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Set out other ways in which the volatility smile can be described.

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

The idea

The smile can be described with variables other than the strike K. Common choices are moneyness K/S0, forward moneyness K/F0, and the option's delta. Let r be the domestic rate and q the dividend yield or foreign rate. The forward price used in forward moneyness is then given by F0 = S0 × exp((r − q)T). The reason is stability. When spot moves, a smile plotted against fixed strikes shifts, whereas the same smile plotted against moneyness or delta changes much less over time and is easier to compare across assets and dates. FX desks quote three numbers. The first is the at-the-money volatility. The risk reversal is the call volatility minus the put volatility at the same delta, such as 25-delta, and it measures skew. The strangle is the average of the call and put volatilities minus the at-the-money volatility, and it measures curvature. Remember that a call and a put with the same strike share one implied volatility, so a 25-delta put and a 75-delta call at the same strike use the same number.

Formulas

Forward price used for forward moneyness

F0 = S0 × e^((r − q) × T)

S0
the spot price today
r
the continuously compounded risk-free rate, per year
q
the continuous income yield, such as an index's dividend yield or a currency's foreign rate
T
the time to delivery, in years

Gives the forward moneyness K ÷ F0. For FX, q is the foreign rate; using K ÷ S0 instead ignores the carry.

FX 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

Measures skew: negative when the put side is dearer.

FX strangle

Strangle = (σcall + σput) ÷ 2 − σATM

σcall
the implied volatility of the call at a given delta, such as 25-delta
σput
the implied volatility of the put at the same delta
σATM
the at-the-money implied volatility

Measures curvature: positive when the wings sit above the at-the-money level.

A practice question

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

Besides the strike price K, which variables are commonly used on the horizontal axis when describing a volatility smile?

  1. The ratio K/S0 (moneyness), the ratio K/F0 against the forward price, or the option's delta. - correct
  2. The option premium, the risk-free rate, or the underlying's historical volatility.Why not: None of these is a measure of how far in or out of the money an option is; they are not smile axes.
  3. The time to maturity, the option's theta, or the underlying's dividend yield.Why not: Maturity gives the term structure, and theta and dividends do not measure moneyness.
  4. The option's vega, the strike multiplied by the spot, or the spot price alone.Why not: Vega and the spot do not identify how far in or out of the money an option sits.

A smile may be plotted against K, K/S0, K/F0 or delta. The last three measure moneyness, which tends to give a more stable curve than strike alone.

A harder one

In FX markets a smile is often quoted by three numbers: the at-the-money volatility, the risk reversal (25-delta call volatility minus 25-delta put volatility) and the strangle (average of the 25-delta call and put volatilities minus the at-the-money volatility). The at-the-money volatility is 8.0%, the 25-delta call volatility is 9.0% and the 25-delta put volatility is 10.5%. What are the risk reversal and the strangle, in volatility points?

  1. Risk reversal +1.5; strangle +1.75Why not: Subtracts the call volatility from the put volatility; the convention is call minus put.
  2. Risk reversal -1.5; strangle +1.75 - correct
  3. Risk reversal -1.5; strangle +9.75Why not: Averages the wings but forgets to subtract the at-the-money volatility.
  4. Risk reversal -1.5; strangle +11.5Why not: Adds the two wings (19.5) and subtracts 8.0 without halving; the strangle uses the average of the wings.

Risk reversal = 9.0 - 10.5 = -1.5 (the put side is dearer: a skew). Strangle = (9.0 + 10.5)/2 - 8.0 = 1.75 (the wings are above the at-the-money level: curvature).

The hardest tier

An index trades at 100. The smile is stable in moneyness: implied volatility (in percent) depends only on K/S as follows: K/S 0.90: 25.0; 0.95: 22.5; 1.00: 20.0; 1.05: 18.5; 1.10: 17.5. A 95-strike option therefore has an implied volatility of 22.5%. The index rises to 104 and the smile keeps its moneyness shape. Using linear interpolation between table points, the new implied volatility of the 95-strike option is closest to:

  1. 24.3% - correct
  2. 22.5%Why not: Keeps the strike's volatility unchanged, which is a fixed-strike assumption, not the moneyness assumption.
  3. 17.6%Why not: Looks up S/K = 1.095 instead of K/S = 0.913, reading the table from the wrong end.
  4. 25.0%Why not: Rounds K/S = 0.913 to the 0.90 table point without interpolating between 0.90 and 0.95.

New K/S = 95/104 = 0.9135, between 0.90 (25.0) and 0.95 (22.5). Interpolating: 25.0 - (0.0135/0.05) x 2.5 = 24.33%. The option is further in the money relative to spot, so its volatility rises under a skew in moneyness.

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