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Show how one jump in an asset's price affects a volatility smile.

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

The idea

Suppose the market expects a single large jump, such as a court ruling or takeover announcement, which will move a stock from 50 to either 40 or 60. The price distribution at expiry is then bimodal, with less probability in the middle and thinner tails than a lognormal. Strikes beyond the two outcomes are worth very little, so their implied volatility is low, while options near the money carry the jump risk. The implied volatility smile is therefore hump-shaped, highest at the money. That is the reverse of the U-shaped smile of fat tails. A jump adds its variance to the total variance of an option's life, so variances add, not volatilities. The effect is greatest for short maturities. Options that expire before the event do not carry the jump, so the term structure steps up after it.

Formulas

Implied volatility with a single jump

Implied = √((σd² × T + J²) ÷ T)

σd
the base diffusion volatility per year
T
the option's life in years
J
the size of the jump, as a proportion

Add variances, not volatilities. Only options expiring after the event include J², and its effect shrinks as T grows.

A practice question

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

A stock trades at 50 and the market expects an announcement within weeks that will send the price to roughly 40 or 60. What does this anticipated single jump do to the risk-neutral distribution of the price at expiry?

  1. It makes the distribution lognormal with a much higher volatility, because the jump simply adds risk.Why not: A single known jump creates two clusters; it does not merely rescale a lognormal.
  2. It makes the distribution bimodal, with less probability in the middle and thinner tails than a lognormal. - correct
  3. It gives a fatter left tail and a thinner right tail, as in equity skew, because large falls are more feared.Why not: A skew needs a negative price-volatility link; a symmetric two-outcome jump gives symmetric bimodality.
  4. It leaves the distribution unchanged, because the mean price is unaffected by the announcement.Why not: The mean is unchanged but the shape is not; the variance is concentrated in the two outcomes.

With two likely outcomes, the density has two peaks (40 and 60) and almost nothing beyond them, so it is bimodal with thin tails, the opposite of the fat-tailed currency distribution.

A harder one

A company will announce the result of a major court case in two months, which can move its stock price sharply either way. The stock's 1-month and 3-month at-the-money options have the same strike. Which statement about their implied volatilities is correct?

  1. The 1-month option should have the higher implied volatility, since time value is greatest for the shortest maturity.Why not: Time value in dollars is lower for short maturities; the 1-month option expires before the event and carries no jump.
  2. They should be equal, since a one-off jump is a diversifiable event and carries no risk premium.Why not: Option prices reflect the scenario distribution, not a risk premium; the jump raises the total variance of the longer option.
  3. Neither includes the jump, because implied volatility is calibrated to historical data.Why not: Implied volatility is backed out of current option prices, so it reflects anticipated events such as the ruling.
  4. The 3-month option should have the higher implied volatility, since only its life spans the announcement. - correct

An option that expires before the announcement does not see the jump, while one that expires after it does. The term structure therefore shows a step up in implied volatility beyond the event date.

The hardest tier

A stock's base annual volatility is 20%. A court ruling due before the 3-month expiry will move the stock by 6% up or down with equal probability. Treat the total variance of the 3-month return as the diffusion variance plus the jump variance (6% squared). Which at-the-money 3-month implied volatility does that give, closest to?

  1. 26.0%Why not: Adds the jump size to the volatility (20% + 6%), but it is variances that add.
  2. 20.9%Why not: Adds the jump variance to a full year of diffusion variance, which is the answer for a 12-month option, not a 3-month one.
  3. 23.3% - correct
  4. 20.0%Why not: Ignores the jump entirely, treating it as already part of the 20% diffusion volatility.

Total variance = 0.20² x 0.25 + 0.06² = 0.0100 + 0.0036 = 0.0136. Annualised: 0.0136 / 0.25 = 0.0544, so volatility = 23.3%. The same jump would add only about 0.9 points to a 12-month option.

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