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### SABR approximations - best practice?

I drew a line between around ATM and OTM because for the first we have liquid quotes but not for the latter. Therefore, as you correctly stated, the OTM option prices/vols are belief based. And I personally prefer to believe in a model with (reasonably) sound dynamics than some approximation which cannot be connected to any diffusion process ...Can you elaborate a bit on your statement, that no stochastic vol model performs well dynamically? And concerning CMS adjustment. I'm not a trader but I thought that the CMS market is rather liquid and you also have totem consensus prices that you can use to fine-tune your SABR model.
Last edited by BerndSchmitz on December 8th, 2015, 11:00 pm, edited 1 time in total. mtsm
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### SABR approximations - best practice? BerndSchmitz
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### SABR approximations - best practice?

Thanks doe the interesting insights mtsm - and the colourful language  Alan
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### SABR approximations - best practice?

QuoteOriginally posted by: mtsmThe problem with a numerical solver is that unless it operates in a split second, it's going to be a tough call to make that work in a production environment. Thinking some more about this, perhaps some nice tables could be produced, solving the SABR approximation problem once and for all.The vol-of-vol $\nu$ can be removed by scaling. That is, if $\Sigma(T,K,F_0,\sigma_0;\nu,\beta)$ is the Black-Scholes implied vol, then$\Sigma(T,K,F_0,\sigma_0;\nu, \beta) = \nu \, \Sigma(\nu^2 T,K,F_0,\frac{\sigma_0}{\nu};1,\beta)$Then, one more variable can be removed by scaling with the use of the effective lognormal volatility $\sigma_0^{LN} \equiv \sigma_0 F_0^{\beta-1}$.That leaves $(T,K/F_0, \sigma_0^{LN},\beta)$ as the only remaining independent parameters.The implied vol is smooth in all the parameters, so it can be accurately interpolated from a suitable table of 'exact' (i.e., non-asymptotic) results. I wonder if this would be useful? Say I create some tables, host them somewhere on the web, and provide some web APIs.Is there any interest in this? If so, how would people like to see it work?
Last edited by Alan on December 9th, 2015, 11:00 pm, edited 1 time in total. Pat
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### SABR approximations - best practice?

The derivation of the SABR formula requires that sqrt(1 + 2*rho*volvol*z/alpha + volvol*volvol*z*z/alpha*alpha) be near to 1, and the formulas are not guaranteed to be valid otherwise, even though they are used routinely in such regions. To curtail the high strike vols in an arb free way, the best way is to bound this factpr, using, say E(z) = min{sqrt(1 + 2*rho*volvol*z/alpha + volvol*volvol*z*z/alpha) , Emax}Taking Emax of 2 or 3 works pretty well, but even taking Emax to be 5 or 10 helps control the high strike tails enough to aid the CMS pricing.If you're willing to solve a PDE, this can be done easily using the Arb Free SABR approach. Analytically is a bit more difficult, as the asymptotic formulas become more complicated. To be published soon (or write to me privately). Alan
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### SABR approximations - best practice?

Pat,What do you think of the idea to just create -- once and for all -- some 'exact' tables for the implied vol and let users interpolate?Since there are only 5 independent parameters, $(T,K/F_0, \sigma_0^{LN}, \beta, \rho)$, where $\sigma_0^{LN} \equiv \sigma_0 F_0^{\beta-1}$, perhaps this is feasible for plausible parameter ranges and might put the approximation problem to rest.
Last edited by Alan on December 11th, 2015, 11:00 pm, edited 1 time in total. Lapsilago
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### SABR approximations - best practice?

The PDE solution is worthwhile to do...This also works for the new free boundary SABR and you can easily adapt it to move the stickyness around by using |F+a|^beta and even place a lower bound on the rates to recover the "knee". Thus, lots of freedom for modelling.Best, Lapsi Alan
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### SABR approximations - best practice?

Thanks for the encouragement and alerting me to the 'free boundary SABR', which is a very interesting extension of the model.For the regular SABR which lives on F>0, if I built tables what ranges would you like to see for the parameters? Alan
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### SABR approximations - best practice?

