## Methodology

Left Tail Volatility (LTV): An option-implied measure of return volatility generated by the left tail of the one-week risk-neutral return distribution.

Left Tail Probability (LTP): An option-implied probability of a left tail event in the one-week risk-neutral return distribution.

The measures are constructed from out-of-the-money short-dated options. The parameters employed in estimation are the tail shape $$\alpha^-$$ and level shift $$\phi^-$$:

\begin{equation*} \begin{split} &\hat{\alpha_t}^- ~=~ \underset{2\leq i\leq N}{\mathrm{median}} \, \left|\,1-\frac{\ln{\frac{O_{t,\tau}(k_{t,i})}{O_{t,\tau}(k_{t,i-1})}}}{k_{t,i}-k_{t,i-1}}\,\right|\\ &\hat{\phi_t}^- ~=~ \underset{1\leq i\leq N}{\mathrm{median}} \, \left|\,\ln{\frac{e^{r_{t,\tau}}O_{t,\tau}(k_{t,i})}{\tau F_{t,\tau}}}-(1+\hat{\alpha_t}^-)k_{t,i}+\ln(\hat{\alpha_t}^-+1)+\ln(\hat{\alpha_t}^-)\,\right| \end{split} \end{equation*} where \begin{equation*} \begin{split} &N ~~~=~~~ \text{number of valid option quotes}\\ &O_{t,\tau}(k_{t,i}) ~= ~\text{mid-price of i-th option}\\ &k_{t,i} ~~~=~~~ \text{unadjusted log-moneyness of i-th option}\\ &r_{t,\tau} ~~~=~~~ \text{riskless interest rate with maturity }\tau \end{split} \end{equation*}
Given $$\hat{\alpha_t}^-$$ and $$\hat{\phi_t}^-$$, our tail measures are computed according to:

\begin{equation*} \begin{split} &\textbf{Left Tail Volatility} ~~~=~~~100 \times \sqrt{~ \hat{\phi_t}^-\times e^{-\hat{\alpha_t}^-|\theta_t|}\times\frac{\hat{\alpha_t}^-|\theta_t|(\hat{\alpha_t}^-k_t+2)+2}{(\hat{\alpha_t}^-)^3 }~}\\ &\textbf{Left Tail Probability} ~=~100 \times \hat{\phi_t}^- \times\,\frac{e^{-\hat{\alpha_t}^-|\theta_t|}}{\hat{\alpha_t}^-} \end{split} \end{equation*} where $$\theta_t$$ is a threshold of negative tail jump. For LTV, we set it to a 10 standard deviation move over one week. For LTP, we fix it to measure the probability of a plunge more than or equal to a constant 10% barrier over one week.