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Describe how the smile looks for options on stocks and for options on currencies, and put forward plausible explanations of why.

FRM® Part II · Market Risk Measurement and Management · Volatility Smiles and Volatility Surfaces · FRM2.MR.17.e

The idea

Equity and equity index options show a downward-sloping skew. Implied volatility falls as the strike rises, so low-strike out-of-the-money puts carry the highest implied volatility. That means a heavier left tail and a thinner right tail than lognormal. Currency options show a roughly symmetric U-shaped smile. Two explanations are given for the equity skew. The leverage effect says that when a share price falls the firm's debt-to-equity ratio rises, the equity becomes riskier and its volatility increases, so volatility moves against the price. Crashophobia says that since the October 1987 crash investors pay up for out-of-the-money puts to guard against another crash, and the skew was not pronounced before that date. Currencies have no natural crash direction, so no skew arises. Match each asset class to its pattern and know both explanations.

A practice question

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

Which pairing of underlying and typical volatility pattern is correct?

  1. Equity options show a symmetric smile; currency options show a downward-sloping skew.Why not: Swaps the two patterns; the skew is the signature of equities and the symmetric smile of currencies.
  2. Equity options show a downward-sloping skew; currency options show a roughly symmetric smile. - correct
  3. Both equity and currency options show a downward-sloping skew with high-strike options the dearest.Why not: Reverses the slope: in equities it is low-strike options that carry the highest volatility.
  4. Equity options show a flat pattern after 1987, while currency options show a hump.Why not: Neither is right: equities have a pronounced skew and currencies a U-shape, not a hump.

For equities and equity indices the implied volatility declines as strike rises (skew or smirk). For currencies the smile is U-shaped and roughly symmetric.

A harder one

Which statement is the leverage explanation of the equity volatility skew?

  1. After a market crash, investors pay a premium for out-of-the-money puts in case another crash happens soon.Why not: That is crashophobia, the behavioural explanation, not the structural leverage argument.
  2. Volatility is a random process that is independent of the share price, so the tails are fat on both sides of the distribution.Why not: Independent stochastic volatility gives a symmetric smile; the skew needs a negative price-volatility link.
  3. When the share price rises, more dividends are paid out, which reduces equity volatility and also lowers call values.Why not: Misattributes the effect to dividends; leverage concerns debt and equity, not dividend payments.
  4. When a share price falls, leverage rises, the equity is riskier and its volatility rises; when it rises, volatility falls. - correct

The negative correlation between price and volatility implied by leverage produces a heavy left tail and a light right tail, hence the downward-sloping skew.

The hardest tier

Why do the leverage and crashophobia arguments for the equity skew not produce a similar one-sided skew in a major currency pair such as EUR/USD?

  1. Currencies have no jumps, so the distribution of exchange rates is exactly lognormal and the smile is flat.Why not: Exchange rates do jump, and these jumps and stochastic volatility create the symmetric smile.
  2. A fall in EUR/USD is a rise in USD/EUR, so neither direction is the crash direction and the smile is symmetric. - correct
  3. Central banks stabilise exchange rates, so the left tail of returns is cut off and the right tail is fat.Why not: Even where interventions occur, their effect is fat tails in both directions, not a cut-off tail on one side.
  4. The interest rate differential tilts the forward rate, which cancels any asymmetry in the tails.Why not: The differential moves the centre of the distribution through the forward; it does not offset tail asymmetry.

Equities have a natural downside (price falls raise leverage and fear). An exchange rate is a ratio of two currencies, so the 'crash' of one is the rally of the other, and a symmetric smile follows from the fat tails on both sides.

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