# 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.

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.

• It is ok if I get the same price .. I focus on risk management and price validation rather than trading this position. For me it looks easier to gather and maintain a local vol matrix than all those option prices. Although this more amounts to the same (extrapolation volas and forward prices to get the option prices, ..) Commented Jul 14, 2016 at 13:57
• Caveat: variance swaps can be statically replicated using plain vanilla (hence path-independent) options only under a pure diffusive setting and in the continuous time limit (realised log-returns variance replaced by quadratic variation of log-returns) Commented Jul 14, 2016 at 14:34
• Good point, Quantuple (and upvoted). In the case of local vol models everything is diffusive and continuous but that's not always true of reality or the stochastic models we desire to apply. Commented Jul 14, 2016 at 15:35

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.
• *ignoring that the market doesn't match theoretical varswap prices :p
– will
Commented Jul 14, 2016 at 14:34
• Depends on what you're really doing indeed. Model-free approach is usually not for pricing variance swaps but rather for building "volatility indices" à la VIX. Pricing variance swaps is more of an art than a science. Actually, we could ask ourselves if those contracts are really worth pricing since they are so liquid that they have a life of their own (supply/demand)... instead of pricing them we should rather view them as inputs/elementary building blocks of our in-house models, as we do with vanilla options. Commented Jul 14, 2016 at 14:48
• There was a nice article about this by the guys at ITO33: nefle.ito33.com/sites/default/files/articles/0609_ayache.pdf. Commented Jul 14, 2016 at 14:51
• You mean can you perfectly calibrate the classic Heston model (5 params) to the vanilla market? Clearly you will have a hard time doing that in general. Double Heston is more flexible, but has more parameters to be stored. At the other end of the spectrum, local volatility is infinitely flexible, but you need to store matrices (or a parametric form?). Commented Jul 14, 2016 at 15:55
• If you're pricing the varswap using the portfolio of strangles, then if you match the vanillas, you'll not match the varswap - that's my understanding. Do you know of a model that can be calibrated to simultaneously replicate varswaps and vanilla options?
– will
Commented Jul 15, 2016 at 15:45