Stochastic Collocation for SABR Volatility and Calendar Arbitrage
Summary
The document discusses using stochastic collocation to improve on Hagan’s approximate SABR implied volatilities, which the author says can become less accurate at longer maturities. The described approach maps a reliable portion of a problematic distribution onto a simpler normal distribution, then uses that mapping to extend or replace unreliable regions and derive option prices. The author reports reproducing results for standard cases but encountering invalid Black implied volatilities for some long-maturity, low-strike cases.
The apparent cause was an implementation error when inverting the interpolation function: outside its well-defined region, the inverse could have multiple solutions, and the numerical solver sometimes selected the wrong one. The author expected that correction to fix the observed prices, but the post leaves its central theoretical question unanswered: whether the method’s resulting density is always free of calendar arbitrage as well as butterfly arbitrage. It offers no proof or general resolution, so the arbitrage properties should not be inferred from the reported standard-case results.
Key ideas
- Stochastic collocation can use a simpler reference distribution to extend regions where an approximate SABR distribution is unreliable.
- Hagan’s SABR volatility approximation may become less accurate at longer maturities.
- An interpolation inverse can be non-unique outside the region where it is well-defined, causing a solver to return invalid option prices.
- A density that avoids butterfly arbitrage is not thereby shown to avoid calendar arbitrage.
- The document reports no proof that the method always produces calendar-arbitrage-free prices.
Tags
Full text
# SABR, Stochastic collocation and calendar arbitrage # SABR, Stochastic collocation and calendar arbitrage Ok, this is a bit of a long read, so be warned.. I am currently learning about the so called "Stochastic collocation" technique which seem to have been quite popular during recent years for different applications in finance. To get some feeling for the method I have made an attempt to implement the method used in the article https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2529684. Basically, the problem is that the well known Hagan analytical formulas to produce Black volatilities for options in the SABR model becomes more inaccurate for larger maturities. The approximative "Hagan" volatility is then, in some cases, no longer close to the true theoretical black volatilities implied by the SABR model. The main idea, very simply speaking, of the article is to project the good part of the distribution of a "bad" variable (the one implied by Hagans formulas here) onto a "good" one with a simple distribution (the normal distribution in this case), which is then used in a clever way to extrapolate and replace the bad sections of the bad variable. The authors claim that they this way create an underlying density implying "arbitrage free implied volatilities". I think I understand everything in the article and I have tried to implement the method. For standard cases I seem to be able to replicate the results in the article. However, in some more extreme cases for larger maturities and smaller strikes, I cannot calculate any implied volatility. Unless I have some rather bad numerical issues in my implementation (which is certainly a possibility), the problem seems to be in the interpolated density using the stochastic collocation method. It gives me call option prices that are not attainable in the Black model. The reason for the problem is arbitrage, more precisely calendar arbitrage: I get call option price which at time 0 is greater than the option price at maturity for small strikes in some cases. The produced density in the article seems free of butterfly arbitrage though. However, the calendar arbitrage makes it impossible to create the arbitrage free volatility since it does not even exist in this case because of precisely arbitrage. Sure, the problem only appears for longer maturities and small strikes, but since the whole point of the article was to produce an arbitrage free volatility in the problematic situations, I am a bit confused. Maybe I am missing something obvious here, but I see no proof or motivation as to why the produced density of the underlying should be free of calendar arbitrage? (I have also implemented the collocation method so that the expected value of the underlying at maturity, given the collocated density, match the forward value) So my question is simply if the density calculated in the article actually always truly both lacks butterfly and calendar arbitrage and I therefore made some mistake/have some numerical issues, or if it actually sometimes has arbitrage? I can think of a few ideas on how to fix this, if it really isn't a simple mistake by me. Knowing the theoretical (unconditional) absorption probability at zero of the SABR distribution would help, but I don't think this is known, right? Or do there exist some good approximations of this? Or maybe there are some well known procedure when it comes to "removing" calendar arbitrage in such situations in some "natural" way? (This situation also sometimes occur during Monte Carlo simulations, I have noticed, when you have some numerical errors causing some slight calendar arbitrage in the sample, making it impossible to calculate the implied vol for some strikes) Edit: After some more investigations, I realized I made an error. A part of the option pricing procedure consist in inverting the main interpolation function used. This is well defined on the "good part" of the function, but outside that part the inverse is typically not uniquely defined. In some cases my solver jumped to the wrong value of the inverse causing the option prices to become invalid. I Will fix this and see if it solves the problem (which I have no doubt it will) I still don't know the answer to my original question though, as to why the method produces option prices free of calendar arbitrage. Probably there is some obvious reason why, but I don't see it at the moment.
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.