.9451.000
Panel C. 25% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0291.000––ΔVOL−0.7691.000––
SKEW−0.2780.1881.000–ΔSKEW−0.017−0.1421.000–
KURT0.309−0.221−0.9761.000ΔKURT0.0180.094−0.9541.000
Panel C. 25% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0291.000––ΔVOL−0.7691.000––
SKEW−0.2780.1881.000–ΔSKEW−0.017−0.1421.000–
KURT0.309−0.221−0.9761.000ΔKURT0.0180.094−0.9541.000
Panel D. 50% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0251.000––ΔVOL−0.7711.000––
SKEW−0.2350.2141.000–ΔSKEW−0.030−0.1461.000–
KURT0.256−0.269−0.9701.000ΔKURT0.0260.107−0.9491.000
Panel D. 50% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0251.000––ΔVOL−0.7711.000––
SKEW−0.2350.2141.000–ΔSKEW−0.030−0.1461.000–
KURT0.256−0.269−0.9701.000ΔKURT0.0260.107−0.9491.000
Panel E. 75% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0211.000––ΔVOL−0.7741.000––
SKEW−0.1980.2251.000–ΔSKEW−0.056−0.1271.000–
KURT0.211−0.302−0.9611.000ΔKURT0.0550.091−0.9401.000
Panel E. 75% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0211.000––ΔVOL−0.7741.000––
SKEW−0.1980.2251.000–ΔSKEW−0.056−0.1271.000–
KURT0.211−0.302−0.9611.000ΔKURT0.0550.091−0.9401.000
Panel F. 100% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0171.000––ΔVOL−0.7831.000––
SKEW−0.1770.2771.000–ΔSKEW−0.1790.0111.000–
KURT0.210−0.335−0.9531.000ΔKURT0.151−0.004−0.9201.000
Panel F. 100% stabilization
Level
First-order difference
ln(S)VOLSKEWKURTΔln(S)ΔVOLΔSKEWΔKURT
ln(S)1.000–––Δln(S)1.000–––
VOL0.0171.000––ΔVOL−0.7831.000––
SKEW−0.1770.2771.000–ΔSKEW−0.1790.0111.000–
KURT0.210−0.335−0.9531.000ΔKURT0.151−0.004−0.9201.000

Notes: This table presents the correlation coefficient estimates between the levels and first-order differences of the daily underlying log price and the implied moment estimates. S, VOL, SKEW, and KURT represent the underlying price and the implied volatility, skewness, and kurtosis estimates, respectively. Δ denotes the first-order difference operator. Panel A provides the correlation coefficient estimates for the no stabilization case, while Panels B–F present the estimates for DStab at intensity levels of 0%, 25%, 50%, 75%, and 100%, respectively.

Second, the correlation between the first-order difference of the underlying log-price and the implied moments tends to increase as the intensity level increases. This suggests that the relationship between the dynamics of the underlying price and the implied moment estimates becomes less affected by the noise component as DStab is applied more intensively. Third, the impact of DStab on the correlations among implied moment estimates is inconsistent and depends on which implied moment pair is examined. Interestingly, the correlation between the level of implied volatility and the higher moments increases as the intensity level increases. This may be because DStab, while not significantly affecting the implied volatility estimate, reduces the noise component of the implied higher moments, thereby revealing the true relationship among the moments more clearly.

4.3 In-sample Performance

To test whether the noise reduction by DStab leads to an enhancement of information contained in implied moment estimates, we conduct an in-sample test of the underlying return predictive ability for implied moments. We estimate two regression models and examine how the model’s explanatory power changes as DStab intensity increases. The first predictive regression model includes the first-order difference of implied higher moments as the main independent variables, employing DStab with various intensity levels:
(19)
where r(t) is the underlying S&P 500 index log-return on day t⁠, ΔVOLt⁠, ΔSKEWt⁠, and ΔKURTt denote the first-order differences of implied volatility, skewness, and kurtosis estimates on day t⁠, respectively. rt−1 is also included as an independent variable to account for daily return reversals.
A feature that should be considered when interpreting Table 6 is the high correlation between the BKM skewness and kurtosis estimates, which is reported by relevant studies (Diavatopoulos et al. 2012; Pan, Shiu, and Wu, 2022, 2024). Table 5 also displays that the correlation coefficient between the implied higher moments is significantly high, regardless of the DStab intensity. Hence, we further introduce principal component analysis to examine the common information content conveyed by the higher moments and investigate the return predictive and forecasting ability of higher moment dynamics. In addition to the baseline predictive regression model defined in Equation (19), we also consider the following second model:
(20)
where PCt represents the first principal component of ΔSKEWt and ΔKURTt⁠.
Table 6
Open in new tab

In-sample return predictive ability of implied moments after stabilization

Panel A. Each implied higher moment as the independent variable
Lagged return onlyNo stabilization0% stabilization25% stabilization50% stabilization75% stabilization100% stabilization
r(t−1)−0.164**−0.039−0.039−0.017−0.0080.0010.018
(−2.05)(−0.36)(−0.36)(−0.16)(−0.07)(0.02)(0.16)
ΔVOL(t−1)5.0565.0335.8356.1676.5267.066
(1.35)(1.37)(1.53)(1.58)(1.64)(1.64)
ΔSKEW(t−1)0.467**0.466**0.822***0.937**1.039**1.518
(2.25)(2.41)(2.37)(2.03)(1.99)(1.60)
ΔKURT(t−1)0.0240.0230.066*0.0860.1080.218
(1.36)(1.48)(1.71)(1.32)(1.20)(0.84)
Intercept0.066**0.059*0.059*0.058*0.058*0.057*0.056*
(2.10)(1.84)(1.84)(1.80)(1.78)(1.76)(1.72)
# of obs.1,5261,5261,5261,5261,5261,5261,526
R20.02680.04260.04320.04560.04670.04750.0473
Panel A. Each implied higher moment as the independent variable
Lagged return onlyNo stabilization0% stabilization25% stabilization50% stabilization75% stabilization100% stabilization
r(t−1)−0.164**−0.039−0.039−0.017−0.0080.0010.018
(−2.05)(−0.36)(−0.36)(−0.16)(−0.07)(0.02)(0.16)
ΔVOL(t−1)5.0565.0335.8356.1676.5267.066
(1.35)(1.37)(1.53)(1.58)(1.64)(1.64)
ΔSKEW(t−1)0.467**0.466**0.822***0.937**1.039**1.518
(2.25)(2.41)(2.37)(2.03)(1.99)(1.60)
ΔKURT(t−1)0.0240.0230.066*0.0860.1080.218
(1.36)(1.48)(1.71)(1.32)(1.20)(0.84)
Intercept0.066**0.059*0.059*0.058*0.058*0.057*0.056*
(2.10)(1.84)(1.84)(1.80)(1.78)(1.76)(1.72)
# of obs.1,5261,5261,5261,5261,5261,5261,526
R20.02680.04260.04320.04560.04670.04750.0473
Panel B. Principal component of implied higher moments as the independent variable
Lagged return onlyNo stabilization0% stabilization25% stabilization50% stabilization75% stabilization100% stabilization
r(t−1)−0.164**−0.053−0.054−0.038−0.024−0.0100.010
(−2.05)(−0.48)(−0.49)(−0.34)(−0.22)(−0.09)(0.09)
ΔVOL(t−1)4.4714.3715.0115.5346.0356.899
(1.20)(1.21)(1.34)(1.45)(1.54)(1.61)
PC(t−1)−0.078**−0.081***−0.078***−0.081***−0.081***−0.071***
(−3.05)(−3.04)(−3.10)(−3.21)(−3.12)(−2.75)
Intercept0.066**0.060*0.060*0.059*0.059*0.058*0.057*
(2.10)(1.86)(1.86)(1.83)(1.81)(1.78)(1.74)
# of obs.1,5261,5261,5261,5261,5261,5261,526
R20.02680.04010.04060.04160.04330.04450.0454
Panel B. Principal component of implied higher moments as the independent variable
Lagged return onlyNo stabilization0% stabilization25% stabilization50% stabilization75% stabilization100% stabilization
r(t−1)−0.164**−0.053−0.054−0.038−0.024−0.0100.010
(−2.05)(−0.48)(−0.49)(−0.34)(−0.22)(−0.09)(0.09)
ΔVOL(t−1)4.4714.3715.0115.5346.0356.899
(1.20)(1.21)(1.34)(1.45)(1.54)(1.61)
PC(t−1)−0.078**−0.081***−0.078***−0.081***−0.081***−0.071***
(−3.05)(−3.04)(−3.10)(−3.21)(−3.12)(−2.75)
Intercept0.066**0.060*0.060*0.059*0.059*0.058*0.057*
(2.10)(1.86)(1.86)(1.83)(1.81)(1.78)(1.74)
# of obs.1,5261,5261,5261,5261,5261,5261,526
R20.02680.04010.04060.04160.04330.04450.0454

Notes: This table presents the results of in-sample return prediction tests conducted using DStab at various intensity levels. The dependent variable, r(t)⁠, represents the S&P 500 index log-return on day t⁠, expressed as a percentage. Panel A reports the test results where the first-order differences of each implied moment estimate are considered independent variables. Panel B presents the test results in which the first-order differences of implied skewness and kurtosis estimates are replaced with their first principal component. ΔVOL(t)⁠, ΔSKEW(t)⁠, and ΔKURT(t) represent the daily first-order differences of implied volatility, skewness, and kurtosis estimates on day t, respectively. PC(t) denotes the first principal component of the first-order differences of implied skewness and kurtosis estimates. The Huber–White sandwich estimator is used to estimate standard errors; therefore, the unadjusted R2 is reported. t-statistics are presented in parentheses. ***, **, and * indicate statistical significance at the 1%, 5%, and 10% levels, respectively.

