South Korea Hyundai Motor Option-Implied 30-Day Expected-Return Lower Bound

The groupEQXR series are single-stock option-implied risk-neutral quantities for two KOSPI 200 constituents, computed model-free from that day's KRX equity-option settlement chain. They fall into two estimand families. The first is the risk-neutral moment family of Bakshi, Kapadia, and Madan (2003), namely the thirty-day risk-neutral variance, skewness, and kurtosis of the stock's log return, recovered from a static portfolio of out-of-the-money option prices by the state-price spanning of Breeden and Litzenberger (1978) and the twice-differentiable-payoff replication of Bakshi and Madan (2000) and Carr and Madan (2001). The second is the forward-looking expected-return lower bound family, namely a lower bound on the stock's expected excess return read off the same option surface (Martin 2017; Kadan and Tang 2020; Martin and Wagner 2019).

The moment quantities describe the whole risk-neutral distribution of the one-month return through its second, third, and fourth standardized moments, and the variance is reported on the annualized volatility-point basis that KRVKOSPI and KRSVIX share while the skewness and kurtosis are dimensionless and horizon-scale-invariant (Dennis and Mayhew 2002; Jiang and Tian 2005). The bound is issued two ways. The Kadan and Tang (2020) bound uses only the stock's own chain and stays live where the market SVIX term goes null, so it is the market-free primary line, and the Martin and Wagner (2019) bound adds the market SVIX term reused from KRSVIX and so inherits the KRSVIX honest gap, going null in the 2026 extreme regime rather than being silently patched. The bound is a lower bound on the expected excess return and is not a forecast of the return itself.

Two properties of the bound are disclosed rather than smoothed over. First, the bound is valid and informative but its tightness is not asserted for the Korean single-stock surface, since a bound valid in all market conditions need not be tight in all of them (Back, Crotty, and Kazempour 2022). Second, both the strike truncation at the outermost genuine strike and the discretisation of the sparse ladder bias the bound downward, so the published number is a conservative and understated lower bound. In the 2026 rally-blackout regime the annualized bound reaches extreme levels above ninety percent, corroborated by the same-day official V-KOSPI 200 near eighty-seven, and those extreme readings are genuine high-volatility regime values shipped as-is rather than capped.

The cross-sectional risk-neutral moment layers of Conrad, Dittmar, and Ghysels (2013) and Huang and Li (2019) were built over the eligible universe but do not replicate on Korean data, so they are withheld and only the two-name moment-and-bound surface publishes. The option-implied risk-neutral skewness and kurtosis of Bakshi, Kapadia, and Madan (2003) were likewise constructed and evaluated over both names but are withheld as measurement-limited on the single-stock ladder, since a median 32 percent of the spanning-integral mass of the skewness contract and 70 percent of the kurtosis contract lies beyond the roughly plus-or-minus 15 percent moneyness band that trades, so the higher moments are pinned by settlement-implied rather than traded strikes (Jiang and Tian 2005; Dennis and Mayhew 2006). Refitting the smile with the arbitrage-free stochastic-volatility-inspired parameterization of Gatheral and Jacquier (2014) cuts the decimation bias by 48 percent for the skewness and by 35 percent for the kurtosis yet leaves the cross-day tracking correlation weak at 0.21 and 0.08 respectively, so neither higher moment is a reliable point estimate and only the variance-based quantities, whose untraded integrand mass is 5 percent, publish as KRRNV volatility and KRERB bound.

Every groupEQXR quantity is computed each day from that day's KRX equity-option settlement chain, with no cross-day smoothing, and the whole panel is recomputed on every run.

(1) The out-of-the-money smile and the spanning core. At the constant thirty-day tenor the out-of-the-money settlement-implied-volatility cross-section is converted to European Black-Scholes prices, interpolated as a cubic spline in log-moneyness inside the genuine-traded band and flat-extrapolated beyond it, and sampled on a fine strike grid, in the interpolation convention of Jiang and Tian (2005) and Chang, Christoffersen, and Jacobs (2013). The single shared quadrature core evaluates the Bakshi, Kapadia, and Madan (2003) volatility, cubic, and quartic contracts by the trapezoidal strip of Dennis and Mayhew (2002), with a minimum of two out-of-the-money strikes per wing and a low-confidence flag when the outer strikes are settlement-implied-volatility-only (Dennis and Mayhew 2006).

(2) The risk-neutral moments. The risk-neutral variance is the annualization base, the variance contract net of the squared risk-neutral mean of the log return.

