Valuation models

VIX, dividend and emissions futures and options

VIX and index dividend futures and options on them by Black-76, or VIX futures and options from the Heston variance at expiry, sampled exactly, with the implied Black-76 volatility, delta and vega.

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Methodology

Conventions

  • Futures are settled daily: worth the future less the trade price, times contract size and quantity.
  • Options are Black-76 on the future, discounted from expiry.
  • VIX from Heston: VIX² is the expected variance over the next 30 days given the variance at expiry, which is sampled exactly.

References

  • Black (1976)
  • Zhu and Zhang (2007), Variance term structure and VIX futures pricing

Formulas

Black-76
e^{−rT}(F N(d₁) − K N(d₂))
VIX
VIX² = θ + (v_T − θ)(1 − e^{−κτ})/(κτ), τ = 30/365