Table 6
Open in new tab

In-sample return predictive ability of implied moments after stabilization

Panel A. Each implied higher moment as the independent variable
Lagged return onlyNo stabilization0% stabilization25% stabilization50% stabilization75% stabilization100% stabilization
r(t−1)−0.164**−0.039−0.039−0.017−0.0080.0010.018
(−2.05)(−0.36)(−0.36)(−0.16)(−0.07)(0.02)(0.16)
ΔVOL(t−1)5.0565.0335.8356.1676.5267.066
(1.35)(1.37)(1.53)(1.58)(1.64)(1.64)
ΔSKEW(t−1)0.467**0.466**0.822***0.937**1.039**1.518
(2.25)(2.41)(2.37)(2.03)(1.99)(1.60)
ΔKURT(t−1)0.0240.0230.066*0.0860.1080.218
(1.36)(1.48)(1.71)(1.32)(1.20)(0.84)
Intercept0.066**0.059*0.059*0.058*0.058*0.057*0.056*
(2.10)(1.84)(1.84)(1.80)(1.78)(1.76)(1.72)
# of obs.1,5261,5261,5261,5261,5261,5261,526
R20.02680.04260.04320.04560.04670.04750.0473
Panel A. Each implied higher moment as the independent variable
Lagged return onlyNo stabilization0% stabilization25% stabilization50% stabilization75% stabilization100% stabilization
r(t−1)−0.164**−0.039−0.039−0.017−0.0080.0010.018
(−2.05)(−0.36)(−0.36)(−0.16)(−0.07)(0.02)(0.16)
ΔVOL(t−1)5.0565.0335.8356.1676.5267.066
(1.35)(1.37)(1.53)(1.58)(1.64)(1.64)
ΔSKEW(t−1)0.467**0.466**0.822***0.937**1.039**1.518
(2.25)(2.41)(2.37)(2.03)(1.99)(1.60)
ΔKURT(t−1)0.0240.0230.066*0.0860.1080.218
(1.36)(1.48)(1.71)(1.32)(1.20)(0.84)
Intercept0.066**0.059*0.059*0.058*0.058*0.057*0.056*
(2.10)(1.84)(1.84)(1.80)(1.78)(1.76)(1.72)
# of obs.1,5261,5261,5261,5261,5261,5261,526
R20.02680.04260.04320.04560.04670.04750.0473
Panel B. Principal component of implied higher moments as the independent variable
Lagged return onlyNo stabilization0% stabilization25% stabilization50% stabilization75% stabilization100% stabilization
r(t−1)−0.164**−0.053−0.054−0.038−0.024−0.0100.010
(−2.05)(−0.48)(−0.49)(−0.34)(−0.22)(−0.09)(0.09)
ΔVOL(t−1)4.4714.3715.0115.5346.0356.899
(1.20)(1.21)(1.34)(1.45)(1.54)(1.61)
PC(t−1)−0.078**−0.081***−0.078***−0.081***−0.081***−0.071***
(−3.05)(−3.04)(−3.10)(−3.21)(−3.12)(−2.75)
Intercept0.066**0.060*0.060*0.059*0.059*0.058*0.057*
(2.10)(1.86)(1.86)(1.83)(1.81)(1.78)(1.74)
# of obs.1,5261,5261,5261,5261,5261,5261,526
R20.02680.04010.04060.04160.04330.04450.0454
Panel B. Principal component of implied higher moments as the independent variable
Lagged return onlyNo stabilization0% stabilization25% stabilization50% stabilization75% stabilization100% stabilization
r(t−1)−0.164**−0.053−0.054−0.038−0.024−0.0100.010
(−2.05)(−0.48)(−0.49)(−0.34)(−0.22)(−0.09)(0.09)
ΔVOL(t−1)4.4714.3715.0115.5346.0356.899
(1.20)(1.21)(1.34)(1.45)(1.54)(1.61)
PC(t−1)−0.078**−0.081***−0.078***−0.081***−0.081***−0.071***
(−3.05)(−3.04)(−3.10)(−3.21)(−3.12)(−2.75)
Intercept0.066**0.060*0.060*0.059*0.059*0.058*0.057*
(2.10)(1.86)(1.86)(1.83)(1.81)(1.78)(1.74)
# of obs.1,5261,5261,5261,5261,5261,5261,526
R20.02680.04010.04060.04160.04330.04450.0454

Notes: This table presents the results of in-sample return prediction tests conducted using DStab at various intensity levels. The dependent variable, r(t)⁠, represents the S&P 500 index log-return on day t⁠, expressed as a percentage. Panel A reports the test results where the first-order differences of each implied moment estimate are considered independent variables. Panel B presents the test results in which the first-order differences of implied skewness and kurtosis estimates are replaced with their first principal component. ΔVOL(t)⁠, ΔSKEW(t)⁠, and ΔKURT(t) represent the daily first-order differences of implied volatility, skewness, and kurtosis estimates on day t, respectively. PC(t) denotes the first principal component of the first-order differences of implied skewness and kurtosis estimates. The Huber–White sandwich estimator is used to estimate standard errors; therefore, the unadjusted R2 is reported. t-statistics are presented in parentheses. ***, **, and * indicate statistical significance at the 1%, 5%, and 10% levels, respectively.

Table 6 summarizes the test results. Panel A reports the results for the baseline model specified in Equation (19), which reveals three notable findings. First, the explanatory power of the predictive regression model tends to increase as DStab intensity increases. When compared to the R2 value of 0.0426 without DStab, the R2 increases by 11.0% to 0.0473 when DStab is applied at one hundred percent intensity. Given that R2 is 0.0268 when only the lagged returns are considered and increases by 0.0158 when the implied moments are added without DStab, it can be argued that the contribution of implied moments to explanatory power increases by 36.1% when 100% DStab is employed. Second, the statistical significance of lagged return tends to be subsumed more completely as DStab intensity increases. When the intensity is 75–100%, the t-statistics for the lagged returns is almost zero. This subsumption may be primarily due to the high correlation between returns and volatility changes. Because of the asymmetric volatility phenomenon, there is a significantly negative return–volatility relationship, and both of these variables exhibit reversals as a consequence. In an unreported analysis, we confirm that there is almost no change in R2 when we do not account for the lagged returns.

Third, the t-statistics of implied volatility dynamics consistently increase as DStab intensity increases, whereas it is not the case for the higher moment estimates. This pattern suggests that, although the nominal effect of DStab is largely concentrated on higher moment estimates, implied volatility estimate is also relevant when underlying return predictive ability is considered.

Panel B of Table 6 presents the return regression results for the second model, with a principal component specified in Equation (20), revealing three notable findings. First, the explanatory power of the model increases as DStab intensity increases, though it remains slightly lower overall compared to the results in Panel A. This increase again suggests that DStab can enhance the information content of model-free implied moment estimates. Second, the principal component of the implied higher moment dynamics retains its statistical significance regardless of DStab intensity. This significance implies that the skewness and kurtosis estimates contribute to return predictive ability, with most of the contribution stemming from the common component shared by the higher moments. Third, the increasing t-statistics for the implied volatility estimate and more complete subsumption of lagged returns, as DStab intensity increases, observed in Panel A, are also demonstrated in Panel B, reinforcing the reliability of these results. Overall, the in-sample test results suggest that DStab effectively improves the information content of implied moment estimates by letting the estimators calculate the value of TRND moments, thereby enhancing return predictive ability. The findings also indicate that the intensity level of DStab plays a significant role; increasing the intensity level can further enhance return predictive ability.

4.4 Out-of-Sample Performance

Although the empirical results in Section 4.3 demonstrate that DStab improves underlying return predictive ability, particularly at higher intensity levels, these findings are limited to in-sample tests. To address this limitation, we conduct a set of out-of-sample return forecasting ability tests on the model specified in Equation (19) to further assess whether DStab enhances the return forecasting ability of implied moment estimates. For each day t>h⁠, where h is the length of the rolling window, we predict the index return r(t) while estimating model parameters with the observations spanning the period [t-h−1,t−1]⁠. To measure the magnitude of improvement in out-of-sample return forecasting ability, we calculate a modified version of the ROS2 statistic, as proposed by Campbell and Thompson (2008), defined as follows:
(21)
where rt^ is the fitted value derived from a predictive regression estimated through the rolling window ending at time t−1⁠, and rt¯ is the benchmark return for the rolling window. A positive value of ROS2 indicates that the predictive regression produces a lower mean squared prediction error than the benchmark return. Since we are interested in whether DStab improves the return forecasting ability of implied moment estimates, we choose the fitted value estimated without DStab as the benchmark return.

We test with various rolling window lengths to investigate whether the out-of-sample test results depend on the rolling window specification. We select twelve window lengths, ranging from 5 to 60 months, with 5-month intervals. Given the multiple window lengths, it is important to determine which window length, when used with the predictive model, results in better return forecasting. This enables a more effective assessment of the increased return forecasting ability due to DStab. Accordingly, we conduct an additional set of out-of-sample tests, using the historical average return as the benchmark, consistent with the original definition of Campbell and Thompson (2008).