The skewness and kurtosis follow from the same three contracts by the Bakshi, Kapadia, and Madan (2003) Theorem 1, in the canonical form whose put leg carries the log of the spot over the strike (Breeden and Litzenberger 1978; Bakshi and Madan 2000).

(3) The Kadan and Tang (2020) market-free bound. The primary bound is the single-name risk-neutral variance of the simple return divided by the gross rate, integrated over out-of-the-money puts below the parity forward and calls above it (Martin 2017; van Binsbergen, Diamond, and Grotteria 2022).

The forward is recovered from put-call parity so it is dividend-robust, and the bound holds as a lower bound under the Negative Correlation Condition, whose class is served alongside it; every name with a published bound classifies conservative on the delta-hat gate. Because the strip is truncated at the outermost genuine strike and the ladder is discrete, the bound is biased downward and is therefore conservative and understated.

(4) The Martin and Wagner (2019) market-inclusive bound. The companion bound adds the market SVIX term, reused verbatim from KRSVIX rather than recomputed, to half the difference between the stock's own SVIX-squared and the cross-sectional average.

The cross-sectional average uses interim equal weights over the eligible names rather than free-float market-cap weights, pending the spot-data grant, so the level of the Martin and Wagner bound is provisional and will revise when float-cap weights land while the market-free Kadan and Tang bound is unaffected. Because the market term is KRSVIX, the Martin and Wagner bound inherits the KRSVIX honest gap and is null in the 2026 extreme-volatility regime rather than extrapolated. European exercise is confirmed on the KRX settlement surface, so the de-Americanization corrections are dropped (Broadie and Detemple 1996; Barone-Adesi and Whaley 1987).

The groupEQXR quantities belong to the model-free risk-neutral moment tradition, in which option prices across the strike range pin down the risk-neutral distribution of the underlying without a parametric model. Breeden and Litzenberger (1978) show that the second strike derivative of the call price is the risk-neutral density, and Bakshi, Kapadia, and Madan (2003) turn that density into closed-form risk-neutral variance, skewness, and kurtosis, so a thirty-day single-stock risk-neutral moment read off the option surface is a well-defined object independent of any pricing model (Jiang and Tian 2005). The risk-neutral skewness in particular is the object that the cross-sectional single-stock option literature prices, in which more negatively skewed names earn different subsequent returns (Conrad, Dittmar, and Ghysels 2013; Boyer, Mitton, and Vorkink 2010; Bali and Murray 2013).

The economic content of the bound family is a lower bound on the expected excess return. Martin (2017) derives the market bound from the risk-neutral variance of the simple return, Kadan and Tang (2020) show that under the Negative Correlation Condition the same construction bounds an individual stock's expected excess return from below using only its own options, and Martin and Wagner (2019) add the market term to sharpen it. The bound is valid and informative but its tightness is not asserted for the Korean surface, and both truncation and discretisation bias it downward, so it is read as a conservative and understated floor rather than as a point forecast (Back, Crotty, and Kazempour 2022; Kilic and Shaliastovich 2019).

For the two Korean names specifically, the moments describe the shape of the priced one-month return distribution while the bound describes the floor the option market places under the expected excess return, and neither is a return forecast. The Kadan and Tang bound is market-free and continuous, the Martin and Wagner bound gaps honestly where the market SVIX term is null, and the two-name surface is a descriptive risk-neutral read on genuinely-traded single-stock options rather than a cross-sectional trading signal (Harrison and Kreps 1979; Stilger, Kostakis, and Poon 2017).

On a markets desk the groupEQXR quantities are read as a model-free risk-neutral characterization of two single stocks, and they are used for monitoring, hedging, and relative-value work rather than as a directional signal. The risk-neutral variance is the option market's price of near-term uncertainty on the name and the natural benchmark for a single-stock variance or volatility book whose static-replication logic Bakshi, Kapadia, and Madan (2003) and Jiang and Tian (2005) set out, while the skewness and kurtosis read the asymmetry and tail weight the surface prices into the one-month distribution (Dennis and Mayhew 2002; Chang, Christoffersen, and Jacobs 2013).

The expected-return bound is read as a floor rather than a forecast. The Kadan and Tang (2020) bound is market-free so it is the line a desk relies on when the market SVIX term is unavailable, and the Martin and Wagner (2019) bound adds the market term when it is live, so their spread against each other is a market-inclusion read rather than a mispricing (Martin 2017). The bound is valid and informative but not asserted tight, and its downward truncation and discretisation bias make it conservative, so a desk reads it as a lower floor on the expected excess return rather than as a precise level (Back, Crotty, and Kazempour 2022; van Binsbergen, Diamond, and Grotteria 2022).

