It is well-known that the original Hagan et al (2002) approximation formula for implied volatility in the SABR model is imprecise for extreme parameter values, typically high volatility and long maturities.Hagan et al (2002) - Managing Smile RiskA number of papers have been written on improvements of the approximation. Among these are:Oblój (2008) - Fine-tune your smile: Correction to Hagan et al Wilmott Magazine May/June 2008Henry-Labordère (2005) - A General Asymptotic Implied VolatilityBerestycki Busca and Florent (2004) - Computing the Implied Volatility in Stochastic Volatility ModelsPaulot (2009) - Asymptotic Implied Volatility at the Second Order with Application to the SABR ModelHagan Lesniewski and Woodward (2005) - Probability Distribution in the SABR Model of Stochastic VolatilityR Rebonato (with JP Barjaktarevic) (2010): Approximate Solutions for the SABR Model: Improving on the Hagan Expansion (Talk at ICBI Global Derivatives Trading and Risk Management Conference, May 2010)I would like to know if there is a consensus about which approximation is the most precise or best to use. Is there an established best practice or hasn't the dust settled yet?Best regards,Niels

Niels, good question.I haven't seen the Rebonato/Barjaktarevic. Based on the others, AFAIK, Paulot offers the most accurate expansion: it's exact to O(T^2) for all strikes. Also, the model can be solved numerically*. Paulot's only fails when nu^2 T gets relatively large, notbecause of any strike. (nu is the vol. of vol.) So, my advice is:1. Feel reasonably comfortable to use Paulot's expansion for nu^2 T < 1 (any strike), but otherwise use numerical solns. 2. When you use *any* expansion, always check against full model numerics (or analytic soln)If you want me to comment on whether or not the Rebonato talk changes this advice, pleasesend me a copy (email in profile). --------------------------------------------------------------------------------------------------------------------------------------------------------* The model can also be solved analytically when (i) beta = rho = 0, or (ii) beta =1 (any rho); or (iii) rho = 0 (any beta)

Last edited by Alan on June 29th, 2010, 10:00 pm, edited 1 time in total.

Thank you for your insightful response, Alan.Our initial testing also shows that Paulot seems quite interesting. We didn't yet implement a Monte Carlo simulation or finite difference scheme to test it up again, though.As for the Rebonato method, it isn't described in sufficient detail in the slides to be directly implementable. Also, it seems that the implementation would be a bit involved including caching percentiles of the terminal distribution of CEV processes. So the question is whether is it so much more precise to be worth the trouble.Do you know if people who are "in the loop" know of some unpushlished way of achieving even better approximation than Paulot?Niels

Another SABR formula has been published by Commerzbank quants.Johnson and Nonas (2009) - Arbitrage-free construction of the swaption cubeI would like to hear how it compares to Paulot.

QuoteOriginally posted by: nielsesThank you for your insightful response, Alan.Our initial testing also shows that Paulot seems quite interesting. We didn't yet implement a Monte Carlo simulation or finite difference scheme to test it up again, though.As for the Rebonato method, it isn't described in sufficient detail in the slides to be directly implementable. Also, it seems that the implementation would be a bit involved including caching percentiles of the terminal distribution of CEV processes. So the question is whether is it so much more precise to be worth the trouble.Do you know if people who are "in the loop" know of some unpushlished way of achieving even better approximation than Paulot?NielsWell, I am doing a lengthy chapter on this model in my so far unpublished new book -- mostly focusing on theexact cases I mention above. Regarding the Paulot expansion, it is pretty predictable somebody will extend itto say O(T^4), so I will guess some grad student out there is plugging away at that. Also, if you drop the requirementof exactness at all strikes, it is possible to automate a double expansion in T and moneyness to very highorder, let's ay O(T^10). There is a big physics literature on this (systematic heat kernel expansion). Personally,I would advise against this route because the exact pde numerics are more worth your time, more accurate.

