FRM Part II · FRM Exam Part II · Volatility Smiles and Volatility Surfaces
A bank's volatility surface for a stock index shows a pronounced skew at the 1-month maturity but a nearly flat profile at the 5-year maturity. Which explanation is most consistent with this term structure of the smile, assuming the underlying follows a jump-diffusion process with identical jump parameters at all horizons?
Jumps matter most over short horizons and their relative effect fades as maturity lengthens, since aggregated returns tend toward normality. So short-dated options display a pronounced skew while long-dated options show a flatter implied volatility profile.
- AJump effects are large relative to diffusion over short horizons but average out over long horizons as the distribution moves toward normalCorrect
- BJump effects strengthen with maturity so long-dated options show more skew
- CBlack-Scholes assumptions hold exactly for long maturities because rates are constant
- DLong-dated options have no vega, so their implied volatility cannot vary by strike
Explanation
With a fixed jump process, the effect of jumps on the return distribution relative to diffusion diminishes as horizon lengthens, because by the central limit theorem the sum of many small effects approaches normality. Hence short-dated options show pronounced smiles and long-dated ones are flatter. Long-dated options actually have high vega, so the vega distractor is false.
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