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Characterise volatility term structures and volatility surfaces, and show how option pricing can make use of them.

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

The idea

The volatility term structure is the implied volatility of at-the-money options plotted against maturity. Volatility mean-reverts. When short-term volatility is low relative to the long-run level the curve slopes up, and when it is unusually high, for example after turmoil, it slopes down. The volatility surface combines the smile and the term structure, giving an implied volatility for each strike (or moneyness, or delta) and each maturity. The smile is typically more pronounced at short maturities and flatter at long ones. To price an option whose strike and maturity are not quoted, interpolate the surface, usually linearly in strike and maturity, and put the interpolated volatility into Black-Scholes-Merton. Interpolate volatilities, not prices. Variances, not volatilities, add over time, so with σ1 and σ2 the at-the-money volatilities for maturities T1 and T2, the forward volatility between them follows from the forward variance: Variance = (σ2² × T2 − σ1² × T1) ÷ (T2 − T1).

Formulas

Forward variance between two maturities

FwdVar = (σ2² × T2 − σ1² × T1) ÷ (T2 − T1)

σ2
the implied volatility to the later maturity
T2
the later maturity in years
σ1
the implied volatility to the earlier maturity
T1
the earlier maturity in years

Variances, not volatilities, add over time. Take the square root for the forward volatility.

A practice question

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

What is a volatility surface?

  1. The implied volatility of at-the-money options only, tabulated against time to maturity.Why not: That is the volatility term structure, a single slice of the surface.
  2. The implied volatility of options on an asset, tabulated against both strike and maturity. - correct
  3. The implied volatility of options at one maturity, tabulated against strike.Why not: That is a single smile, one slice of the surface at a fixed maturity.
  4. The historical volatility of the underlying, tabulated against the length of the sampling window.Why not: Describes realised volatility over different windows; the surface is made of implied volatilities.

The surface combines the smile (variation with strike) and the term structure (variation with maturity), giving an implied volatility for each strike and maturity pair.

A harder one

A vol surface (implied volatility in percent) is: 6-month maturity, strike 90: 24, strike 100: 20; 12-month maturity, strike 90: 22, strike 100: 19. Using linear interpolation in both strike and maturity, the implied volatility for a 9-month option with strike 94 is closest to:

  1. 21.6% - correct
  2. 21.25%Why not: Takes the plain average of the four nodes; strike 94 is closer to 90 than to 100, so the weights are not equal.
  3. 22.4%Why not: Interpolates in strike only on the 6-month slice, ignoring the 12-month points that also drive a 9-month option.
  4. 20.8%Why not: Interpolates in strike only on the 12-month slice; the 9-month maturity needs both the 6-month and 12-month points.

At 9 months: strike 90 = 23.0, strike 100 = 19.5. At strike 94 (40% of the way from 90 to 100): 23.0 - 0.4 x 3.5 = 21.6%. Price the option with Black-Scholes-Merton at that volatility.

The hardest tier

The 1-year at-the-money implied volatility is 25% and the 2-year at-the-money implied volatility is 20%, after a period of market stress. What forward volatility between year 1 and year 2 is implied by this term structure, closest to?

  1. 15.0%Why not: Extrapolates the volatilities linearly (2 x 20 - 25), but variances, not volatilities, are additive over time.
  2. 22.5%Why not: Averages the two volatilities, which does not isolate the second year.
  3. 20.0%Why not: Takes the 2-year volatility as the forward volatility, ignoring that the 2-year figure averages both years.
  4. 13.2% - correct

Total variance: 0.20² x 2 = 0.0800 for two years; 0.25² x 1 = 0.0625 for the first. Forward variance = 0.0175 for one year, so the forward volatility is 13.2%.

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