please upload the talk by rebonato. interesting thread. you guys really have to say which expansion you are talking about? Is it the normal or lognormal expansion? This is really important, especially in fixed income. I think that academically oriented people consistently misunderstand the importance of Bachelier'smodel, again in fixed income (dunno about equity). it is unbelievable that the bachelier model gets so little attention compared to the black model in the literature. It is soooo important. The standard argument that rates can go negative just doesn't get that much attention in practice, at least not on the market making side. The Bachelier model is much more important in fixed income.So, that the lognormal sabr by hagan stinks is a well known fact in the banking world, and I don't think that anybody is using it anymore. The fact that the Hagan expansion is inaccurate compared to Monte Carlo is true. So for given SABR parameters, if you compare Monte Carlo simulations against analytic computations, you will see that the two don't match as niels says, especially for long dated contracts under high volas. But does thatreally matter?Again one of the main mistakes I see especially on this forum is that people believe that market finance is a natural science and that particular attention should be given to particular models. This misunderstandingleads some people into a spiral of irrelevance. So let's just be sure about one thing. An options market is not governed by some silly diffusive or jumpyprocess. And never will be. This is not physics, chemistry or biology. There is nothing special about the SABR or any other model. This becomes especially true on the buy side, in case it is not already clear on the sell side.So back to Hagan's expansion. And then does it matter that the expansion is accurately representing theactual solutions of the SABR model? I am not sure at all. What would be very useful, would be to reverse engineer the actual stoch vol process that the Hagan SABR expansion represents. Once this is done, then use this as a starting point and operate with the exact expansion. It is much mor important to know that the expansion can fit market skew well and the resulting risk management practice is approximately OKfor trading book management. Who gives a shit about hyperbolic geometry and what not? Again, there is nothing special about the SABR model, it is not a magic conservation law or something like this. finance doesn't work like this.

Good points, unkpath.I don't know if Rebonato wants his slides from the ICBI conference to be distributed so I am hesistant to upload it. But send me an e-mail (see my profile) and I can send you a copy.I care less about the mathematics used in the derivation of the approximation. I am just interested in the approximation that gives the value which is closest to the real price (as could be calculated by Monte Carlo or finite difference). Ideally, the approximation should be valid for all allowed parameters (including 0 <= beta <= 1 and rho != 0) and for the whole swaption grid (all expiries and tenors) and for all strikes.As you mention, if SABR is simply used as a smile parameterisation it matters less to have an exact approximation. In fact, it is more important to use SABR as an interpolation tool in the same way as your brokers or counterparties. This would probably mean the original Hagan et al (2002) formula.However, if you implement a Monte Carlo model to value spread options or build it into a SABR/LMM model along the lines of Rebonatos (2009) book it is nice that the analytical price and the Monte Carlo price don't deviate (too much).Rebonato (2009) - The SABR/LIBOR Market ModelI agree that in the interest rate world the Bachelier (normal) model is as relevant as the Black (log-normal) model, but for this very reason CEV-style models (SABR) where you can vary beta between 0 (normal) and 1 (log-normal), and displaced log-normal style models (shifted Heston) where you can vary the mix parameter between 0 (normal) and 1 (log-normal), are quite common.Niels

QuoteOriginally posted by: unkpath What would be very useful, would be to reverse engineer the actual stoch vol process that the Hagan SABR expansion represents. Once this is done, then use this as a starting point and operate with the exact expansion. It is much mor important to know that the expansion can fit market skew well and the resulting risk management practice is approximately OKfor trading book management. Please elaborate on this comment, as I don't understand it. What do you mean by "the actual stoch. volprocess that the Hagan SABR expansion represents"?

yes, I also thought this is interesting. Find a model which exact solution is Hagans approximate solution for SABR. What is that good for?

hi nielses, please send to unkpath@gmail.com.You are right, it will matter for you if you are operating in a model that is genuinely tied to the SABR model. It is unpleasant to have your pricing approximations diverge too much from your MC prices. In that case you have to resort to some tricks during model calibration. Been there. However, see below, you may not need to use the SABR model at all, but maybe someapproximate SABR model would do the trick. Yes I agree with your comments on the local vol models. I was only insisting on the normal limit, but my view was a bit simplistic.I was really only speaking my mind somewhat loosely Alan. I was suggesting that for an options desk, given the SABR expansionand assuming one is satisfied by its performance, etc... one could attempt to find the diffusion process that will have as exact solution the SABR expansion. I am barely familiar with the asymptotic expansion they used in the original paper, but maybe this can taken and pieced together from there. Once we have that then, let's call this the approximate SABR model and use this thenas a starting point. Does that make sense? I was thinking back to my lectures on finite difference methods for pde's at school, when we were shown that the continuous time limit of an appropriately rewritten numerical scheme for a pde will result for some schemes in a pde with an extra diffusion term, i.e. a numerical diffusion term. Don't know if you know what I am talking about. In other words I am saying that if you can't find an integral to an sde you like, pick an integral you like and find the sde that goes with it.QuoteOriginally posted by: nielsesGood points, unkpath.I don't know if Rebonato wants his slides from the ICBI conference to be distributed so I am hesistant to upload it. But send me an e-mail (see my profile) and I can send you a copy.I care less about the mathematics used in the derivation of the approximation. I am just interested in the approximation that gives the value which is closest to the real price (as could be calculated by Monte Carlo or finite difference). Ideally, the approximation should be valid for all allowed parameters (including 0 <= beta <= 1 and rho != 0) and for the whole swaption grid (all expiries and tenors) and for all strikes.As you mention, if SABR is simply used as a smile parameterisation it matters less to have an exact approximation. In fact, it is more important to use SABR as an interpolation tool in the same way as your brokers or counterparties. This would probably mean the original Hagan et al (2002) formula.However, if you implement a Monte Carlo model to value spread options or build it into a SABR/LMM model along the lines of Rebonatos (2009) book it is nice that the analytical price and the Monte Carlo price don't deviate (too much).Rebonato (2009) - The SABR/LIBOR Market ModelI agree that in the interest rate world the Bachelier (normal) model is as relevant as the Black (log-normal) model, but for this very reason CEV-style models (SABR) where you can vary beta between 0 (normal) and 1 (log-normal), and displaced log-normal style models (shifted Heston) where you can vary the mix parameter between 0 (normal) and 1 (log-normal), are quite common.Niels

