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FRM Part II · FRM Exam Part II · Volatility Smiles and Volatility Surfaces

A jump-diffusion model assumes the asset price follows geometric Brownian motion plus Poisson jumps with intensity 0.5 per year. Jump sizes are such that the expected jump contribution to the instantaneous return is zero after compensation. The jump intensity is doubled to 1.0 per year while jump size distribution and diffusive volatility stay fixed. What is the most likely effect on the three-month implied volatility smile?

Implied volatility levels rise because jump variance doubles, but with more frequent jumps of the same size, excess kurtosis per unit of variance falls, since returns look more normal. So smile curvature relative to level is somewhat reduced rather than amplified.

  1. AThe smile flattens because more frequent jumps make returns more normal at all horizons
  2. BTotal variance rises by a factor of exactly four
  3. CImplied volatility levels rise, and with unchanged jump size the excess kurtosis of short-horizon returns per unit variance falls, so the smile curvature relative to level is somewhat reducedCorrect
  4. DImplied volatility falls because the compensator offsets the jumps

Explanation

Doubling intensity doubles the jump variance contribution, which raises overall implied volatility. For a Poisson process the excess kurtosis of the jump component over a horizon is inversely proportional to intensity times horizon, so with more frequent jumps returns are closer to normal per unit variance. Hence the smile curvature is relatively lower, although levels rise. Variance does not quadruple since it is linear in intensity.

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