After playing around some, I think extensive tables would require much more effort than I want to invest.However, I can create some interactive Mathematica demonstrations that will show the difference between theexact solutions for the implied vols (IV) and the asymptotic formulas. These will use Mathematica's free CDF playerand will work for anybody -- you don't need to be a Mathematica user. While warming up to do that,here is a gif (we'll see if it works) showing the exact IV smile vs. Paulot's O(T^2) expansion for the lognormal SABR caseas T ranges from 1 to 10 (with vol-of-vol = 1, $\rho = -0.7$, and $\sigma_0=0.2$, and $F_0=100$).It starts at T=1 where the expansion is good and runs until T = 10, where the expansion gets silly, and then loops to the beginning.If you focus on the vertical axis, you'll see that both the exact IV and Paulot IV are changing with T, with the asymptotic expansion changing the fastest:The CDF demo will ultimately look similar, with some controls that let you adjust the parameters.
Last edited by Alan on December 16th, 2015, 11:00 pm, edited 1 time in total. Islacanela
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### SABR approximations - best practice?

Dear Gurus,If in the original paper by Hagan the SABR model is introduced with 4 parameters (alpa0, beta, v, rho) and usually 1 parameters is fixed (like beta), is there some literature offering an alternative model with smaller number of parameters, say 3, (and perhaps a slightly different stochastic process to start with) and with with some closed form approximations? Or is it too much to ask for? Lapsilago
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### SABR approximations - best practice?

Dear Alan,what FD solver do you use for the exact solution? Is there a Mathematica file available for that?I wish to compare the Hagan ArbFree Solution from Wilmott magazine with the exact solution and also for the free SABR (approximate solutions, PDE solutions based on Hagan appraoch).Grateful for some hints, best,Lapsi Alan
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### SABR approximations - best practice?

Lapsi,The lognormal SABR, which is shown in the animation I posted, does not need a pde solver, but is a closed-form in the spirit of the Heston model.For general $\beta$, you do need a solver; I use NDSolve. You can proceed directly, using the two spatial variables; butmy preferred method is what I call the PDE-Fourier hybrid. Essentially, the idea is that you reduce the system to "standard stochastic volatility form",by introducing the effective lognormal vol: $\zeta = \sigma \, S^{\beta-1}$. This leads to a 1D PDE for the characteristic function and then anintegration to get option values, again similar to what you do with the Heston model. The difference is that, here, you indeed need to employa solver to get the characteristic function, so it is an order of magnitude more computationally expensive. (Say a couple of minutes for SABR vs a couple ofseconds for Heston). Details are in Ch. 8 of my "Option Valuation under Stochastic Volatility II" (forthcoming Q1, 2016). p.s. Re Mathematica files. Once I get the book out the door, I hope to post some complete Mathematica files online, but as to exactly what, I haven'tdecided yet.
Last edited by Alan on December 16th, 2015, 11:00 pm, edited 1 time in total. Alan
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### SABR approximations - best practice?

QuoteOriginally posted by: IslacanelaDear Gurus,If in the original paper by Hagan the SABR model is introduced with 4 parameters (alpa0, beta, v, rho) and usually 1 parameters is fixed (like beta), is there some literature offering an alternative model with smaller number of parameters, say 3, (and perhaps a slightly different stochastic process to start with) and with with some closed form approximations? Or is it too much to ask for?Well, the system$dS_t = \sqrt{V_t} \, S_t \, dB_t$,$dV_t = \xi \, V_t^p \, dW_t$,$dB_t \, dW_t = \rho \, dt$ is solvable for $p=1/2$ (Heston), $p=1$ (lognormal SABR), and $p=3/2$ (3/2-model). Once you fix p, the parameters are $(\rho,\xi,V_0)$.The case p=1 was just discussed. The Heston and 3/2 model solutions are available in many places.
Last edited by Alan on December 16th, 2015, 11:00 pm, edited 1 time in total. scottstephens
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### SABR approximations - best practice?

