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What is the instantaneous P&L of a variance swap.

Is it $(\sigma^{2}_{t}-\sigma^{2}_{implied})dt$?

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  • $\begingroup$ Your question would benefit from a link or a short reminder of how a variance swap works, possibly including formulas in the post and indication how you got to your solution. $\endgroup$ – SRKX Sep 28 '12 at 15:20
  • $\begingroup$ Perhaps he means something like the instantaneous credit risk of a variance swap? $\endgroup$ – John Sep 28 '12 at 21:17
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A variance swap has a set of fixing times, and the volatility between those times has no specified effect. Therefore you end up wanting to apply a model. For a model-free approximation, though, your formula works up to a constant.

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definition of a variance swap is

$ \int^{T+\Delta}_T \mathbb{E}_t[v_s] ds $

where $v_s$ is the variance and $\mathbb{E}_t[v_s]$ is the expectation of the variance of time s at time t.

therefore, pnl is: $ (\int^{T+\Delta}_T \mathbb{E}_t[v_s] ds - \int^{T+\Delta}_{T} \mathbb{E}_{t-\delta}[v_s] ds)*d\delta $

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