Table 7 summarizes the test results. Panel A presents the main out-of-sample test results, in which three notable findings can be observed. First, the out-of-sample return forcasting ability of implied higher moments tends to improve as DStab is applied more intensively. The test results show that the value of ROS2 generally increases with intensity level across most rolling window lengths, except when the window length is either significantly long or short. Second, despite the upward trend, the forecasting ability is strongest when DStab intensity is moderate, particularly when ROS2 is notably high. For the rolling window lengths between 30 and 50 months—where ROS2 reaches its highest levels—the predictive power peaks at fifty or 75% DStab intensity. In contrast, at 100% DStab intensity, ROS2 is lower than at 25% intensity. Third, DStab may reduce return forecasting ability if the rolling window is too long or short. Panel A shows that ROS2 can turn negative when the rolling window length is shorter than 30 months or longer than 50 months.

Table 7
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Out-of-sample return forecasting ability of implied moments after stabilization

Panel A. With versus without stabilization
ROS2
Rolling window length (months)No stabilization0% stabilization25% stabilization50% stabilization75% stabilization100% stabilization
50.00−0.370.300.270.27−0.25
100.00−0.370.430.540.520.14
150.00−0.240.380.620.670.42
200.00−0.130.320.430.440.25
250.00−0.070.370.450.460.29
300.000.201.011.020.970.57
350.000.130.850.900.880.60
400.000.180.840.910.900.64
450.000.140.640.710.700.51
500.000.160.600.710.770.60
550.000.07−0.07−0.26−0.46−1.05
600.000.02−0.08−0.27−0.49−1.02
Panel A. With versus without stabilization
ROS2
Rolling window length (months)No stabilization0% stabilization25% stabilization50% stabilization75% stabilization100% stabilization
50.00−0.370.300.270.27−0.25
100.00−0.370.430.540.520.14
150.00−0.240.380.620.670.42
200.00−0.130.320.430.440.25
250.00−0.070.370.450.460.29
300.000.201.011.020.970.57
350.000.130.850.900.880.60
400.000.180.840.910.900.64
450.000.140.640.710.700.51
500.000.160.600.710.770.60
550.000.07−0.07−0.26−0.46−1.05
600.000.02−0.08−0.27−0.49−1.02
Panel B. Model versus historical mean
ROS2
Rolling window length (months)No stabilization0% stabilization25% stabilization50% stabilization75% stabilization100% stabilization
5−7.42−7.79−7.12−7.15−7.15−7.67
10−2.33−2.70−1.90−1.79−1.81−2.20
150.18−0.050.560.800.850.61
200.940.811.261.371.381.19
251.081.011.451.531.541.37
301.541.742.552.562.512.11
352.002.132.842.902.882.59
402.332.513.183.253.232.98
452.612.753.253.323.313.12
503.083.243.683.793.853.68
55−4.37−4.30−4.44−4.63−4.83−5.42
60−7.38−7.36−7.46−7.65−7.86−8.40
Panel B. Model versus historical mean
ROS2
Rolling window length (months)No stabilization0% stabilization25% stabilization50% stabilization75% stabilization100% stabilization
5−7.42−7.79−7.12−7.15−7.15−7.67
10−2.33−2.70−1.90−1.79−1.81−2.20
150.18−0.050.560.800.850.61
200.940.811.261.371.381.19
251.081.011.451.531.541.37
301.541.742.552.562.512.11
352.002.132.842.902.882.59
402.332.513.183.253.232.98
452.612.753.253.323.313.12
503.083.243.683.793.853.68
55−4.37−4.30−4.44−4.63−4.83−5.42
60−7.38−7.36−7.46−7.65−7.86−8.40

Notes: This table presents the results of the out-of-sample return forecasting ability test. Following Campbell and Thompson (2008), we report the ROS2 statistic, which is defined as ROS2=1-∑t=1Trt-rt^2/∑t=1Trt-rt¯2⁠, where rt^ is the fitted value derived from a predictive regression estimated through the rolling window that ends at time t−1⁠, and rt¯ is the benchmark value for the rolling window. Benchmark value is defined as the fitted value estimated without stabilization for Panel A, and the historical mean log-return for Panel B. A positive value of ROS2 indicates that the predictive regression produces a lower mean squared prediction error than the benchmark value. The value of ROS2 is expressed as a percentage.

Table 7
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Out-of-sample return forecasting ability of implied moments after stabilization

Panel A. With versus without stabilization
ROS2
Rolling window length (months)No stabilization0% stabilization25% stabilization50% stabilization75% stabilization100% stabilization
50.00−0.370.300.270.27−0.25
100.00−0.370.430.540.520.14
150.00−0.240.380.620.670.42
200.00−0.130.320.430.440.25
250.00−0.070.370.450.460.29
300.000.201.011.020.970.57
350.000.130.850.900.880.60
400.000.180.840.910.900.64
450.000.140.640.710.700.51
500.000.160.600.710.770.60
550.000.07−0.07−0.26−0.46−1.05
600.000.02−0.08−0.27−0.49−1.02
Panel A. With versus without stabilization
ROS2
Rolling window length (months)No stabilization0% stabilization25% stabilization50% stabilization75% stabilization100% stabilization
50.00−0.370.300.270.27−0.25
100.00−0.370.430.540.520.14
150.00−0.240.380.620.670.42
200.00−0.130.320.430.440.25
250.00−0.070.370.450.460.29
300.000.201.011.020.970.57
350.000.130.850.900.880.60
400.000.180.840.910.900.64
450.000.140.640.710.700.51
500.000.160.600.710.770.60
550.000.07−0.07−0.26−0.46−1.05
600.000.02−0.08−0.27−0.49−1.02
Panel B. Model versus historical mean
ROS2
Rolling window length (months)No stabilization0% stabilization25% stabilization50% stabilization75% stabilization100% stabilization
5−7.42−7.79−7.12−7.15−7.15−7.67
10−2.33−2.70−1.90−1.79−1.81−2.20
150.18−0.050.560.800.850.61
200.940.811.261.371.381.19
251.081.011.451.531.541.37
301.541.742.552.562.512.11
352.002.132.842.902.882.59
402.332.513.183.253.232.98
452.612.753.253.323.313.12
503.083.243.683.793.853.68
55−4.37−4.30−4.44−4.63−4.83−5.42
60−7.38−7.36−7.46−7.65−7.86−8.40
Panel B. Model versus historical mean
ROS2
Rolling window length (months)No stabilization0% stabilization25% stabilization50% stabilization75% stabilization100% stabilization
5−7.42−7.79−7.12−7.15−7.15−7.67
10−2.33−2.70−1.90−1.79−1.81−2.20
150.18−0.050.560.800.850.61
200.940.811.261.371.381.19
251.081.011.451.531.541.37
301.541.742.552.562.512.11
352.002.132.842.902.882.59
402.332.513.183.253.232.98
452.612.753.253.323.313.12
503.083.243.683.793.853.68
55−4.37−4.30−4.44−4.63−4.83−5.42
60−7.38−7.36−7.46−7.65−7.86−8.40

Notes: This table presents the results of the out-of-sample return forecasting ability test. Following Campbell and Thompson (2008), we report the ROS2 statistic, which is defined as ROS2=1-∑t=1Trt-rt^2/∑t=1Trt-rt¯2⁠, where rt^ is the fitted value derived from a predictive regression estimated through the rolling window that ends at time t−1⁠, and rt¯ is the benchmark value for the rolling window. Benchmark value is defined as the fitted value estimated without stabilization for Panel A, and the historical mean log-return for Panel B. A positive value of ROS2 indicates that the predictive regression produces a lower mean squared prediction error than the benchmark value. The value of ROS2 is expressed as a percentage.

Panel B of Table 7 reports the test results with the historical average return as the benchmark. The panel shows that ROS2 exceeds 2% only when the window length is between 30 and 50 months, which aligns with the cases in Panel A where ROS2 reaches its highest levels. This result suggests that DStab is most effective when return forecasting with implied moment estimates is at its peak effectiveness. Furthermore, when DStab reduces return forecasting ability due to excessively short or long window lengths, ROS2 is negative even without DStab applied in most cases. This pattern implies that, while DStab can refine the information content of implied moment estimates, it cannot generate new information when these estimates are inherently uninformative. This is quite plausible, as DStab sacrifices some option-implied information to reduce noise but does not collect information from new sources. Overall, Table 7 suggests that the improvement in return predictive or forecasting ability due to DStab remains consistent, whether tested in-sample or out-of-sample.

4.5 Comparison with Alternative Methods

Although the empirical analysis so far has shown that DStab improves return predictive and forecasting ability, two issues still need to be considered before we can advocate for its use. First, a side effect of DStab, which is a decrease in integration domain width, needs to be further investigated. Since DStab maintains consistency in the integration domain width by discarding OTM price observations, it inevitably narrows down the integration domain, making implied moment estimates more heavily influenced by near-the-money (NTM) observations. According to the literature, NTM options particularly have predictive power (Chang, Hsieh, and Lai 2009), and factors such as the jump risk premium are reflected not only in DOTM options but also in NTM options (Pan, 2002). Given these findings, the improved return predictive and forecasting ability induced by DStab might be attributable to focusing more on the information contained in NTM option prices while not losing much information by discarding OTM options, rather than solely ensuring consistency in the integration domain width.

Second, it is essential to confirm that DStab improves return predictive and forecasting ability more than alternative methods. While existing literature highlights the issues with flat extrapolation and DSym, a more comprehensive comparison requires testing these methods under the same conditions. For flat extrapolation, we should evaluate the effectiveness of full extrapolation conducted to sufficiently distant locations, following the methodology of previous studies. For instance, JT suggests extrapolating up to points that are three standard deviations away from the underlying price. In the case of DSym, to fully understand the effects of symmetrization, we need to examine both symmetries in terms of strike price, and symmetry in terms of d1⁠, the primary measure used in this study.

