Variance Swap Strikes and Fast Pricing Models
Summary
The document outlines how a variance swap works: the buyer exchanges a fixed variance strike for realized variance calculated from returns over specified observation dates. For liquid standard contracts, the market quotes the strike; for nonstandard expiries or strikes, the author considers interpolation before turning to models for faster valuation. It asks which models or approximations are practical for pricing many variance swaps and options on realized variance, including volatility swaps.
The document gives the fair-strike expectation under a forward measure as its pricing framework, with annualization based on a 252-day convention. It does not provide a pricing model, empirical evidence, or a comparison of methods; it is a question about those topics. Its setup also simplifies calendar effects and contains a potentially ambiguous return formula, so the expression should not be treated as a complete implementation specification.
Key ideas
- A variance swap exchanges realized variance for a fixed strike at maturity.
- Liquid standard contracts can have market-quoted strikes, while nonstandard contracts may require interpolation or a model.
- The fair strike is framed as the expected annualized realized variance under a forward measure.
- The document asks about fast models for variance swaps and options on realized variance but does not supply an answer.
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Full text
# Variance swap "fast" models
# 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 ?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.