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Link the form taken by a smile or skew to the price distribution it implies for the underlying, and to the resulting option prices.

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

The idea

Option prices and implied volatilities are two descriptions of one risk-neutral distribution of the underlying at maturity. A flat smile corresponds to the lognormal distribution. Where the market gives more probability than lognormal to a region of prices, options struck there cost more and have a higher implied volatility. A heavier left tail and lighter right tail give a downward skew, so out-of-the-money puts are dearer and out-of-the-money calls cheaper than the model at the at-the-money volatility. Heavier tails on both sides, with a higher peak, give a U-shaped smile and dearer out-of-the-money calls and puts. A heavier right tail gives an upward-sloping skew. The rule is that a heavy tail raises the implied volatility of strikes in that tail. Pricing every strike with one at-the-money volatility therefore misprices the tails.

A practice question

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

If the implied volatility of options on an asset is the same at every strike, what distribution of the asset price at maturity does the market imply?

  1. A normal distribution of the price, with equal probability of rises and falls of the same dollar size.Why not: Confuses lognormal prices with normal prices; the model's log returns, not prices, are normal.
  2. A distribution with fatter tails than lognormal, because every option is priced at one volatility.Why not: A single volatility removes excess tail weight; fat tails produce a non-flat smile.
  3. A uniform distribution over the range of strikes traded, because strikes are priced equally.Why not: Equal volatilities do not imply equal probabilities; the density is still the lognormal bell shape.
  4. A lognormal distribution, which is the Black-Scholes-Merton assumption. - correct

Black-Scholes-Merton assumes lognormal prices and constant volatility. A flat smile means market prices are consistent with that lognormal distribution at all strikes.

A harder one

An index stands at 100 and the at-the-money 6-month implied volatility is 20%. The 90-strike 6-month put has an implied volatility of 24%. With a risk-free rate of 3% and no dividends, by how much does Black-Scholes-Merton at the at-the-money volatility understate the price of the 90-strike put, closest to?

  1. 0.78 - correct
  2. 2.24Why not: Reports the full put price at 24% instead of the understatement relative to the 20% model value.
  3. 1.46Why not: Reports the put's value at the at-the-money volatility, not the shortfall versus the market price.
  4. 1.11Why not: Scales the 4-point volatility gap by the at-the-money option's vega; the 90-strike put has a much smaller vega.

The put is worth 2.24 at 24% and 1.46 at 20%, so the model at the at-the-money volatility understates it by 0.78. The skew means out-of-the-money puts cost more than flat-volatility values.

The hardest tier

Spot is 100, the risk-free rate is zero, and options expire in 3 months. At a flat volatility of 10%, calls struck at 95, 100 and 105 are worth 5.39, 1.99 and 0.45. The market prices are 5.48, 1.99 and 0.55. A butterfly (long one 95 call, short two 100 calls, long one 105 call) pays off like a triangle around 100, so its price is proportional to the implied probability density near 100. How does the market-implied density at 100 compare with the lognormal one at 10% volatility?

  1. About 10% lower, because the butterfly costs 1.86 in the market against 2.05 in the model.Why not: Inverts the comparison: the market butterfly is the dearer one, so the density is higher, not lower.
  2. About 10% higher, because the butterfly costs 2.05 against 1.86. - correct
  3. About 5% higher, because the three-call spread costs 4.04 against 3.85.Why not: Omits the factor of two on the middle call, which is not a butterfly and gives the wrong ratio.
  4. No different, because the at-the-money call is worth 1.99 under both volatilities.Why not: Looks only at the middle leg; the density at 100 depends on the wings through the butterfly price.

Butterfly = c(95) - 2c(100) + c(105): flat 1.86, market 2.05. Ratio 1.10, so the market-implied density at 100 is about 10% higher: the smile implies a higher peak and fatter tails.

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