The definition of tail risk (risk of 3-standard deviations movement) seems to imply there would be a current market indicator for this.
Is there such an indicator available somewhere?
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Twitter feed of Taleb is a decent predictor.
Other than that, most of the stuff that you find is proprietary and somewhat subjective/vague. Here is a paper that proposes VVIX index as a tail risk indicator:
This paper reports that the volatility-of-volatility implied by VIX options has predictability for tail risk hedging returns. Specifically, an increase in the volatility-of-volatility as measured by the VVIX index raises current prices of tail risk hedging options, such as S&P 500 puts and VIX calls, and lowers their subsequent returns over the next three to four weeks. The results are robust to jump risk, skewness, kurtosis, option liquidity, variance risk premium, and limit of arbitrage. The predictability can be explained by either risk premiums for a time-varying crash risk factor or uncertainty premiums for a time-varying uncertain belief in volatility.
St. Louis Fed Financial Stress Index - https://fred.stlouisfed.org/series/STLFSI2
There are others (e.g. from the sellside) but this is publicly available.
EDIT: looks like some of the sellside indicators are published on Bloomberg (no permissioning required), e.g. GFSI Index from BAML.
Bollerslev and Todorov have a paper on this: Tails, Fears, and Risk Premia, Jrl of Finance, (December 2011), pages 2165-2211 (link) . If you have option data, you can simply apply their method. Very often, academics publish their codes and their data on their website -- maybe either of them, or someone else who quote them have the codes.
We consider seven well-established FCIs: the Bloomberg FCI, the Citi FCI, the Deutsche Bank (DB) FCI, the Goldman Sachs (GS) FCI, the Kansas City Federal Reserve Financial Stress Index (KCFSI), the Macroeconomic Advisers Monetary and Financial Conditions Index, and the OECD FCI.