# What is the instantaneous P&L of a Variance Swap?

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

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

• 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. – SRKX Sep 28 '12 at 15:20
• Perhaps he means something like the instantaneous credit risk of a variance swap? – John Sep 28 '12 at 21:17

## 2 Answers

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.

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