Valuation models

Variance and volatility swaps

Fair variance replicated from out-of-the-money options across the smile, seasoned swaps with realized volatility, the volatility swap convexity adjustment, vega and the replicating strip.

Inputs

Results

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Methodology

Conventions

  • Fair variance is replicated from out-of-the-money options across the smile, weighted by 1/K².
  • Realized variance to date is blended in by elapsed time; the strike is a volatility, the notional vega per vol point.
  • Volatility swaps take a convexity adjustment from the volatility of volatility, with realized variance lognormal.

References

  • Demeterfi, Derman, Kamal and Zou (1999)
  • Carr and Lee (2009), Volatility Derivatives

Formulas

Fair variance
K(var) = (2/T) ∫ OTM(K) / K² dK
Variance swap
V = N(vega)·100 / (2K) · (E[σ²] − K²) · D(T)
Volatility swap
V = N(vega)·100 · (E[σ] − K) · D(T)