Abstract
The nonparametric jump risk measures are sometimes difficult to construct using only the daily closing quotes and prices of short-dated and deep-out-of-the-money options. In this case, a time-varying shape parameter of risk-neutral jump tails in asset returns is usually assumed to be constant from week to week, in order to mitigate the impact of noise. In this context, this study proposes a method for measuring the daily option-implied jump tail risks. We use high-frequency options data with a data cleaning process, which relaxes the constancy assumption to more general cases such that the shape parameter can change at a daily frequency. We also apply the proposed daily tail measure to the high-frequency data of Nikkei 225 options. The results confirm the coherence of the daily tail risk measure with the existing measures but reveal relatively large spikes on particular days during the week associated with the tail events. To demonstrate the usefulness of the proposed measure, we empirically analyze the short-term predictability of the variance risk premium (VRP). The analyses suggest that the daily tail risk measure, which is a jump tail risk component of \({\textit{VRP}}\), has a significant predictive power for future \({\textit{VRP}}\) and that the inclusion of the diffusive and jump risk components of \({\textit{VRP}}\) as separate predictors improves the forecasting accuracy.
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Notes
The parameter estimates of \(\alpha ^{+}_t\) and \(\phi ^{+}_t\) for the right jump tails in Eq. (7) are obtained as
$$\begin{aligned} {\hat{\alpha }}^{+}_t&= \mathop {\mathrm{arg\,min}}\limits _{\alpha ^{+}} \frac{1}{N^{+}_t} \sum ^{N^{+}_t}_{i=2} \left| \text {log} \left( \frac{O_{t,\tau }(k_{t,i})}{O_{t,\tau }(k_{t,i-1})} \right) (k_{t,i} - k_{t,i-1})^{-1} - \, (1 + (-\alpha ^{+}) )\right| , \end{aligned}$$(11)$$\begin{aligned} {\hat{\phi }}^{+}_t&= \mathop {\mathrm{arg\,min}}\limits _{\phi ^{+}} \frac{1}{N^{+}_t} \sum ^{N^{+}_t}_{i=2} \biggr |\text {log} \left( \frac{e^{r_{t}\tau }O_{t,\tau }(k_{t,i})}{\tau F_{t,\tau }} \right) - (1 - {\hat{\alpha }}^{+}_t ) k_{t,i} + \text {log} ({\hat{\alpha }}^{+}_t - 1) + \text {log} ({\hat{\alpha }}^{+}_t ) - \text {log} (\phi ^{+}) \biggr |. \end{aligned}$$(12)In case of the right jump tails, \(O_{t,\tau }(k_{t,i})\) corresponds to OTM call option price or mid-quote for i-th log-moneyness \(k_{t,i}\) and \(N^{+}_t\) is the number of puts used in estimation with moneyness \(0< k_{t,1}< \cdots < k_{t,N^{+}_t}\). Then, the right jump tail variation is expressed as
$$\begin{aligned} {\textit{RJV}}^{{\mathbb {Q}}}_{t,\tau } = \tau \phi ^{+}_t e^{-\alpha ^+_t |k_t|} ( \alpha ^{+}_t k_{t} (\alpha ^{+}_t k_t + 2) +2 ) / (\alpha ^{+}_t)^3 = {{\mathbb {E}}}^{{\mathbb {Q}}}_t [{\textit{RJV}}^{\mathbb Q}_{t,\tau }]. \end{aligned}$$(13)The average number of \(N^-_t\) for the estimation of daily \(\alpha ^-_t\) based on the cleaned deep OTM put options data at the 10–30 min intervals is 178, and the corresponding 25% and 75% empirical quantiles are 104 and 228 in our sample period. They are more than the average number being 45 in Table 2 of Andersen et al. (2021) for the estimation of weekly \(\alpha ^-_t\) based on the daily deep OTM options data. Overall, our sample provides an alternative basis for the computation of the option-implied left jump variation.
We obtain one estimated value of \(\alpha ^{-}_t\) within the day by keeping the mid-quotes from the 14 intradaily intervals for the day. After that, we estimate \(\phi ^{-}_t\) and calculate \({\textit{LJV}}\) for each intradaily interval. Thus, we investigate whether there is a periodicity pattern in \(\phi ^{-}_t\) and \({\textit{LJV}}\) or not. We find that \(\phi ^{-}_t\) and \({\textit{LJV}}\) do not exhibit a remarkable pattern such as well-observed U-shaped intraday periodicity in asset return volatility in financial markets.
We also compute weekly \({\textit{LJV}}\) measures by averaging the daily measures constructed from high-frequency options data within the week. The sample cross-correlation versus the existing measure of weekly \({\textit{LJV}}\) in Andersen et al. (2021) is 0.88.
Using the implied left jump variation, Ubukata (2019) investigates its predictive ability for credit spreads in the US and Japanese corporate bond markets. They find that it could strongly predict lower-rated credit spreads and default spreads in Japan, even when controlled for the traditional predictors and lagged credit spreads, though it might be a weaker predictor of the US credit spreads.
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Acknowledgements
This work is supported by financial support from Grants-in-Aid for Scientific Research 18K01690 and 20H00073. We thank the Editor, Bertrand Candelon, and the anonymous referee for many suggestions that significantly improved the paper. The author thanks participants at the 6th Hitotsubashi Summer Institute (HSI2020) at Hitotsubashi University, October 2020, and the 4th International Conference on Econometrics and Statistics (EcoSta 2021) at the Hong Kong University of Science and Technology, June 2021, for useful comments.
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Ubukata, M. A time-varying jump tail risk measure using high-frequency options data. Empir Econ 63, 2633–2653 (2022). https://doi.org/10.1007/s00181-022-02209-5
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DOI: https://doi.org/10.1007/s00181-022-02209-5