# Variance swap “fast” models

As far as I understand, Variance Swap (VS for short) function as follows :

• no payment when entering the contract
• at maturity the VS buyer pays a strike $K^2$ and is paid (by the VS seller) the realized variance over $[\textrm{today} = 0, \textrm{expiry}=T_N]$ (with prescribed constat. dates $T_i$'s), and "the strike is set according to prevailing market levels so that the swap initially has zero value", that is, $\mathbf{Q}^{T_N}$ being the forward $T$ measure, so that : $$K^2 = \mathbf{E}^{\mathbf{Q}^{T_N}}\left[\frac{252}{N} \sum_{i=0}^{N-1} \left( \ln\left(\frac{S_{T_{i+1}}}{T_i}\right) \right)^2\right]$$

(Yes,to be correct the $\frac{252}{T}$ should be replaced by $1/y$ where $y$ is the year fraction represented by the time period $[\textrm{today}, \textrm{expiry}]$ but I ignore calendar effects.)

Obviously if $S$ is the S&P (having quite a liquid VS market), for a given "standard" expiry $T_N$, the strike is quoted by market, that is determined by the law of the supply and demand, without any model. Right ?

Now, still for liquid names like the S&P but for non-standard strikes or expiries, I can see how we could infer strikes without a model, for instance by interpolation, I guess that it's enough.

But at some point, we'll need a model. My questions is : what are models used to prices VS (approximations authorized) and options on realized variance (square root payoff vol swaps for instance) in the context of algorithmic trading were we potentially have to price a lot of them as well as options on them, quite fast ?

• See this question. The payoff of a VS can be replicated by a portfolio of calls and puts. – Gordon Mar 9 '17 at 15:03
• @Gordon : sorry, forgot to put in the question that I was also interested in option on realized variance, so that their replication through discretized Carr-Madan won't be enough to me. – Olorin Mar 9 '17 at 15:28
• Can you please be more specific on the payoff forms of your options? I do have problems to find a good approximation for pricing a VIX futures. – Gordon Mar 9 '17 at 15:44
• Have you looked at the Bergomi model(s) of variance? – noob2 Mar 9 '17 at 15:52
• @noob2: not seriously. I may ask a question later on, and you can probably provide an answer for it. – Gordon Mar 9 '17 at 15:59