Two operational limits govern the surface. The Martin and Wagner bound inherits the KRSVIX honest gap and goes null in the 2026 extreme-volatility regime, so a desk treats those gaps as missing data rather than as a low bound, while the market-free Kadan and Tang bound stays live, and the interim equal-weight cross-sectional average makes the Martin and Wagner level provisional pending the float-cap revision (Newey and West 1987; Fama and MacBeth 1973). And the sparse next-month wings of the domestic single-stock options market cap the precision of any tail reading on illiquid days, so the deep-tail quantities are flagged low-confidence rather than over-read (Feunou, Jahan-Parvar, and Okou 2017; Amihud 2002).

KRERB005380 publishes the Hyundai Motor expected-excess-return lower bound, whose primary line is the market-free Kadan-Tang bound, and the object tested is the natural logarithm of that strictly-positive continuous bound, where the Martin-Wagner line is null-gapped rather than the battery object. Its innovations scale multiplicatively with the level, so the log is the natural scale and the deterministic term is pinned to a constant. A deterministic trend in the log bound would imply the level diverging to zero or infinity so trend-stationarity is excluded a priori as uninformative rather than tested. The bound is a model-free deterministic transform of each day's option chain computed from that day alone with no two-sided filter and no full-sample re-estimation, so its measured persistence is a property of the option-price dynamics and not a smoother gain (Stock and Watson 1998; Hamilton 2018), and the log removes the zero lower bound so the regulated-process distortion of unit-root inference under a bound does not apply (Cavaliere 2005; Cavaliere and Taylor 2009; Cavaliere and Xu 2014). The bound is a valid lower bound under the conservative Negative Correlation Condition, whose class is conservative at a delta-hat below the pre-registered threshold of 4, and its tightness is not asserted, while the deep tail beyond the traded band is filled by flat-extrapolated settlement implied volatility rather than dropped so the bias direction relative to the true bound is not established, the downward-truncation understatement applying to the raw-strike method and not to this flat-tail-extended path.

Over 2005 published days the unit-root reading is ambiguous. The augmented Dickey-Fuller test reads -2.3856 at p = 0.1459 and does not reject the unit root, and the KPSS test rejects level stationarity at 1.2589 with p < 0.01, so the two concur on a non-stationary reading, while the Phillips-Perron test is the sole dissenter, rejecting the unit root at -4.5083 with p = 0.000190, which reflects the documented Phillips-Perron over-rejection toward spurious stationarity under near-unit moving-average errors rather than independent evidence of stationarity, so no clean order of integration is assigned (Dickey and Fuller 1979; Said and Dickey 1984; Phillips and Perron 1988; Kwiatkowski et al. 1992; Ng and Perron 2001). The generalized-least-squares refinement is reserved for such borderline readings and only confirms the augmented Dickey-Fuller leg (Elliott, Rothenberg, and Stock 1996). The first-order autocorrelation of the log bound is 0.9677 with an implied half-life of about 21.12 trading days. This is a descriptive persistence summary and not an integration claim, and it is read against the near-unity small-sample downward bias of the autoregressive estimate (Andrews 1993).

Serial-dependence and break diagnostics are read on the first difference of the log bound over the 1871 consecutive-trading-day steps that remain once differences spanning the null gaps are excluded by pre-registration, so the KRSVIX-blackout gaps of the Martin and Wagner line and the low-confidence days are excluded from the difference sample, and the persistence read here is not independent of the KRSVIX and KRVKOSPI readings on the same option surface. The Ljung and Box (1978) portmanteau, refining the Box and Pierce (1970) form, rejects white noise at lags 10 and 20 at the five percent level, and the automatic portmanteau of Escanciano and Lobato (2009) rejects at the five percent level, which is the mandated companion because conditional-heteroskedasticity clustering is the defining feature of an option-implied series, and the plain portmanteau over-rejects under that clustering. The Bai and Perron (1998) procedure by the Bai and Perron (2003) algorithm finds no break in the mean of the differenced series (Perron 1989).

The series is daily and carries no low-integer seasonal period, and the constant-thirty-day tenor removes any expiry-cycle periodicity, so the seasonal-unit-root and seasonal-stationarity batteries are inapplicable and are not run (Hylleberg et al. 1990; Canova and Hansen 1995; Beaulieu and Miron 1992; Ghysels and Osborn 2001). Because the bound carries only authentic traded prices and the Martin and Wagner line goes null in the 2026 extreme-volatility regime with settlement implied volatility near ninety percent and V-KOSPI 200 near eighty-seven, these persistence and stability readings are statements about the published spanned days, and the finite 2005-day sample makes them weak rather than strong.