Payoffs, Pricing Challenges, and Risk Management for Corridor Variance Spreads
Summary
The discussion distinguishes an ordinary corridor variance spread, where each index’s variance accrues inside its own corridor, from a cross-index structure where both variance terms accrue according to the same index’s corridor condition. It writes these payoffs as time integrals and notes that the shared corridor indicator changes the pricing problem. The proposed intuition for the lower price is that the relationship between one index and the other index’s volatility can make the short-variance leg more valuable than a standard corridor leg.
The answers emphasize that a clean analytical replication for the cross-index term is not established here. They cite sparse calibration data, variance-market premia, long tenors, and limited transparency in listed prices as practical obstacles; they also caution that vendor marks can be inaccurate. For exposure management, one answer suggests tracking in-range gamma and estimating remaining vega using the probability of being in the corridor, potentially informed by listed volatility. For quote fairness, it suggests comparing two-way quotes from several counterparties. These are practitioner comments, not a validated pricing model or measured result.
Key ideas
- In the discussed structure, both variance legs are gated by the same index’s corridor condition.
- The shared corridor condition makes the cross-index variance term harder to replicate and price analytically.
- The proposed pricing intuition depends on the correlation between the indices and the relationship between one index and the other’s volatility.
- Calibration can be difficult when variance-market data, long-tenor prices, or transparent listed quotes are limited.
- Risk management can focus on in-corridor gamma and vega adjusted by the estimated probability of remaining in range.
- Comparing dealer quotes is offered as a practical check on quoted prices.
Tags
Full text
# Pricing Corridor Variance Spreads # Pricing Corridor Variance Spreads Recently in the equity derivatives market there have been some trades on what are known as "Corridor Variance Spreads." The large equity derivative dealers and investment banks have been promoting it quite heavily. Basically, the structure delivers the buyer short variance on one underlying (for example, SPX) and long variance on another underlying (for example, SX5E), however, the variance only accrues if the SX5E stays within a range of spot values. This structure is quite a bit cheaper than going long one SX5E corridor variance swap and short one SPX corridor variance swap individually, with the same corridor range. The correlation somehow cheapens the structure. I would like to trade it, due to the attractive levels that it offers, but my fund hires somewhat fewer quantitative analysts than say Goldman Sachs and we can't come up with a way to properly price the thing. Does anyone have an analytical pricing formula, an approximation or a way the structure can be replicated with other, less complex, instruments? Thank you very much in advance! ## Answer by user34971 (score 2) https://quant.stackexchange.com/a/40733 I wanted to comment on the question in order to ask more information, but apparently don't have enough points yet to comment, so will just "answer" your question. A normal corridor variance swap spread would have payoff $$ \int_0^T (\sigma_t^X)^2 \theta(X_t - K_X) dt - \int_0^T (\sigma_t^Y)^2 \theta(Y_t - K_Y) dt $$ where $X_t$ is say the SX5E and $Y_t$ is the SPX, $K_X$ the lower corridor for SX5E and $K_Y$ the lower corridor for the SPX, and $\sigma_t^X$ resp. $\sigma_t^Y$ the vols of the two indices, $\theta$ is the Heaviside function. Note that I only have a lower corridor but could just have easily included an upper as well. Do you agree with the above payoff for a normal corr varswap spread? The corr varswap spread you're trying to value on the other hand, has the following payoff: $$ \int_0^T (\sigma_t^X)^2 \theta(X_t - K_X) dt - \int_0^T (\sigma_t^Y)^2 \theta(X_t - K_X) dt $$ Note that the term $\theta(X_t - K_X)$ occurs in both integrals now. Is my interpretation correct, before I continue trying to answer your question? Following your comment below: Well, I think this is a non-trivial problem in the sense that even though a normal corridor variance swap has an analytical replication expression, the 2nd integral above I'm not so sure of. In any case, I think you can see though why the price is cheaper than a normal corr varswap spread: If the SPX is highly negatively correlated to its volatility, it's reasonable to assume that the SX5E is less negatively correlated to the SPX vol. This means that the second integral above (the product of SPX vol with SX5E corridor) will have more value than a pure SPX corridor varswap. Hence the spread (both integrals above) will have less value. I will continue thinking about an approximate analytical expression for the replication of the second integral above, but at this moment I don't see a straightforward answer. [Regarding BBG valuation of corridor or ordinary varswaps: should be OK for SPX, but could be off for SX5E and Asian indices due among others structured products hedging, one of the factors in the strip/varstrike basis.] ## Answer by hjw (score 1) https://quant.stackexchange.com/a/40722 I tried (albeit not very hard) this in-house with very little success for a variety of reasons. The most important of which is the lack of data to calibrate your models. Even if you fit listed well, there is a var premium on top of listed strip particularly in asia so calibration has to be done to the variance market itself. This is compounded by the fact that corridors are typically very long tenor where listed prices themselves are not transparent to buyside. With this in mind I am offering you the following solutions depending on why you feel you need to price this. - My risk department says I can't trade if I can't price it. Bloomberg --> DLIB has a corridor pricer that is just as inaccurate as the one you will build in-house. Bloomberg's quant team is infinitely more capable than me. The fact that the price is off market is very likely not due to modelling inaccuracies but due to calibration reasons I mentioned above - I want to manage my risk All you actually need to know is your gamma when you are in range (This is a constant) and the vega of your position (Remaining vega * probability in range). The only slightly tricky part is probability in range which you can estimate from listed vols. Even if you get this off by 10% its not really material. - I want to make sure the prices I am quoted are fair. Get 2-way quotes from 5 counter-parties and trade with the cheapest.
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.