Variance Swap Mark-to-Market and the Role of Expected Variance

Summary

The note considers how to describe the instantaneous mark-to-market change of a variance swap and asks whether it is simply realized variance minus implied variance, multiplied by elapsed time. One response says a variance swap is defined through a set of fixing times, so volatility between fixings does not directly determine its payoff. A model is therefore needed to describe interim value changes; the proposed expression is presented only as a model-free approximation and is said to hold up to a constant.

A second response frames the swap’s value in terms of the time integral of expected future variance over the remaining contract interval. On that view, a change in value over a short interval depends on the difference between current and previous expectations of that integrated variance. The discussion highlights that a mark-to-market expression must account for the contract’s fixing schedule and evolving expectations, rather than equating instantaneous P&L unconditionally with a spot variance gap. It gives no derivation of the constant, detailed model assumptions, or treatment of realized fixings and other implementation conventions.

Key ideas

• Variance swaps accrue variance through specified fixing times, which shape interim valuation.
• A simple spot variance minus implied variance expression is described as an approximation, up to a constant.
• One valuation framing integrates expected future variance over the remaining contract period.
• Mark-to-market changes reflect how expectations of that integrated variance change through time.

Tags

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


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












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

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

## Answer by Brian B (score 1)

https://quant.stackexchange.com/a/4140

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.

## Answer by Andrew (score 1)

https://quant.stackexchange.com/a/4866

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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