Show how equity and option markets price in geopolitical risk, including shifts in downside risk and tail-risk premiums.

FRM® Part II · Current Issues in Financial Markets · Global Financial Stability Report, Chapter 2 · FRM2.CI.4.e

The idea

Equity and option markets show geopolitical risk through levels and shapes. Implied volatility rises after a shock, and the volatility skew steepens because out-of-the-money puts become more expensive relative to at-the-money options and calls. Risk reversal = call implied volatility − put implied volatility. This simplified symmetric-strike version uses strikes such as 90% and 110% of spot, and becomes more negative after a shock. Market quotes often use matched deltas instead. Downside risk also shows up as a lower left-tail quantile of the return distribution. Here the tail-risk premium is proxied by the risk-neutral minus physical probability wedge, the risk-neutral probability of a large fall minus its physical probability. The wedge indicates priced tail risk and is not a price by itself. Similarly, Variance risk premium = implied variance − expected realized variance. For illustration, expected realized variance is taken as the square of expected realized volatility, a simplification and not a general identity. Do not subtract volatilities when the premium is defined in variance terms. The exam point is that rising skew signals demand for protection, not only higher average volatility.

Formulas

Simplified symmetric-strike risk reversal

RR = σcall − σput

σcall: implied volatility of the out-of-the-money call, such as the 110% strike
σput: implied volatility of the out-of-the-money put, such as the 90% strike

Becomes more negative after a shock as puts become dearer. Market quotes often use matched deltas instead of strikes.

Variance risk premium

VRP = σimplied² − σexpected²

σimplied: the option's implied volatility
σexpected: expected realised volatility

Defined in variance terms, so square before subtracting. Treating expected realised variance as the square of expected volatility is a simplification.

Left-tail 5% quantile of a normal return

q = μ − 1.645 × σ

μ: the mean return
σ: the volatility of the return

Higher volatility lowers the quantile even with the mean unchanged, which is how downside risk rises after a shock.

A practice question

From the question bank, with the reason each wrong option is wrong.

After a major geopolitical shock, which change is expected in the equity option market?

Implied volatility rises, with out-of-the-money puts becoming more expensive relative to at-the-money options. - correct
Implied volatility falls because investors expect calm after the news.Why not: Reverses the effect; uncertainty raises implied volatility.
Out-of-the-money calls become more expensive than out-of-the-money puts.Why not: Reverses the skew; demand for downside protection raises put prices.
Option prices stay unchanged because they follow historical volatility.Why not: Option prices reflect forward-looking implied volatility, which responds immediately.

Geopolitical shocks raise uncertainty and demand for downside protection, lifting implied volatility and steepening the skew toward out-of-the-money puts.

A harder one

The risk-neutral probability of a 20% fall in an equity index over a quarter is 8%, while the physical probability is 5%. The tail-risk premium, measured as the risk-neutral minus the physical probability, is closest to:

13.0 percentage pointsWhy not: Adds the two probabilities instead of subtracting them.
3.0 percentage points - correct
1.6 percentage pointsWhy not: Takes the ratio 8 / 5 and mislabels it as a difference in points.
60.0 percentage pointsWhy not: Uses the percentage excess (3 / 5) rather than the difference in probabilities.

Tail-risk premium (the risk-neutral minus physical probability wedge) = 8% - 5% = 3.0 percentage points. The ratio is 1.6, but the question asks for the difference.

The hardest tier

An index option has implied volatility of 30%. The analyst expects realized volatility of 24%. Using variance terms, the variance risk premium (implied variance minus expected realized variance) is closest to:

0.0600Why not: Subtracts the volatilities (0.30 - 0.24) instead of the variances.
0.0324 - correct
0.0036Why not: Squares the volatility difference, which is not the difference of the variances.
0.1476Why not: Adds the two variances instead of subtracting them.

Treating expected realized variance as the square of expected realized volatility (a simplifying assumption): implied variance = 0.30² = 0.0900 and expected realized variance = 0.24² = 0.0576. Variance risk premium = 0.0900 - 0.0576 = 0.0324. It measures the compensation embedded in option prices.

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