<t>You are welcome.One can use the recursion relation, it's very robust. For performance purposes (when calibrating the model) you may want to cache the numerical values.Yes, there is a close form of the regularization term (I did not put it in the notes).QuoteOriginally posted by: stasThanks for th...

<t>In my experience, cubic splines is all you need. I have been using this methodology for many years in various markets, and have not had issues with oscillations. The oscillations you have noticed may have something to do with the fact that you are not using auxiliary knots (by doing so you loose ...

<t>I believe that equation (19) is correct. You need the auxiliary knot points in order to accomodate the support properties of the basis functions.QuoteOriginally posted by: stasThe thing is that I'm not doing interpolation but a least squares fit. lesniewski, that is precisely my primary reference...

You may want to look at this for a basic outline.

Do not do what frenchX proposes: the symmetrized KL divergence is not a good distance concept. Use .

Thanks, Alan, I'll fix it. Also, the Neumann problem isn't written out in full detail - the argument should be similar to the Dirichlet boundary condition(Hankel transform).

Alan, you are welcome. In my previous posting, I have updated the file with a version that is slightly more readable.

Alan, attached are my (somewhat rough and incomplete) calculations with the CEV model. I believe that what you are asking about is related to formula (31) of my notes. Its proof is written on the upper half of page 5.

<t>I am afraid I don't have really good answers to your questions... These are asymptotic expansions, which means that they provide approximations to the (exact) solutions only within a certain range of parameters. Asymptotic expansions are often very useful, when set up so that that range contains ...

<t>Yes, this is the content of the classic theorem by Varadhan which states roughly the following. The transition probability is given asymptotically (in the limit where the time ) by . Here is the geodesic distance between the points on the state space with respect to a suitable Riemannian metric (...

I am not not aware of any theoretical (asymptotic) results that using antithetic variables with Sobol sequences improves convergence. Anectodal evidence is unconvincing and points in different directions depending on the context.

<t>Yes, they are, to within the validity of the asymptotic expansion (the parameter of the SABR expansion is volvol^2 x time to expiration). A good way of monitoring it is to keep track of the total volume of the asymptotic probability distribution (it should not deviate from 1 too much). Btw, you c...

These are different drafts of the same thing, use the more recent one. For the implied vol, use the original Wilmott article.

<t>You shouldn't use the asymptotic formula for the implied volatility in the SABR model to calculate the probabilities you are interested in. Instead, you can use the asymptotic formula for the probability density derived in the old paper by Pat Hagan, Diana Woodward, and myself "Probability distri...