Therefore, we aim to resolve the two issues mentioned above by conducting additional in-sample and out-of-sample tests on alternative methods. First, to examine the effect of the decrease in integration domain width, we propose another method for adjusting integration domain width, termed domain reduction (DRed). As its name suggests, DRed decreases the integration domain width, matching DStab in reducing the width for each intensity level i⁠. However, unlike DStab, which aims to stabilize the integration domain width, DRed does not pursue this objective at all. DRed is conducted through the following two-step process. First, we determine the extent to which the integration domain should be narrowed by DRed for each intensity level i⁠. We calculate Δd1minDStab,i and Δd1maxDStab,i⁠, defined as follows:
(22)
 
(23)
where n represents the number of days in the sample period; d1min,j and d1max,j represent the values of d1 at the minimum and maximum endpoints of the daily integration domain on day j before DStab, respectively; d1min,jDStab,i and d1max,jDStab,i represent the respective values of d1 after applying DStab with intensity i⁠. Given Equations (22) and (23), Δd1minDStab,i and Δd1maxDStab,i can be interpreted as the mean amount of decrease in the integration width incurred by DStab with intensity i⁠.
Next, we set the new values of d1 at the minimum and maximum endpoints of the daily integration domain on each day by discarding observations so that, for each intensity i and day j⁠, the values of d1 at the endpoints become d1min,jDRed,i and d1max,jDRed,i⁠, which are defined as follows:
(24)
 
(25)
where d1min,j⁠, d1max,j⁠, Δd1minDStab,i⁠, and Δd1maxDStab,i are defined as in Equations (22) and (23). After this adjustment, the values of d1 at the endpoints become closer to zero by fixed amounts Δd1minDStab,i and Δd1maxDStab,i⁠, so that the mean decrease in integration domain width in terms of d1 is identical between DStab and DRed for each intensity. However, in contrast to DStab which makes the domain width more consistent as the intensity level increases, DRed does not stabilize domain width at all regardless of the intensity. Hence, we expect that if the improvement in the predictive and forecasting ability by DStab is due to a decrease in integration domain width, DRed will also enhance the performance of implied higher moments. Due to the highly variable nature of domain width, 100% DRed is not feasible, so DRed is applied only up to 75%.

Next, to examine the effect of flat extrapolation, we first apply DStab at various intensity levels and then conduct full extrapolation up to [S/3, 3S]⁠, where S represents the underlying price, to make truncation trivial. For 0% DStab, the setup is equivalent to the extrapolation procedure in the previous studies, and from 25% onwards, we combine DStab with full extrapolation. For DSym, we discard additional OTM option price observations until |d1| of both domain endpoints become equal for the d1-based DSym, and until the strike price of both endpoints becomes equidistant from the underlying price for the strike price-based DSym. After applying each method, we conduct in-sample and out-of-sample tests as described above.

Figure 5 shows how the integration domain changes when DRed and DSym are applied. In the case of DRed, as the intensity level increases, the integration domain width decreases, but the domain endpoint dynamics remain unchanged. Therefore, even at higher intensity levels, the original dynamics of the integration domain endpoints observed in the data are preserved. On the other hand, with DSym, although the integration domain is not as stabilized as with 100% DStab, it is still significantly stabilized. This can be attributed to two reasons. First, since OTM put observations always cover a broader region of the integration domain than OTM call observations in our sample, DSym discards only DOTM put price observations. Second, as shown in Panel C of Figure 2, the location of the call-side endpoints is much less volatile than that of the put-side endpoints. As a result, the highly volatile put-side endpoints are adjusted based on the more stable call-side endpoints, leading to a moderate stabilization of the integration domain.

Graphical representation of the time-series dynamics for the integration domain in terms of Black–Scholes d1 after applying alternative treatments—domain reduction and symmetrization—illustrating how each method shapes the integration domain differently.
Figure 5

Integration domain after alternative treatments.

Notes: This figure illustrates the time-series dynamics of the integration domain in terms of Black–Scholes d1 after applying alternative integration domain treatments. Panels A and B depict the time-series dynamics when domain reduction is applied at intensity levels of 25% and 75%, respectively. Panels C and D present the time-series dynamics when domain symmetrization is applied in terms of Black–Scholes d1 and strike price, respectively. An n percent reduction is implemented by uniformly discarding OTM option price observations by the average reduction amount of n percent stabilization. A symmetrization is employed by further discarding OTM option price observations to make the two integration domain endpoints equidistant from the underlying price level, either in terms of d1 or strike price.

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Table 8 presents the in-sample return predictive ability test results for the alternative methods. Panel A reports the results for full extrapolation, highlighting three noteworthy features. First, compared to DStab, the explanatory power of the model is lower when only full extrapolation is applied without DStab. The R2 is 0.0430 for full extrapolation with no stabilization, which is even lower than the R2 for 0% DStab (0.0432). Second, when DStab is combined with full extrapolation, the explanatory power is slightly lower at moderate stabilization intensity levels compared to the original DStab with limited extrapolation. This underperformance may be due to the instability of the reference points for extrapolation when stabilization is not conducted at 100% intensity, resulting in a larger number of DOTM option prices generated from these unstable reference points with full extrapolation. Third, the explanatory power slightly improves when 100% DStab is employed with full extrapolation. This improvement may be attributed to the implied volatility observations measured at fully stabilized endpoints, allowing for the effective replenishment of OTM price observations that were heavily reduced due to intensive stabilization.

Table 8
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In-sample return predictive ability of implied moments after alternative treatments

Panel A. Full extrapolation (FE)
FE with 0% stabilizationFE with 25% stabilizationFE with 50% stabilizationFE with 75% stabilizationFE with 100% stabilization
r(t−1)−0.040−0.028−0.008−0.0120.012
(−0.37)(−0.26)(−0.07)(−0.11)(0.11)
ΔVOL(t−1)4.9955.4956.1676.1156.968*
(1.36)(1.47)(1.58)(1.59)(1.71)
ΔSKEW(t−1)0.443**0.664***0.937**0.782**0.831*
(2.38)(2.62)(2.03)(1.99)(1.77)
ΔKURT(t−1)0.0200.066**0.0860.0640.093
(1.46)(1.96)(1.32)(1.13)(0.76)
Intercept0.059*0.059*0.058*0.058*0.056*
(1.85)(1.82)(1.78)(1.79)(1.74)
# of obs.1,5261,5261,5261,5261,526
R20.04300.04440.04600.04670.0482
Panel A. Full extrapolation (FE)
FE with 0% stabilizationFE with 25% stabilizationFE with 50% stabilizationFE with 75% stabilizationFE with 100% stabilization
r(t−1)−0.040−0.028−0.008−0.0120.012
(−0.37)(−0.26)(−0.07)(−0.11)(0.11)
ΔVOL(t−1)4.9955.4956.1676.1156.968*
(1.36)(1.47)(1.58)(1.59)(1.71)
ΔSKEW(t−1)0.443**0.664***0.937**0.782**0.831*
(2.38)(2.62)(2.03)(1.99)(1.77)
ΔKURT(t−1)0.0200.066**0.0860.0640.093
(1.46)(1.96)(1.32)(1.13)(0.76)
Intercept0.059*0.059*0.058*0.058*0.056*
(1.85)(1.82)(1.78)(1.79)(1.74)
# of obs.1,5261,5261,5261,5261,526
R20.04300.04440.04600.04670.0482
Panel B. Domain reduction (DRed) and symmetrization (DSym)
25% DRed50% DRed75% DRedDSym (d1)DSym (strike price)
r(t−1)−0.042−0.051−0.080−0.0010.024
(−0.38)(−0.47)(−0.73)(0.02)(0.21)
ΔVOL(t−1)4.9724.6673.5926.7457.848*
(1.31)(1.21)(0.92)(1.53)(1.78)
ΔSKEW(t−1)0.471**0.481***0.504**1.629***1.600***
(2.26)(2.30)(2.39)(2.65)(2.83)
ΔKURT(t−1)0.0250.0270.0320.0720.031
(1.37)(1.41)(1.52)(0.91)(0.41)
Intercept0.060*0.060*0.062**0.057*0.056*
(1.85)(1.86)(1.91)(1.78)(1.74)
# of obs.1,5261,5261,5261,5261,526
R20.04220.04130.03930.05190.0561
Panel B. Domain reduction (DRed) and symmetrization (DSym)
25% DRed50% DRed75% DRedDSym (d1)DSym (strike price)
r(t−1)−0.042−0.051−0.080−0.0010.024
(−0.38)(−0.47)(−0.73)(0.02)(0.21)
ΔVOL(t−1)4.9724.6673.5926.7457.848*
(1.31)(1.21)(0.92)(1.53)(1.78)
ΔSKEW(t−1)0.471**0.481***0.504**1.629***1.600***
(2.26)(2.30)(2.39)(2.65)(2.83)
ΔKURT(t−1)0.0250.0270.0320.0720.031
(1.37)(1.41)(1.52)(0.91)(0.41)
Intercept0.060*0.060*0.062**0.057*0.056*
(1.85)(1.86)(1.91)(1.78)(1.74)
# of obs.1,5261,5261,5261,5261,526
R20.04220.04130.03930.05190.0561

Notes: This table presents the results of in-sample return prediction tests conducted under alternative integration domain treatments. Panel A reports the test results where flat extrapolation is applied up to strike prices equivalent to one-third and three times the underlying price. Panel B presents the test results where domain reduction and symmetrization are applied as truncation treatments. The dependent variable, r(t)⁠, represents the S&P 500 index log-return on day t⁠, expressed as a percentage. ΔVOL(t)⁠, ΔSKEW(t)⁠, and ΔKURT(t) denote the daily first-order differences of implied volatility, skewness, and kurtosis estimates on day t, respectively. The Huber–White sandwich estimator is used to estimate standard errors; therefore, the unadjusted R2 is reported. t-statistics are presented in parentheses. ***, **, and * indicate statistical significance at the 1%, 5%, and 10% levels, respectively.

