Hi allI am a student and am currently working on creating a local volatility surface using Dupires formula. I have created an implied vol surface using market quotes in the FX market (strangles, risk reversals etc). I am looking for a really good, and hopefully simple article on dupires formula. I have read just about all the posts on the forum on dupire and local vol and it seems that there are a lot of problems with dupire(excluding the fact that it is numberically unstable!)If anyone knows of a good article on local vol and/or Dupire please can you reply to this post.. The paper I am writing on is mainly dealing with the theory of local vol using Dupire and the numerous problems that one faces - and also why in the face of all these problems, it is still being used. Thanks!

Local volatility is a term used in quantitative finance to denote the set of diffusion coefficients, σ(ST,T), that are consistent with the set of market prices for all option prices on a given underlier. This models are used to calculate values of exotic options which are consistent with observed prices of vanilla options.The concept of a local volatility was originated by Emanuel Derman and Iraj Kani as part of the implied volatility tree model.[1] As described by Derman and Kani, the local volatility function attempts to model the instantaneous volatility to use at each node in a binomial options pricing model such that the tree will produce a set of option valuations that are consistent with the option prices observed in the market for all strikes and expirations.[1]The key equations used in local volatility models were developed by Bruno Dupire in 1994. Dupire's equation statesWhile constant local volatilities are inconsistent with the dynamics of the implied volatility surface,[2] they are nonetheless useful in the formulation of stochastic volatility models.[3]Local volatility models have a number of attractive features. Because the only source of randomness is the stock price, local volatility models are easy to calibrate.Local volatility models are not very well used to price cliquet options or forward starting options.References[1] a b Derman, E., Iraj Kani (January, 2004). "The Votility Smile and its Implied Tree" (PDF). Goldman-Sachs. Retrieved on 2007-06-01. [2] Dumas, B., J. Fleming, R. E. Whaley (1998). "Implied volatility functions: Empirical tests". The Journal of Finance 53. [3] Gatheral, J. (2006). The Volatility Surface: A Practioners's Guide. Wiley Finance. ISBN 13 978-0-471-79251-2.

> While constant local volatilities are inconsistent with the dynamics of the implied volatility surface,[2] they are nonetheless useful in the formulation of stochastic volatility models.[3]How come a deterministic constant can help you study a dynamic variable?> hopefully simple article on dupires formulaI am sorry to dash your hope of finding such article - that formula is easy but its implementation given a set of discrete option pricies is a nightmare.>The paper I am writing on is mainly dealing with the theory of local vol using Dupire and the numerous problems that one faces - and also why in the face of all these problems, it is still being used. Nothing bad with that theory but recall that they always say "...given a continuum of arbitrage-fee vanilla option prices..." - this assumption is never satisfied in practice.More importantly, using local vol you cannot hedge changes in the vol surface itself, so that you cannot use it for vol products including forward-starts and cliquetsIt is clear to me however it would be nice to proof it theoretically that with local vol surface volatility you cannot hedge VIX-type products (products on the future realized volatility)

Thank you both for your repliesI am extremely new to local volatility and am finding that the implementation is a nightmare! I have found some code on quantcode for local vol, and it uses a formula for dupire that I am unfamiliar with.It comes from a thesis by Elder, John, 2002. ”Hedging for Financial Derivatives”. University of Oxford, Ph.D. Thesisthat I am having trouble finding. Has anyone used this code (attached)?Any help would be much appreciatedTextextremelt

i find the implementation via dupire's paper (using prices) to be very unstablegatheral's recommendation in his book is much better (at least for me)

Hello all. I'm sorry to have joined so late into the conversation, I hope is not too late. Right now I'm implementing the model in matlab, its pretty similar to the one post upside, except for some differences in the numerical approximations of the derivatives. The fact is that with the current data, and in my implementation which I think its alright at least from the point of view of the literature, what would it be the "d2" in the code above, or the denominator, passes through zero at some point what it means, that my local volatility reaches values that tend to infinity and then, imaginary values once it goes negative... I was wondering if any of you have faced this problem, and if you find some solution.I would really appreciate any respond.Thanks a lot.P.s: I guess it would be too late, but for Calli, the best article I've found is form atur seep: "Pricing Barrier Options under Local Volatility"P.s.2: Any corrections on my english also would be quite appreciated; off course, if it doesn't take the whole page, he, thanks!

Hi all,I've worked on this model for a while and I've seen quite some "instabilities", but I don't think it is a fundamental problem. As far as I can see, it's mostly related to the quality of the implied volatility surface. If you start from a good implied volatility surface, you get a good local volatility. But of course the question is what good means and how to get such a surface.If you want more details I can suggest these notes I put onlinehttp://papers.ssrn.com/sol3/papers.cfm?abstract_id=2112819http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1779463The first is about the hybrid Dupire model and it's focused on the hybrid calibration, so there's probably too much in there, but you will find some comments about the implied volatility surface and a few tricks and formulas that are relevant for the non-hybrid version (which I guess is the one you're talking about).The second reference shows you an example of implied volatility (IV) surface that tends to give good local volatilities (at least in my opinion, which is why I wrote those notes).But without going into the details of these notes, I'd say the key is to get an IV surface that is smooth enough and that satisfies the non-arbitrage properties as much as possible. What you describe, pablogarciaj, looks similar to what I have experienced when using bad IVs. The denominator in Dupire's formula is basically proportional to the probability density. So it's supposed to be a strictly positive number. But if your implied volatility surface does not satisfy non-arbitrage (which, in particular, requires strict positivity of the density), then your denominator can be equal to 0 or even negative, which makes your local vol blow up or just become non-existent.You can see other kinds of problems happening. For example, your numerator may become negative, which would signal an other form of arbitrage. Also, even if you satisfy non-arbitrage, if your implied vol moves too much because you start from a strongly irregular market and you fit everything, then your IV will also shows lack of smoothness, and when you differentiate it, well, you can guess what happens.So the key is the IV surface. It's pretty difficult to find an IV surface satisfying all smoothness and non-arbitrage conditions, while at the same time having a good fit to the market data and being intuitive. The second reference I give you above produces a nice IV in my opinion, but it doesn't satisfy all the above requirements. In particular if you want something that satisfies the non-arbitrage in the strike direction, you should look at the mixtures of log normals. But what I use in the second reference is based on Gatheral's SVI surface, which, although not guaranteed to be free of strike arbitrage, comes pretty close to it in practice.sebgur

And recently Gatheral has published material on SVI and avoiding arbitrage, so looking for these papers (and related papers as well) will help in deciding how to choose between exact fitting on IVs and smooth LVs.