Pricing Variance Swaps with Local Volatility Monte Carlo

# Pricing variance swaps using Monte Carlo


# Pricing variance swaps using Monte Carlo












For pricing variance swaps there is the well-known formula as sum of OTM options weighted by the inverse of the squared strike (see e.g. here).

Would it also be valid to derive the local-volatility surface from option prices and then do a Monte-Carlo simulation of future paths and calculate the variance from these prices?

In case you have a local-vol surface ready this would be a nice way to calculate the var-swap-rate.

## Answer by Brian B (score 2, accepted)

https://quant.stackexchange.com/a/28077

It is valid to do that, but if your local volatility surface is calibrated to the same OTM options, then your price will converge to the same answer.

A local volatility surface is mainly a way of treating path-dependent options consistently with the option volatility surface. Variance swaps are path dependent on the face of it, but as you note the math works out such that they have a representation as a portfolio of path-independent options.

## Answer by Quantuple (score 1)

https://quant.stackexchange.com/a/28078

The only problem I see with this approach, which remains completely valid from a theoretical perspective, is the embedded (and probably not accounted for) calibration risk: what if your LV surface does not allow you to correctly reproduce the observed vanilla option prices in the first place? In that case, you'll have lost information in the process and always produce biased variance swap prices.

Some remarks:

- The "model-free" method you refer to in your link is an approximation of its own: if you want to incorporate real-life details such as discrete variance sampling or cash dividends, you're better off using MC simulations based on your in-house (jump-)diffusion model than using the Carr-Madan based formula, which only holds in the continuous time limit, for pure diffusive processes.

- A corollary of the Carr-Madan approach is that, if you take 2 purely diffusive models perfectly calibrated to the vanilla options market, be it a local volatility model and a stochastic volatility model such as Heston, then both of these models should give exactly the same variance swap par rates. So you could equivalently store Heston parameters. Since there are only 5 of them, there is even less data to store/maintain than with a local volatility model.

- Finally, while pricing variance swaps using Monte Carlo is the most general approach, for some models, e.g. Heston, closed-form formulas exist. This can be helpful when computational burden becomes a concern.