Table 8
Open in new tab

In-sample return predictive ability of implied moments after alternative treatments

Panel A. Full extrapolation (FE)
FE with 0% stabilizationFE with 25% stabilizationFE with 50% stabilizationFE with 75% stabilizationFE with 100% stabilization
r(t−1)−0.040−0.028−0.008−0.0120.012
(−0.37)(−0.26)(−0.07)(−0.11)(0.11)
ΔVOL(t−1)4.9955.4956.1676.1156.968*
(1.36)(1.47)(1.58)(1.59)(1.71)
ΔSKEW(t−1)0.443**0.664***0.937**0.782**0.831*
(2.38)(2.62)(2.03)(1.99)(1.77)
ΔKURT(t−1)0.0200.066**0.0860.0640.093
(1.46)(1.96)(1.32)(1.13)(0.76)
Intercept0.059*0.059*0.058*0.058*0.056*
(1.85)(1.82)(1.78)(1.79)(1.74)
# of obs.1,5261,5261,5261,5261,526
R20.04300.04440.04600.04670.0482
Panel A. Full extrapolation (FE)
FE with 0% stabilizationFE with 25% stabilizationFE with 50% stabilizationFE with 75% stabilizationFE with 100% stabilization
r(t−1)−0.040−0.028−0.008−0.0120.012
(−0.37)(−0.26)(−0.07)(−0.11)(0.11)
ΔVOL(t−1)4.9955.4956.1676.1156.968*
(1.36)(1.47)(1.58)(1.59)(1.71)
ΔSKEW(t−1)0.443**0.664***0.937**0.782**0.831*
(2.38)(2.62)(2.03)(1.99)(1.77)
ΔKURT(t−1)0.0200.066**0.0860.0640.093
(1.46)(1.96)(1.32)(1.13)(0.76)
Intercept0.059*0.059*0.058*0.058*0.056*
(1.85)(1.82)(1.78)(1.79)(1.74)
# of obs.1,5261,5261,5261,5261,526
R20.04300.04440.04600.04670.0482
Panel B. Domain reduction (DRed) and symmetrization (DSym)
25% DRed50% DRed75% DRedDSym (d1)DSym (strike price)
r(t−1)−0.042−0.051−0.080−0.0010.024
(−0.38)(−0.47)(−0.73)(0.02)(0.21)
ΔVOL(t−1)4.9724.6673.5926.7457.848*
(1.31)(1.21)(0.92)(1.53)(1.78)
ΔSKEW(t−1)0.471**0.481***0.504**1.629***1.600***
(2.26)(2.30)(2.39)(2.65)(2.83)
ΔKURT(t−1)0.0250.0270.0320.0720.031
(1.37)(1.41)(1.52)(0.91)(0.41)
Intercept0.060*0.060*0.062**0.057*0.056*
(1.85)(1.86)(1.91)(1.78)(1.74)
# of obs.1,5261,5261,5261,5261,526
R20.04220.04130.03930.05190.0561
Panel B. Domain reduction (DRed) and symmetrization (DSym)
25% DRed50% DRed75% DRedDSym (d1)DSym (strike price)
r(t−1)−0.042−0.051−0.080−0.0010.024
(−0.38)(−0.47)(−0.73)(0.02)(0.21)
ΔVOL(t−1)4.9724.6673.5926.7457.848*
(1.31)(1.21)(0.92)(1.53)(1.78)
ΔSKEW(t−1)0.471**0.481***0.504**1.629***1.600***
(2.26)(2.30)(2.39)(2.65)(2.83)
ΔKURT(t−1)0.0250.0270.0320.0720.031
(1.37)(1.41)(1.52)(0.91)(0.41)
Intercept0.060*0.060*0.062**0.057*0.056*
(1.85)(1.86)(1.91)(1.78)(1.74)
# of obs.1,5261,5261,5261,5261,526
R20.04220.04130.03930.05190.0561

Notes: This table presents the results of in-sample return prediction tests conducted under alternative integration domain treatments. Panel A reports the test results where flat extrapolation is applied up to strike prices equivalent to one-third and three times the underlying price. Panel B presents the test results where domain reduction and symmetrization are applied as truncation treatments. The dependent variable, r(t)⁠, represents the S&P 500 index log-return on day t⁠, expressed as a percentage. ΔVOL(t)⁠, ΔSKEW(t)⁠, and ΔKURT(t) denote the daily first-order differences of implied volatility, skewness, and kurtosis estimates on day t, respectively. The Huber–White sandwich estimator is used to estimate standard errors; therefore, the unadjusted R2 is reported. t-statistics are presented in parentheses. ***, **, and * indicate statistical significance at the 1%, 5%, and 10% levels, respectively.

Panel B of Table 8 reports the test results for DRed and DSym, highlighting two key findings. First, the explanatory power of the model with DRed decreases as DRed intensity increases. This decline suggests that reducing the integration domain alone diminishes the information content of implied moment estimates, indicating that the improvement in return predictive ability with DStab is driven by stabilization rather than reduction. Second, the explanatory power of the model significantly increases with DSym, and the R2 for DSym is higher than in all DStab cases, whether the symmetry is defined in terms of d1 or strike price. A possible explanation is that, in addition to its moderate stabilization effect, DSym balances stabilization with retaining heavily weighted observations by not discarding any OTM call observations—which are much more heavily truncated than puts—while more aggressively filtering OTM put observations, as illustrated in Figure 5. Although DSym does not explicitly aim for this, its result appears to be a reasonably stabilized integration domain.

Table 9 presents the out-of-sample return forecasting ability test results for the alternative methods, which yield implications similar to those found in Table 8 but also reveal two noteworthy differences. First, although the enhancement in out-of-sample performance with full extrapolation alone is not superior to DStab, as seen in the in-sample test, full extrapolation contributes to improved return forecasting ability when applied alongside DStab at 50% intensity or higher. Second, while the improvement in out-of-sample performance with DSym is as impressive as in the in-sample test, the enhancement appears to be more dependent on the selection of rolling window length. For instance, although ROS2 for DSym in terms of strike price exceeds that of 75% DStab at its peak when the rolling window length is 45 or 50 months, the ROS2 is higher for 75% DStab in all other cases. Given that the integration domain remains somewhat unstable, the performance of DSym may depend on how abrupt changes in the integration domain affect regression results, particularly when the sample size is small.

Table 9
Open in new tab

Out-of-sample return forecasting ability of implied moments after alternative treatments

Panel A. Model versus historical mean: Full extrapolation
ROS2
Rolling window length (months)Without stabilizationWith 25% stabilizationWith 50% stabilizationWith 75% stabilizationWith 100% stabilization
5−7.76−7.17−6.99−6.86−7.26
10−2.68−2.05−1.67−1.55−1.86
15−0.020.350.810.920.99
200.831.131.401.421.58
251.011.351.611.611.75
301.632.402.692.562.51
352.032.713.002.932.95
402.432.983.303.243.34
452.693.113.393.353.42
503.183.563.823.823.94
55−4.32−4.25−4.47−4.62−5.01
60−7.38−7.22−7.46−7.58−7.95
Panel A. Model versus historical mean: Full extrapolation
ROS2
Rolling window length (months)Without stabilizationWith 25% stabilizationWith 50% stabilizationWith 75% stabilizationWith 100% stabilization
5−7.76−7.17−6.99−6.86−7.26
10−2.68−2.05−1.67−1.55−1.86
15−0.020.350.810.920.99
200.831.131.401.421.58
251.011.351.611.611.75
301.632.402.692.562.51
352.032.713.002.932.95
402.432.983.303.243.34
452.693.113.393.353.42
503.183.563.823.823.94
55−4.32−4.25−4.47−4.62−5.01
60−7.38−7.22−7.46−7.58−7.95
Panel B. Model versus historical mean: Domain reduction (DRed) and symmetrization (DSym)
ROS2
Rolling window length (months)25% DRed50% DRed75% DRedDSym (d1)DSym (strike price)
5−7.35−7.35−8.21−7.92−7.61
10−2.28−2.28−2.86−3.22−2.83
150.220.20−0.15−0.140.16
200.930.850.41−0.120.14
251.050.930.410.160.41
301.451.230.651.561.92
351.921.731.162.072.46
402.231.981.342.633.12
452.522.301.753.093.57
502.972.722.073.634.10
55−4.28−3.97−2.91−4.60−5.06
60−7.27−6.94−5.87−8.08−8.74
Panel B. Model versus historical mean: Domain reduction (DRed) and symmetrization (DSym)
ROS2
Rolling window length (months)25% DRed50% DRed75% DRedDSym (d1)DSym (strike price)
5−7.35−7.35−8.21−7.92−7.61
10−2.28−2.28−2.86−3.22−2.83
150.220.20−0.15−0.140.16
200.930.850.41−0.120.14
251.050.930.410.160.41
301.451.230.651.561.92
351.921.731.162.072.46
402.231.981.342.633.12
452.522.301.753.093.57
502.972.722.073.634.10
55−4.28−3.97−2.91−4.60−5.06
60−7.27−6.94−5.87−8.08−8.74

Notes: This table presents the results of the out-of-sample return forecasting ability tests conducted while applying alternative integration domain treatments. Panel A reports the test results where flat extrapolation is applied up to strike prices equivalent to one-third and three times the underlying price. Panel B presents the test results where domain reduction and symmetrization are applied as truncation treatments. Following Campbell and Thompson (2008), we report the ROS2 statistic, which is defined as ROS2=1-∑t=1Trt-rt^2/∑t=1Trt-rt¯2⁠, where rt^ is the fitted value derived from a predictive regression estimated through the rolling window that ends at time t−1⁠, and rt¯ is the benchmark value for the rolling window. Benchmark value is defined as the historical mean log-return. A positive value of ROS2 indicates that the predictive regression produces a lower mean squared prediction error than the benchmark value. The value of ROS2 is expressed as a percentage.