again, I may just not be making sense here. It is good for your piece of mind. You then have a consistent approach, a diffusive process you can use and study, do monte carlo with, pdes, extend, etc...QuoteOriginally posted by: pcaspersyes, I also thought this is interesting. Find a model which exact solution is Hagans approximate solution for SABR. What is that good for?

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I thought the (worrying) problem with Hagan's approximation to SABR was that at low strikes it can give rise to what amounts to negative densities (and this gets progressively worse at longer option expiries?). That being the case, is it possible for an SDE that corresponds exactly to Hagan's approximation to exist? Also, given the negative density / arbitrage issue would there be any point in trying to reversse engineer it?

Yes, I agree -- strongly suspect there does not exist any stochastic processthat produces the SABR asymptotic expansion truncated at any order O(T^n).Even it such a process existed, it would have some properties you might not like.For example, the exact SABR process has a (Black-Scholes) implied volatility thattends to 0 as T -> infinity. The truncated SABR short time expansion has an impliedvolatilitysig_imp(T,f0,sigma_0,K) = a_0( ) + a_1( ) T + ... + a_N( ) T^Nwhere coefs depend upon f0,sigma_0, K and the other SABR parameters.Now if a_N is negative for any parameter combo, the sig_imp will tend to negative infinityas T-> infinity, which is impossible and no SDE process exists. This means, for the idea to work, (i) a_N has to be positive everywhere,and (ii) the other a_n have to keep the whole expression non-negative everywhere -- which is unlikely. But even if you are lucky and get that, it now means the replicating process has an implied volatility that diverges as T^N as T -> infinity with a positive coef. So, the lack of mean-reversion in the vol. process comes back to bite youin a different way. In the exact process, it causes the implied vol. to fall to zero,but now in the putative truncated process, it diverges like a polynomial.This may cause other problems that you weren't anticipating in some newlong-dated instrument.

Last edited by Alan on July 2nd, 2010, 10:00 pm, edited 1 time in total.

another "stream" seems to be to get inspiration from a model but then create a totally model free parametrization of smiles and volatility surfaces with properties that you really want and need and which are useful for calibration (and which the "real" models do not (fully) give you).E.g. CMS pricing seem to work much better in the current market conditions with these parametrizations than with models like SABR. So forget about SABR and use easy formulae to extrapolate your swaption smile and replicate your cms coupons?What do you need a model for in the end, if your task is "just" to price in consistency with the market and try to get near "the others"?

Thank you for the Johnson and Nonas (2009) reference, PierreG.I implemented the formula. It doesn't seem to be better than the Paulot approximation.As the authors write, it is only valid for 0 < beta < 1, which makes it give strange values for beta < 0. We sometimes get negative beta when calibrating, so it is nicer if the approximation formula also works in this case as it seems the Paulot formula does.Also Johnson and Nonas seems to have a tendency to give way too high prices and thus vols for high strikes.And I tried with the following benign parameters:t = 10, K = 0.8, F0 = 1.0, sigma (usually alpha) = 0.2, beta = 0.8, alpha (usually nu) = 1e-9, rho = 1e-9This gives sigmaCEV = 0.2080 (at least in my implementation)As the alpha (vol. of vol.) and rho are zero, this should reduce to the CEV model with sigma = 0.2, so I'm puzzled that it gives 0.2080 != 0.2.My implementation uses the Sankaran (1963) via Schröder (1989) approximation for the non-central chi square CDF in the CEV pricing. It can be made more precise using an infinite sum, but I don't think that would be enough to beat Paulot.Any inputs appreciated.Niels

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