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Sorry, I should have though more before posting this question. By the way, the payoff of a call option on VIX index, priced at time $t$, with maturity at time $T$, is \begin{equation} (VIX_{T} - K)^+ \end{equation} and since the time $t$ strike of a VIX futures with same maturity $T$ is \begin{equation} F_{t,T} = E^{Q}[VIX_T \big| \mathcal{F}_t] ...


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Forward implied volatility smile is implied from forward start options. For example call options have payoff $$ g_{T+\theta} = \left( \frac{S_{T+\theta}}{S_T} -K\right)_+ $$ If you are in a stochastic volatility model this can be rewritten $$ g_{T+\theta} = \left( e^{ \int_T^{T+\theta} r - \frac{1}{2}\sigma_t^2 dt + \int_T^{T+\theta}\sigma_tdW^S_t } ...


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I'm not an equities guy so I don't know anything about volume-weighting, but if I was set this problem the approach that I would take would be to work out the implied volatility of each strike for that day, so that I have a graph of implied volatilty against strike, then interpolate on that graph to get the implied volatility for the ATM strike. Do that for ...



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