Table 9
Open in new tab

Out-of-sample return forecasting ability of implied moments after alternative treatments

Panel A. Model versus historical mean: Full extrapolation
ROS2
Rolling window length (months)Without stabilizationWith 25% stabilizationWith 50% stabilizationWith 75% stabilizationWith 100% stabilization
5−7.76−7.17−6.99−6.86−7.26
10−2.68−2.05−1.67−1.55−1.86
15−0.020.350.810.920.99
200.831.131.401.421.58
251.011.351.611.611.75
301.632.402.692.562.51
352.032.713.002.932.95
402.432.983.303.243.34
452.693.113.393.353.42
503.183.563.823.823.94
55−4.32−4.25−4.47−4.62−5.01
60−7.38−7.22−7.46−7.58−7.95
Panel A. Model versus historical mean: Full extrapolation
ROS2
Rolling window length (months)Without stabilizationWith 25% stabilizationWith 50% stabilizationWith 75% stabilizationWith 100% stabilization
5−7.76−7.17−6.99−6.86−7.26
10−2.68−2.05−1.67−1.55−1.86
15−0.020.350.810.920.99
200.831.131.401.421.58
251.011.351.611.611.75
301.632.402.692.562.51
352.032.713.002.932.95
402.432.983.303.243.34
452.693.113.393.353.42
503.183.563.823.823.94
55−4.32−4.25−4.47−4.62−5.01
60−7.38−7.22−7.46−7.58−7.95
Panel B. Model versus historical mean: Domain reduction (DRed) and symmetrization (DSym)
ROS2
Rolling window length (months)25% DRed50% DRed75% DRedDSym (d1)DSym (strike price)
5−7.35−7.35−8.21−7.92−7.61
10−2.28−2.28−2.86−3.22−2.83
150.220.20−0.15−0.140.16
200.930.850.41−0.120.14
251.050.930.410.160.41
301.451.230.651.561.92
351.921.731.162.072.46
402.231.981.342.633.12
452.522.301.753.093.57
502.972.722.073.634.10
55−4.28−3.97−2.91−4.60−5.06
60−7.27−6.94−5.87−8.08−8.74
Panel B. Model versus historical mean: Domain reduction (DRed) and symmetrization (DSym)
ROS2
Rolling window length (months)25% DRed50% DRed75% DRedDSym (d1)DSym (strike price)
5−7.35−7.35−8.21−7.92−7.61
10−2.28−2.28−2.86−3.22−2.83
150.220.20−0.15−0.140.16
200.930.850.41−0.120.14
251.050.930.410.160.41
301.451.230.651.561.92
351.921.731.162.072.46
402.231.981.342.633.12
452.522.301.753.093.57
502.972.722.073.634.10
55−4.28−3.97−2.91−4.60−5.06
60−7.27−6.94−5.87−8.08−8.74

Notes: This table presents the results of the out-of-sample return forecasting ability tests conducted while applying alternative integration domain treatments. Panel A reports the test results where flat extrapolation is applied up to strike prices equivalent to one-third and three times the underlying price. Panel B presents the test results where domain reduction and symmetrization are applied as truncation treatments. Following Campbell and Thompson (2008), we report the ROS2 statistic, which is defined as ROS2=1-∑t=1Trt-rt^2/∑t=1Trt-rt¯2⁠, where rt^ is the fitted value derived from a predictive regression estimated through the rolling window that ends at time t−1⁠, and rt¯ is the benchmark value for the rolling window. Benchmark value is defined as the historical mean log-return. A positive value of ROS2 indicates that the predictive regression produces a lower mean squared prediction error than the benchmark value. The value of ROS2 is expressed as a percentage.

Overall, although both full extrapolation and DSym demonstrate their own strengths, the use of DStab can still be advocated. Full extrapolation, when applied on its own, does not improve underlying return predictive and forecasting ability to the same extent as DStab. While DSym shows impressive enhancement in performance, the out-of-sample forecasting ability appears to be sensitive to specification. Moreover, since DSym possesses effective features that could potentially be applied to DStab, there is a possibility that the performance of DStab could be further improved.

4.6 Asymmetric DStab

As noted above, a possible reason for the superior performance of DSym is its asymmetric effect on the integration domain. Given that the observed integration domains are consistently asymmetric, with broader coverage on the put side, DSym retains OTM call observations while extensively discarding OTM put observations. In contrast, DStab always discards some of the OTM call observations if their intensity level is nonzero. To account for the asymmetric feature of DSym, we introduce a generalized version of DStab that allows for different intensity levels for puts and calls, and investigate whether this generalization improves return predictive and forecasting ability.

Table 10 presents the in-sample test results for fifteen put-call intensity level pairs. Panels A, B, and C report the results for 0, 50, and 100% intensity levels for OTM puts, respectively. The table reveals that underlying return predictive ability can be further improved when the put-side endpoints are extensively stabilized but the call-side endpoints are only moderately stabilized, similar to the case of DSym. Specifically, Panel C shows that the R2 peaks when the intensity levels for puts and calls are 100% and 50%, respectively. These results suggest that DSym’s superior performance may be attributed to its asymmetric feature.

Table 10
Open in new tab

In-sample return predictive ability of implied moments after asymmetric stabilization

Panel A. 0% stabilization for OTM puts
Puts 0%, calls 0%Puts 0%, calls 25%Puts 0%, calls 50%Puts 0%, calls 75%Puts 0%, calls 100%
r(t−1)−0.039−0.034−0.032−0.032−0.035
(−0.36)(−0.31)(−0.29)(−0.28)(−0.31)
ΔVOL(t−1)5.0335.1835.1895.1664.860
(1.37)(1.40)(1.39)(1.38)(1.29)
ΔSKEW(t−1)0.466**0.536***0.552**0.561**0.575**
(2.41)(2.45)(2.43)(2.39)(2.18)
ΔKURT(t−1)0.0230.028*0.029*0.030*0.030
(1.48)(1.67)(1.69)(1.68)(1.59)
Intercept0.059*0.059*0.059*0.059*0.059*
(1.84)(1.83)(1.83)(1.83)(1.83)
# of obs.1,5261,5261,5261,5261,526
R20.04320.04360.04350.04330.0415
Panel A. 0% stabilization for OTM puts
Puts 0%, calls 0%Puts 0%, calls 25%Puts 0%, calls 50%Puts 0%, calls 75%Puts 0%, calls 100%
r(t−1)−0.039−0.034−0.032−0.032−0.035
(−0.36)(−0.31)(−0.29)(−0.28)(−0.31)
ΔVOL(t−1)5.0335.1835.1895.1664.860
(1.37)(1.40)(1.39)(1.38)(1.29)
ΔSKEW(t−1)0.466**0.536***0.552**0.561**0.575**
(2.41)(2.45)(2.43)(2.39)(2.18)
ΔKURT(t−1)0.0230.028*0.029*0.030*0.030
(1.48)(1.67)(1.69)(1.68)(1.59)
Intercept0.059*0.059*0.059*0.059*0.059*
(1.84)(1.83)(1.83)(1.83)(1.83)
# of obs.1,5261,5261,5261,5261,526
R20.04320.04360.04350.04330.0415
Panel B. 50% stabilization for OTM puts
Puts 50%, calls 0%Puts 50%, calls 25%Puts 50%, calls 50%Puts 50%, calls 75%Puts 50%, calls 100%
r(t−1)−0.020−0.011−0.008−0.005−0.004
(−0.18)(−0.10)(−0.07)(−0.05)(−0.04)
ΔVOL(t−1)5.8696.1226.1676.1695.887
(1.54)(1.58)(1.58)(1.58)(1.49)
ΔSKEW(t−1)0.633*0.866**0.937**0.988**1.164*
(1.94)(2.37)(2.03)(2.01)(1.89)
ΔKURT(t−1)0.0440.0770.0860.0920.108
(0.89)(1.26)(1.32)(1.34)(1.40)
Intercept0.058*0.058*0.058*0.057*0.057*
(1.80)(1.79)(1.78)(1.77)(1.76)
# of obs.1,5261,5261,5261,5261,526
R20.04550.04670.04670.04650.0443
Panel B. 50% stabilization for OTM puts
Puts 50%, calls 0%Puts 50%, calls 25%Puts 50%, calls 50%Puts 50%, calls 75%Puts 50%, calls 100%
r(t−1)−0.020−0.011−0.008−0.005−0.004
(−0.18)(−0.10)(−0.07)(−0.05)(−0.04)
ΔVOL(t−1)5.8696.1226.1676.1695.887
(1.54)(1.58)(1.58)(1.58)(1.49)
ΔSKEW(t−1)0.633*0.866**0.937**0.988**1.164*
(1.94)(2.37)(2.03)(2.01)(1.89)
ΔKURT(t−1)0.0440.0770.0860.0920.108
(0.89)(1.26)(1.32)(1.34)(1.40)
Intercept0.058*0.058*0.058*0.057*0.057*
(1.80)(1.79)(1.78)(1.77)(1.76)
# of obs.1,5261,5261,5261,5261,526
R20.04550.04670.04670.04650.0443
Panel C. 100% stabilization for OTM puts
Puts 100%, calls 0%Puts 100%, calls 25%Puts 100%, calls 50%Puts 100%, calls 75%Puts 100%, calls 100%
r(t−1)0.0090.0100.0120.0150.018
(0.08)(0.09)(0.11)(0.13)(0.16)
ΔVOL(t−1)7.259*7.411*7.455*7.455*7.066
(1.72)(1.75)(1.75)(1.75)(1.64)
ΔSKEW(t−1)0.607**0.935**1.065**1.170**1.518
(2.26)(2.19)(2.13)(2.05)(1.60)
ΔKURT(t−1)−0.0360.0970.1360.1630.218
(−0.30)(0.61)(0.77)(0.85)(0.84)
Intercept0.057*0.057*0.056*0.056*0.056*
(1.75)(1.74)(1.74)(1.74)(1.72)
# of obs.1,5261,5261,5261,5261,526
R20.04890.05020.05040.05030.0473
Panel C. 100% stabilization for OTM puts
Puts 100%, calls 0%Puts 100%, calls 25%Puts 100%, calls 50%Puts 100%, calls 75%Puts 100%, calls 100%
r(t−1)0.0090.0100.0120.0150.018
(0.08)(0.09)(0.11)(0.13)(0.16)
ΔVOL(t−1)7.259*7.411*7.455*7.455*7.066
(1.72)(1.75)(1.75)(1.75)(1.64)
ΔSKEW(t−1)0.607**0.935**1.065**1.170**1.518
(2.26)(2.19)(2.13)(2.05)(1.60)
ΔKURT(t−1)−0.0360.0970.1360.1630.218
(−0.30)(0.61)(0.77)(0.85)(0.84)
Intercept0.057*0.057*0.056*0.056*0.056*
(1.75)(1.74)(1.74)(1.74)(1.72)
# of obs.1,5261,5261,5261,5261,526
R20.04890.05020.05040.05030.0473

