What is the instantaneous P&L of a variance swap.
Is it $(\sigma^{2}_{t}-\sigma^{2}_{implied})dt$?
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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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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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