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!

• I used to have an excel sheet that could fairly accurately price these at inception, provided the varswap par rates, option chains, and correlations (spot spot between underlyings, and spot/vol for each asset). Someone remind me later and I'll write something up this weekend. Fair warning, it was very handwavey, but it worked.
– will
Oct 12 '18 at 12:18
• Hey yes please. That would be great - i'm still looking for a solution. Why does it only work at inception? If you updated the option chains and correlations as the trade progressed, you would be able to get an updated price no? Or am I missing something. Thanks a lot in advance! Oct 13 '18 at 13:29
• it's more that it worked better when you were farther from the barriers. I'm on a boat at the moment so will update maybe tomorrow.
– will
Oct 13 '18 at 16:42

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?

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.]

• Yes! That's precisely what I mean. The corridor of both the short and the long leg of the product is contingent on just one index, say the SX5E. Basically, I would like some analytical or closed-form formula or a replication with simpler instruments so that I can see the risk of the structure as I trade it. Thank you for your response! Jul 12 '18 at 7:46
• See my edited answer above. Jul 13 '18 at 0:55

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.

1. 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

1. 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.

1. 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.

• I think DLIB only applies for regular corridor swaps and not corridor swap spreads. Jul 12 '18 at 9:29
• That would be inaccurate. Please try it. Regardless, my whole point is that its not the model that is difficult, its the calibration of the model. BBG has extremely good models but the price is completely off market
– hjw
Jul 12 '18 at 12:42
• @hjw please go have a look at the local vol surface generated by bbg in dlib and then come back and tell me they have excellent models again.
– will
Oct 12 '18 at 11:57
• @will I actually think thats my point though. BBG doesn't have access to IDB flows nor traded OTC values to calibrate their surfaces beyond liquid listed points nor to calibrate variance premiums above listed strip. If you plugged a good vol surface into their model you should get a good price. (Although I'm not sure why I'm defending them) Regardless my point is that getting good data to calibrate from the OP's perspective would be a lot trickier than writing the monte carlo/pde engine to price the contract.
– hjw
Oct 15 '18 at 7:09