Notes: This table presents the results of in-sample return prediction tests conducted using asymmetric DStab at various intensity levels. Panels A, B, and C report the test results where the intensity levels for OTM puts are set at 0%, 50%, and 100%, respectively. For OTM calls, intensity levels of 0%, 25%, 50%, 75%, and 100% are considered within each panel. The dependent variable, r(t)⁠, represents the S&P 500 index log-return on day t⁠, expressed as a percentage. ΔVOL(t)⁠, ΔSKEW(t)⁠, and ΔKURT(t) denote the daily first-order differences of implied volatility, skewness, and kurtosis estimates on day t, respectively. The Huber–White sandwich estimator is used to estimate standard errors; therefore, the unadjusted R2 is reported. t-statistics are presented in parentheses. ***, **, and * indicate statistical significance at the 1%, 5%, and 10% levels, respectively.

Table 10
Open in new tab

In-sample return predictive ability of implied moments after asymmetric stabilization

Panel A. 0% stabilization for OTM puts
Puts 0%, calls 0%Puts 0%, calls 25%Puts 0%, calls 50%Puts 0%, calls 75%Puts 0%, calls 100%
r(t−1)−0.039−0.034−0.032−0.032−0.035
(−0.36)(−0.31)(−0.29)(−0.28)(−0.31)
ΔVOL(t−1)5.0335.1835.1895.1664.860
(1.37)(1.40)(1.39)(1.38)(1.29)
ΔSKEW(t−1)0.466**0.536***0.552**0.561**0.575**
(2.41)(2.45)(2.43)(2.39)(2.18)
ΔKURT(t−1)0.0230.028*0.029*0.030*0.030
(1.48)(1.67)(1.69)(1.68)(1.59)
Intercept0.059*0.059*0.059*0.059*0.059*
(1.84)(1.83)(1.83)(1.83)(1.83)
# of obs.1,5261,5261,5261,5261,526
R20.04320.04360.04350.04330.0415
Panel A. 0% stabilization for OTM puts
Puts 0%, calls 0%Puts 0%, calls 25%Puts 0%, calls 50%Puts 0%, calls 75%Puts 0%, calls 100%
r(t−1)−0.039−0.034−0.032−0.032−0.035
(−0.36)(−0.31)(−0.29)(−0.28)(−0.31)
ΔVOL(t−1)5.0335.1835.1895.1664.860
(1.37)(1.40)(1.39)(1.38)(1.29)
ΔSKEW(t−1)0.466**0.536***0.552**0.561**0.575**
(2.41)(2.45)(2.43)(2.39)(2.18)
ΔKURT(t−1)0.0230.028*0.029*0.030*0.030
(1.48)(1.67)(1.69)(1.68)(1.59)
Intercept0.059*0.059*0.059*0.059*0.059*
(1.84)(1.83)(1.83)(1.83)(1.83)
# of obs.1,5261,5261,5261,5261,526
R20.04320.04360.04350.04330.0415
Panel B. 50% stabilization for OTM puts
Puts 50%, calls 0%Puts 50%, calls 25%Puts 50%, calls 50%Puts 50%, calls 75%Puts 50%, calls 100%
r(t−1)−0.020−0.011−0.008−0.005−0.004
(−0.18)(−0.10)(−0.07)(−0.05)(−0.04)
ΔVOL(t−1)5.8696.1226.1676.1695.887
(1.54)(1.58)(1.58)(1.58)(1.49)
ΔSKEW(t−1)0.633*0.866**0.937**0.988**1.164*
(1.94)(2.37)(2.03)(2.01)(1.89)
ΔKURT(t−1)0.0440.0770.0860.0920.108
(0.89)(1.26)(1.32)(1.34)(1.40)
Intercept0.058*0.058*0.058*0.057*0.057*
(1.80)(1.79)(1.78)(1.77)(1.76)
# of obs.1,5261,5261,5261,5261,526
R20.04550.04670.04670.04650.0443
Panel B. 50% stabilization for OTM puts
Puts 50%, calls 0%Puts 50%, calls 25%Puts 50%, calls 50%Puts 50%, calls 75%Puts 50%, calls 100%
r(t−1)−0.020−0.011−0.008−0.005−0.004
(−0.18)(−0.10)(−0.07)(−0.05)(−0.04)
ΔVOL(t−1)5.8696.1226.1676.1695.887
(1.54)(1.58)(1.58)(1.58)(1.49)
ΔSKEW(t−1)0.633*0.866**0.937**0.988**1.164*
(1.94)(2.37)(2.03)(2.01)(1.89)
ΔKURT(t−1)0.0440.0770.0860.0920.108
(0.89)(1.26)(1.32)(1.34)(1.40)
Intercept0.058*0.058*0.058*0.057*0.057*
(1.80)(1.79)(1.78)(1.77)(1.76)
# of obs.1,5261,5261,5261,5261,526
R20.04550.04670.04670.04650.0443
Panel C. 100% stabilization for OTM puts
Puts 100%, calls 0%Puts 100%, calls 25%Puts 100%, calls 50%Puts 100%, calls 75%Puts 100%, calls 100%
r(t−1)0.0090.0100.0120.0150.018
(0.08)(0.09)(0.11)(0.13)(0.16)
ΔVOL(t−1)7.259*7.411*7.455*7.455*7.066
(1.72)(1.75)(1.75)(1.75)(1.64)
ΔSKEW(t−1)0.607**0.935**1.065**1.170**1.518
(2.26)(2.19)(2.13)(2.05)(1.60)
ΔKURT(t−1)−0.0360.0970.1360.1630.218
(−0.30)(0.61)(0.77)(0.85)(0.84)
Intercept0.057*0.057*0.056*0.056*0.056*
(1.75)(1.74)(1.74)(1.74)(1.72)
# of obs.1,5261,5261,5261,5261,526
R20.04890.05020.05040.05030.0473
Panel C. 100% stabilization for OTM puts
Puts 100%, calls 0%Puts 100%, calls 25%Puts 100%, calls 50%Puts 100%, calls 75%Puts 100%, calls 100%
r(t−1)0.0090.0100.0120.0150.018
(0.08)(0.09)(0.11)(0.13)(0.16)
ΔVOL(t−1)7.259*7.411*7.455*7.455*7.066
(1.72)(1.75)(1.75)(1.75)(1.64)
ΔSKEW(t−1)0.607**0.935**1.065**1.170**1.518
(2.26)(2.19)(2.13)(2.05)(1.60)
ΔKURT(t−1)−0.0360.0970.1360.1630.218
(−0.30)(0.61)(0.77)(0.85)(0.84)
Intercept0.057*0.057*0.056*0.056*0.056*
(1.75)(1.74)(1.74)(1.74)(1.72)
# of obs.1,5261,5261,5261,5261,526
R20.04890.05020.05040.05030.0473

Notes: This table presents the results of in-sample return prediction tests conducted using asymmetric DStab at various intensity levels. Panels A, B, and C report the test results where the intensity levels for OTM puts are set at 0%, 50%, and 100%, respectively. For OTM calls, intensity levels of 0%, 25%, 50%, 75%, and 100% are considered within each panel. The dependent variable, r(t)⁠, represents the S&P 500 index log-return on day t⁠, expressed as a percentage. ΔVOL(t)⁠, ΔSKEW(t)⁠, and ΔKURT(t) denote the daily first-order differences of implied volatility, skewness, and kurtosis estimates on day t, respectively. The Huber–White sandwich estimator is used to estimate standard errors; therefore, the unadjusted R2 is reported. t-statistics are presented in parentheses. ***, **, and * indicate statistical significance at the 1%, 5%, and 10% levels, respectively.

