Valuation of Corridor Variance Swaps

Question

Given that the payout of the Corridor Variance Swap (CVS) is $V \left(\frac{\sum_{n=0}^{N}I}{T_2 - T_0} (\sigma^2 - K^2) \right)$, where $\sigma^2$ is the realized variance within the pre-specified corridor (e.g. 90-110 of the spot) and $I$ is the number of days in the range.

I was wondering what would be the price of this CVS at time $T_1$, such that $T_0<T_1<T_2$? In my understanding, the $\sigma^2$ will be split into two pieces:

1. Realized variance for the time period of $T_0<T_1$, which is calculated from the actual prices of the underlying;
2. Implied variance for the time period of $T_1<T_2$.

The second part is not clear to me -- since we will be projecting the variance for the remainder of the time, do we look at various strikes between 90 and 110 with the maturity equal to $T_2-T_1$ and then derive the implied variance? Are there any other approaches?

Per FINCAD:

Vanilla variance and volatility swaps can be replicated with a portfolio of European options. The valuation of the next-generation variance derivatives described above cannot be done without a model for the volatility process, because conditional and corridor variance swaps depend on the price path of the underlying asset and because of the optionality in the option on variance. In the Heston model both the price S with drift and the variance V follow a stochastic process.

Answer 1

Generally, FINCAD is correct. I do have some reservations though.

Yes, variance swaps have a theoretical replication. A vanilla option trader following a delta-hedging strategy is essentially replicating the payoff of a weighted variance swap where the daily squared returns are weighted by the option’s dollar gamma. Taking this argument one step further, a fair variance swap can be shown to equal the integral of weighted prices of out-of-the-money options over all strikes. These weights are being inversely proportional to squared strikes, an application of the BlackScholes closed-form formula for gamma.

One obvious problem here is that options markets are composed of a discrete set of option prices for a given maturity. Therefore, it is common to first compute a vol surface, usually using Black Scholes again (ignoring complications involved in creating vol surfaces, like de-Americanizing option prices, finding implied forwards and dividends and the like if we think of index or equity VS, FX, for example, is generally quoted in vol which makes surface construction easier). Practically, you may also want to limit the integration region (strike range) to avoid issues with the weights (especially very small strikes are a concern due to the weighting).

Due to practical difficulties in replicating the actual log payout across strikes, the market for equity index varswaps usually trades at a basis to the replicating portfolio. Hence, why I do not fully agree with FINCAD here.

The range of strikes needed to replicate most corridors is well inside the liquidity spectrum. A conditional variance swap is hedged in much the same way as a normal variance swap. Because variance exposure is only required at certain underlying levels, however, the portfolio of options is truncated at the trigger level. This would imply they tend to trade at the replication price. Closed Form Pricing Formulas for Discretely Sampled Generalized Variance Swaps shows how a conditional variance swap can be decomposed into a corridor variance swap with the same upper barrier plus a range accrual note. The advantage of such a decomposition is that it removes any dependency on a model. Hence it is possible, and because of the aforementioned basis, also likely to be a more reliable replication compared to vanilla (index) variance swaps.

However, as often, supply/demand is the primary driver of where corridor varswaps trade relative to replication. There is usually a (large) demand from buy-side clients which means corridor varswaps also tend to trade at a premium.

There are two documents from JP Morgan Variance Swaps and Just what you need to know about Variance Swaps with the latter being more concise. This is mainly for vanilla varswaps but gets me to the final point. The way delta and gamma are defined in the first paper is a simplification. It will only work intraday. Ideally, the Greeks are directly derived from the replicating portfolio. However, such a full decomposition is not something (many) vendors offer and mainly tier 1 banks have implemented.

The same applies to FINCAD, Bloomberg and co. They do not (to my knowledge) offer a solution that allows a complete decomposition of such complex trades. Bloomberg will price them in DLIB with BLAN (some OCAML based scripting language) which is agnostic to the actual trade type. It will simply use MC runs based on your model of choice. That will neither match a vanilla replication (what OVME should do, in terms of pricing, not Greeks) nor provide a fully satisfying solution for corridor var swaps. Technically, you should be able to book the decomposed trade yourself by packaging it. I doubt that vendors will be able to help with that though. At least not standard help desks. Either way, it should not be "miles" off the actual quotes and even with full decomposition you would still face the issue with the liquidity premium.