Thanks -- if my remaining tasks go smoothly, look for it in the first quarter (2016).

Here is an old blog post around the free boundary SABR with Hagan's PDE approach http://chasethedevil.github.io/post/arb ... ted-sabr/I could not find a real world example where the free boundary SABR matches much better the market however. I was able to calibrate shifted SABR to give much better results than the ones in Antonov et al. paper. The absence of a shift parameter is however interesting, even if the density looks a bit funny at 0.I also had a look at Pat proposal to cap the "equivalent" local volatility as I was curious what would be the outcome. It seems like finding a good analytical approximation is not so simple due to the non differentiability of the equivalent local vol.http://chasethedevil.github.io/post/con ... hagan-pde/

Thanks for that clarification Alan, it really helped a lot. I'm looking forward to reading the new book now.

Hi everyone,I'm still interested what you think about just using the I^0 term of the 2nd order expansion, i.e.[$]\sigma_{LN} = I^0[$] instead of [$]\sigma_{LN} = I^0 \cdot (1+I^1 T)[$]Obviously one needs to investigate whether one can always reasonably fit the market with that stub formula. And the dynamics are even further away from the true SABR dynamics. However, the way I see it I^0 is always positive, so the implied vol can never get negative.Bernd

QuoteOriginally posted by: BerndSchmitzHi everyone,I'm still interested what you think about just using the I^0 term of the 2nd order expansion, i.e.[$]\sigma_{LN} = I^0[$] instead of [$]\sigma_{LN} = I^0 \cdot (1+I^1 T)[$]Obviously one needs to investigate whether one can always reasonably fit the market with that stub formula. And the dynamics are even further away from the true SABR dynamics. However, the way I see it I^0 is always positive, so the implied vol can never get negative.BerndMight not be too crazy an idea. What you would lose would be the tendency of the smile to flatten with maturity.(See the 'exact' smile I posted and watch it somewhat flatten with maturity).If you just use it as an interpolation formula at fixed maturities, and it fits your market, maybe it makes sense.

Does anybody have a numerical example at hand that leads to a negative density if I use Paulot's first order expansion? Apparently I'm too stupid to find one either in the forum or with google.Thanks,Berndp.s.: I would be also very interested in seeing a few vols numbers for some parametrization (for comparison to my implementation)[Edit]:I have found an example in Hagan's "Arbitrage Free SABR" paper (forward=1, alpha=35%, beta=0.25, nu=100%, rho=-10%) and I get exactly the same pictures - so everything seems to be fine

Another example: the gif I posted will go negative using the first order expansion at T=20, K < 25 or K > 275

I can privately send a PDF which derives the analytical formulas for implied vols with and without a capped stochastic vol effect ... and discusses the issues.Best to use pathagan1954@yahoo.comThe 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).

QuoteIf 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). I suppose the PDE is insensitive to the parameter size in the singular perturbation expansion?

The one space dimensional PDE used in the arb free SABR approach is always gives self consistent (ie, arb free) prices, no matter how large the parameters.However, if one used finite differences to solve the exact two dimensional SABR model, one would be see that the prices obtained from the Arb Free SABR approach drift away from the exact SABR prices as the VolVol *sqrt(Tex) gets large.The asymptotic formulas for the smile does well for high strikes, even under pretty extreme conditions. It cannot do well for the ultra low strikes, because there is usually a boundary layer adjacent to the bottom boundary which needs to be incorporated in the low strike analysis.What's amusing is that unless the conditions are too severe, the smiles of most other stochastic volatility models (Heston, ZABR, exponential, ...) are given by the same formulas ... article to be published in Wilmott ...

QuoteOriginally posted by: PatThe one space dimensional PDE used in the arb free SABR approach is always gives self consistent (ie, arb free) prices, no matter how large the parameters.Does it also hold for small parameters? I did my PhD research on singular perturbations and exponentially fitted methods which were also use by Roelof Sheppard in his thesis (advisor; the late Graeme West) for Heston (and SABR to some extent). the goal is to have a FD scheme for all parameter values. Roelof used Soviet Splitting method.Most FD schemes will work for large numbers (Peclet/Reynolds not an issue in that case).QuoteHowever, if one used finite differences to solve the exact two dimensional SABR model, one would be see that the prices obtained from the Arb Free SABR approach drift away from the exact SABR prices as the VolVol *sqrt(Tex) gets large.Which FD was used? QuoteThe asymptotic formulas for the smile does well for high strikes, even under pretty extreme conditions. It cannot do well for the ultra low strikes, because there is usually a boundary layer adjacent to the bottom boundary which needs to be incorporated in the low strike analysis.The strike is part of the payoff only I suppose so you are referring to t = T? If the method is A_0 stable only then oscillations can appear. When V = 0 in Heston Roelof had to solve for a 1st order hyperbolic PDE.QuoteWhat's amusing is that unless the conditions are too severe, the smiles of most other stochastic volatility models (Heston, ZABR, exponential, ...) are given by the same formulas ... article to be published in Wilmott ...Looking forward to it.

The arb free SABR approach leads to solving a PDE in time and the forward F with a delta function initial condition. Crank-Nicholson is absolutely stable, but not A0 stable (the decay rate of short wavelength perturbations goes to zero as the wavelength goes to zero ... so the finer the spatial discretization, the slower the decay rate. Upshot: The delta function initial condition is resolved very slowly. There are several variations of Crank-Nicholson which overcome this: Lawson-Swayne, LMG, .... However the simplest fix is just to start the scheme with two fully implicit half-sized steps and then switch to CN, and this worked fine, very nearly as well as the fancier schemes. I think this is called starting with a Ranacher step.

To be honest, I'm allergic to Rannacher and Crank Nicolson (too many tweaks needed) and it is now well-known what their several technical issues are. A discussion is here The greeks in particular is an issue.A better solution that achieves desired L0 stability and second-order accuracy is Richardson extrapolation that Roelof Sheppard describes in his thesis for Heston On a related issue, one approach is to discretize the PDE in S only to produce a system of ODEs and then give this system to a robust solver in NDSolve (Mathematica) or Boost library odeint (C++). ODE stuff is a specialism within numerical analysis that I like to leave to ODE experts.Interesting to note that this Method of Lines (MOL) is not widely used by quants, it would seem. It makes life (much) easier and results are almost immediate.Alan Lewis' forthcoming new book has a number of chapters on MOL + applications I believe.

I believe that using fully implicit schemes and then using Richardson extrapolation to up the order of accuracy is essentially the LMG scheme, or a close cousin. See paper by Le Floc'h and Kennedy, Finite Difference Techniques ... SABRThe method of lines mentioned by Cuchulainn works beautifully for most problems of this type, and if you're doing one-offs, or solving problems for research purposes, I highly recommend it. There is the usual caution about hooking up third party softward if you are building or contributing to a system running daily risks and hedges.