Calculating Options and Variance Swap P&L from Volatility Surfaces

Summary

The document asks whether an implied volatility surface can be used to calculate options strategy P&L and variance swap P&L when individual option prices are unavailable. The response says a surface can support these calculations if its construction conventions are known and matched, including calendar time, interest rate compounding, and dividend treatment.

For variance swaps, the response states that P&L can be derived from the surface using an analytical pricing formula based on vanilla European option prices. The surface must therefore be interpreted consistently with the assumptions used to produce it; implied volatility values are linked to option prices through the relevant pricing setup. The exchange gives a high-level direction, but no worked calculation, formula, or treatment of practical issues such as interpolation and discrete strikes.

Key ideas

• Options strategy P&L can be calculated from an implied volatility surface when its pricing conventions are known.
• Calendar time, rate compounding, and dividend conventions must match those used to construct the surface.
• Variance swap P&L can be derived from the surface through a replication method based on European option prices.
• The response gives no worked example or detail on interpolation and other implementation choices.

Tags

# Calculating PnL of Options strategies with Volatility Surface


# Calculating PnL of Options strategies with Volatility Surface












New to Vol trading - wondering if there are any good references on calculating PnL from options strategies if I only have a volatility surface (delta or moneyness) and no individual options prices.

Also wondering is it possible to compute variance swap PnLs directly from implied volatility surfaces? Or do I need a pricer to convert the imputed implied volatilities to prices as inputs to the Variance swap replication process?

## Answer by Rodrigo (score 2)

https://quant.stackexchange.com/a/74195

- The function that converts option prices and implied vols is bijective. So yes, you can compute the PnL given you have the volatility surface and you know the parameters that where used in its construction: calendar time must use the same convention, interest rate compounding with same convention, divided model with same convention and so on.

- It is also possible to compute variance swap PnLs from the surface alone, given you respect the constraints listed above. Variance swaps have a closed analytical formula and can be priced using the vanilla european option prices and it should not be hard for you to find the method online (Would suggest GS's "More than you wanted to know about voltility swaps" for starters.)

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.