QuoteOriginally posted by: AlanWell, the system$dS_t = \sqrt{V_t} \, S_t \, dB_t$,$dV_t = \xi \, V_t^p \, dW_t$,$dB_t \, dW_t = \rho \, dt$ is solvable for $p=1/2$ (Heston), $p=1$ (lognormal SABR), and $p=3/2$ (3/2-model). Once you fix p, the parameters are $(\rho,\xi,V_0)$.The case p=1 was just discussed. The Heston and 3/2 model solutions are available in many places.Two questions:(1) The model above with $p=1$ doesn't really correspond to SABR with $\beta=1$ does it? In SABR the volatility process is a driftless GBM, but in your model with $p=1$ variance is a driftless GBM. Have you just left out a drift term in the variance process for simplicity, but you would actually add it when you solve the model? I think if you added the drift term $\frac{1}{4}\xi^2 V_t dt$ to the variance equation you really would have SABR with $\beta=1$ (and $\nu=\xi/2$).(2) Can you spell out exactly how to solve either your system with $p=1$ or SABR with $\beta=1$ (or both)? A reference to a paper or a chapter of your book would be fine; I haven't been able to figure it out from reading this thread or the table of contents of your book. Alan
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### SABR approximations - best practice?

QuoteOriginally posted by: scottstephensQuoteOriginally posted by: AlanWell, the system$dS_t = \sqrt{V_t} \, S_t \, dB_t$,$dV_t = \xi \, V_t^p \, dW_t$,$dB_t \, dW_t = \rho \, dt$ is solvable for $p=1/2$ (Heston), $p=1$ (lognormal SABR), and $p=3/2$ (3/2-model). Once you fix p, the parameters are $(\rho,\xi,V_0)$.The case p=1 was just discussed. The Heston and 3/2 model solutions are available in many places.Two questions:(1) The model above with $p=1$ doesn't really correspond to SABR with $\beta=1$ does it? In SABR the volatility process is a driftless GBM, but in your model with $p=1$ variance is a driftless GBM. Have you just left out a drift term in the variance process for simplicity, but you would actually add it when you solve the model? I think if you added the drift term $\frac{1}{4}\xi^2 V_t dt$ to the variance equation you really would have SABR with $\beta=1$ (and $\nu=\xi/2$).(2) Can you spell out exactly how to solve either your system with $p=1$ or SABR with $\beta=1$ (or both)? A reference to a paper or a chapter of your book would be fine; I haven't been able to figure it out from reading this thread or the table of contents of your book.(1) Yes, I was being loose with language -- I should have said 'similar to SABR'. Both the model above and the lognormal SABR are solvable, though.(2) For SABR with $\beta=1$, there are two ways to proceed. The first relates the transition density to Brownian motion on a hyperbolic space (H^3).The second, which is used for the animation I posted uses the 'Fundamental Transform' approach of my existent "Option Valuation under StochasticVolatility". Both methods are shown in Ch. 8 of my forthcoming Vol II. I have prepared an excerpt for you which shows the lead-in to theFundamental Transform computation (see attached).=========================================================================p.s. A third way to get the Fundamental Transform solution in (2) is by reduction of the very messy solution on pgs 334-335 of the my existent "Option Valuation under Stochastic Volatility". (year 2000 book). This is quite tedious though; plus, the new derivation in Vol. II is cleaner and more fully elaborated. But, if you want to try that, just make exactly the SDE correspondences you noted in your post.I suggest using $\nu=1$, so $\xi = 2$, since the general $\nu$ soln can always be recovered by scaling. At the top of the mentioned pg. 335, it says "$\beta = 2 \alpha/\xi^2$". This is not the SABR $\beta$, but simply an introduced parameter, which will equal 1/2 for the lognormal SABR reduction and this will result in much simplificationof eqns (3.4) and (3.5) there.
Attachments Lewis.Ch8Excerpt.zip
Last edited by Alan on December 18th, 2015, 11:00 pm, edited 1 time in total.  