Table 11 summarizes the out-of-sample test results for the three top-performing put-call intensity level pairs, alongside the results for DSym for comparison. The table shows that asymmetric DStab outperforms DSym for all rolling window lengths of 50 months or less, which encompass all window lengths with positive ROS2 relative to the historical mean. Additionally, given the 25% intervals between the intensity levels considered in this study, there is potential for further improvement in return forecasting ability by fine-tuning the intensity levels. Overall, the results suggest that DStab enhances the information content of implied moment estimates more effectively than other methods when the sample integration domain properties are appropriately accounted for and assessed with out-of-sample return forecasting ability.

Table 11
Open in new tab

Out-of-sample return forecasting ability of implied moments after asymmetric stabilization

Panel A. With versus without treatment
ROS2
Rolling window length (months)Puts 100%, calls 25%Puts 100%, calls 50%Puts 100%, calls 75%DSym (⁠d1⁠)DSym (strike price)
50.490.410.31−0.50−0.19
100.960.860.73−0.89−0.50
151.191.141.04−0.32−0.02
201.010.960.88−1.06−0.80
251.141.080.98−0.92−0.67
301.391.341.260.020.38
351.361.331.260.070.46
401.381.371.320.300.79
451.201.191.130.480.96
501.271.261.200.551.02
55−0.70−0.73−0.76−0.23−0.69
60−0.47−0.50−0.54−0.70−1.36
Panel A. With versus without treatment
ROS2
Rolling window length (months)Puts 100%, calls 25%Puts 100%, calls 50%Puts 100%, calls 75%DSym (⁠d1⁠)DSym (strike price)
50.490.410.31−0.50−0.19
100.960.860.73−0.89−0.50
151.191.141.04−0.32−0.02
201.010.960.88−1.06−0.80
251.141.080.98−0.92−0.67
301.391.341.260.020.38
351.361.331.260.070.46
401.381.371.320.300.79
451.201.191.130.480.96
501.271.261.200.551.02
55−0.70−0.73−0.76−0.23−0.69
60−0.47−0.50−0.54−0.70−1.36
Panel B. Model versus historical mean
ROS2
Rolling window length (months)Puts 100%, calls 25%Puts 100%, calls 50%Puts 100%, calls 75%DSym (⁠d1⁠)DSym (strike price)
5−6.93−7.01−7.11−7.92−7.61
10−1.37−1.47−1.60−3.22−2.83
151.371.321.22−0.140.16
201.951.901.82−0.120.14
252.222.162.060.160.41
302.922.882.801.561.92
353.363.333.262.072.46
403.713.713.652.633.12
453.813.803.743.093.57
504.354.344.283.634.10
55−5.07−5.10−5.12−4.60−5.06
60−7.85−7.88−7.92−8.08−8.74
Panel B. Model versus historical mean
ROS2
Rolling window length (months)Puts 100%, calls 25%Puts 100%, calls 50%Puts 100%, calls 75%DSym (⁠d1⁠)DSym (strike price)
5−6.93−7.01−7.11−7.92−7.61
10−1.37−1.47−1.60−3.22−2.83
151.371.321.22−0.140.16
201.951.901.82−0.120.14
252.222.162.060.160.41
302.922.882.801.561.92
353.363.333.262.072.46
403.713.713.652.633.12
453.813.803.743.093.57
504.354.344.283.634.10
55−5.07−5.10−5.12−4.60−5.06
60−7.85−7.88−7.92−8.08−8.74

Notes: This table presents the results of the out-of-sample return forecasting ability test for the three asymmetric domain stabilization cases with the strongest predictive power. For OTM puts, the intensity level is fixed at 100%, as this level exhibits the strongest forecasting performance. For OTM calls, intensity levels of 25%, 50%, and 75% are considered in each panel. The test results for domain symmetrization (DSym) are also included for comparison. Following Campbell and Thompson (2008), we report the ROS2 statistic, which is defined as ROS2=1-∑t=1Trt-rt^2/∑t=1Trt-rt¯2⁠, where rt^ is the fitted value derived from a predictive regression estimated through the rolling window that ends at time t−1⁠, and rt¯ is the benchmark value for the rolling window. Benchmark value is defined as the fitted value estimated without stabilization for Panel A, and the historical mean log-return for Panel B. A positive value of ROS2 indicates that the predictive regression produces a lower mean squared prediction error than the benchmark value. The value of ROS2 is expressed as a percentage.

Table 11
Open in new tab

Out-of-sample return forecasting ability of implied moments after asymmetric stabilization

Panel A. With versus without treatment
ROS2
Rolling window length (months)Puts 100%, calls 25%Puts 100%, calls 50%Puts 100%, calls 75%DSym (⁠d1⁠)DSym (strike price)
50.490.410.31−0.50−0.19
100.960.860.73−0.89−0.50
151.191.141.04−0.32−0.02
201.010.960.88−1.06−0.80
251.141.080.98−0.92−0.67
301.391.341.260.020.38
351.361.331.260.070.46
401.381.371.320.300.79
451.201.191.130.480.96
501.271.261.200.551.02
55−0.70−0.73−0.76−0.23−0.69
60−0.47−0.50−0.54−0.70−1.36
Panel A. With versus without treatment
ROS2
Rolling window length (months)Puts 100%, calls 25%Puts 100%, calls 50%Puts 100%, calls 75%DSym (⁠d1⁠)DSym (strike price)
50.490.410.31−0.50−0.19
100.960.860.73−0.89−0.50
151.191.141.04−0.32−0.02
201.010.960.88−1.06−0.80
251.141.080.98−0.92−0.67
301.391.341.260.020.38
351.361.331.260.070.46
401.381.371.320.300.79
451.201.191.130.480.96
501.271.261.200.551.02
55−0.70−0.73−0.76−0.23−0.69
60−0.47−0.50−0.54−0.70−1.36
Panel B. Model versus historical mean
ROS2
Rolling window length (months)Puts 100%, calls 25%Puts 100%, calls 50%Puts 100%, calls 75%DSym (⁠d1⁠)DSym (strike price)
5−6.93−7.01−7.11−7.92−7.61
10−1.37−1.47−1.60−3.22−2.83
151.371.321.22−0.140.16
201.951.901.82−0.120.14
252.222.162.060.160.41
302.922.882.801.561.92
353.363.333.262.072.46
403.713.713.652.633.12
453.813.803.743.093.57
504.354.344.283.634.10
55−5.07−5.10−5.12−4.60−5.06
60−7.85−7.88−7.92−8.08−8.74
Panel B. Model versus historical mean
ROS2
Rolling window length (months)Puts 100%, calls 25%Puts 100%, calls 50%Puts 100%, calls 75%DSym (⁠d1⁠)DSym (strike price)
5−6.93−7.01−7.11−7.92−7.61
10−1.37−1.47−1.60−3.22−2.83
151.371.321.22−0.140.16
201.951.901.82−0.120.14
252.222.162.060.160.41
302.922.882.801.561.92
353.363.333.262.072.46
403.713.713.652.633.12
453.813.803.743.093.57
504.354.344.283.634.10
55−5.07−5.10−5.12−4.60−5.06
60−7.85−7.88−7.92−8.08−8.74

Notes: This table presents the results of the out-of-sample return forecasting ability test for the three asymmetric domain stabilization cases with the strongest predictive power. For OTM puts, the intensity level is fixed at 100%, as this level exhibits the strongest forecasting performance. For OTM calls, intensity levels of 25%, 50%, and 75% are considered in each panel. The test results for domain symmetrization (DSym) are also included for comparison. Following Campbell and Thompson (2008), we report the ROS2 statistic, which is defined as ROS2=1-∑t=1Trt-rt^2/∑t=1Trt-rt¯2⁠, where rt^ is the fitted value derived from a predictive regression estimated through the rolling window that ends at time t−1⁠, and rt¯ is the benchmark value for the rolling window. Benchmark value is defined as the fitted value estimated without stabilization for Panel A, and the historical mean log-return for Panel B. A positive value of ROS2 indicates that the predictive regression produces a lower mean squared prediction error than the benchmark value. The value of ROS2 is expressed as a percentage.

5 Conclusion

This study proposes a new empirical methodology to enhance the information content and underlying return predictive and forecasting ability of implied moment estimates derived from BKM’s model-free implied moment estimators. Specifically, DStab stabilizes the endpoint location of the integration domains where OTM option price observations are available for moment estimation. By enhancing consistency in the width and asymmetry of the integration domain throughout the entire sample period, DStab ensures the BKM estimators continue to estimate the moments of a consistently defined TRND, thereby refining the information content of the estimates. The improved information content results in stronger underlying return predictive and forecasting ability. Both in-sample and out-of-sample test results demonstrate that the return predictive and forecasting ability of implied moments is strengthened by DStab, provided that the availability of the options data is properly reflected when determining the intensity level of DStab. Furthermore, the test results reveal that the out-of-sample performance of implied moments is better with DStab compared to other treatment methods.

Although DStab may not successfully reduce the bias of BKM’s model-free implied moment estimators when the bias is defined in terms of the true implied RND, the method remains significant because it enhances the estimates’ informativeness regarding the underlying return by circumventing the original specification of the estimators. Future research could leverage DStab alongside model-free implied moment estimators to explore the relationship between option-implied moments and other relevant market variables, such as real economic activity (Kim, Cho, and Ryu 2025), with greater accuracy. The relationship across DStab, implied moment estimate dynamics, and underlying returns can be further investigated with more sophisticated models, given the potential nonlinear relationship among implied moments and underlying price (Lee, Ryu, and Yang 2025), the limited return predictability of parsimonious models (Kelly, Malamud, and Zhou 2024), and the inherent complexity of economic systems, which often resists simple measurement approaches (Bybee et al. 2024).

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