So there's alot of research concerning numerical methods for approximations using computers etc. But what about research based on analytical methods? Derivative pricing comes down primarily to evaluating integrals and obviously other branches of science would be grateful for advancements in this area.So my question is: Is there much active research in this area? (analytical evaluation of integrals). If so, can anyone recommend any papers or books? do you think there are any PhD opportunities in this area?

I don't know in general but Wolfram Research would be interested.They made a big deal about a lot of their (analytic) integration improvements when they released Mathematica Ver 5 (V6 is current).Anyway, I'll let you search their site for that, but in googling to find a link for you there, I ran across this, whichyou should find interestinghttp://www.ams.org/notices/200203/fea-moll.pdfregards,

The field you're looking for is Asymptotic approximation of integrals. Google it and you'll get plenty of hits. I don't really know how much it's used in finance -- I'm not a quant yet, am currently job hunting -- but it's almost certainly not used enough.The classic book is the one by Frank Olver.One of the great breakthroughs in recent years was by Sir Michael Berry, on the Stokes effect:M. V. Berry (1989): Uniform asymptotic smoothing of Stokes’ discontinuities. Proc. R. Soc. Lond. A, 4227–21.R. B. Paris and A. D. Wood (1995): Stokes phenomenon demystified. IMA Bulletin, 31 21–28.Any paper by Berry is worth reading.There are many more recent results in the field.There are certainly plenty of possible PhD problems in the field. Apropos Mathematica, it can evaluate symbolically only a small fraction of integrals. Knowing asymptotic methods expands your powers by at least an order of magnitude.The question is whether spending the time on mastering integrals is worth the effort in finance. I suspect not.

Thanks for both replies. cheers for the text and journal links! I'll take a look. I didn't think there would be much direct application to finance really. how come you know about the subject of asymptotic integral approximation? did you do a phd related to this area?

My PhD was broadly in theoretical condensed matter physics; but I published in maths journals too. Came up with a rather nice result in asymptotic approximation of integrals ( ) with a direct physics application.Probably about 2/3 of the thesis would have had some close connection with asymptotic expansions of one sort or another. Probably leaning too far towards applied maths. What's done is done.I think there are some nice, big ideas in asymptotics that impact on all of physics (e.g. through the Born expansion, which is itself an asymptotic expansion).The nice thing about the field is that most mathematicians know little about it (which can also be a disadvantage), and many of the results seem counterintuitive and rather akin black magic. It's certainly a very applicable field, and numerical computation (i.e. on computers) is certainly disadvantaged without a knowledge of analytical asymptotics. It certainly ought to be better known in finance.

Perhaps none may be what you are interested in, butHowison Sam, Mario Steinberg 'Matched Asymptotic Expansions For Discretely Sampled Barrier Options' Bachelier Conference 2004 Kawai Atsushi 'A New Approximate Swaption Formula in the LIBOR Market Model:An Asymptotic Expansion Approach'App.Math.Fin.3/03 7/02 term structure <BGM, volatility,Andersen, Andreasen>Quintanilla Maite 'Asymptotic Expansion for Value at Risk' MS U. Toronto Widdicks Martin, Peter W. Duck, Ari D. Andricopoulos, David P. Newton 'The Black-Scholes Equation Revisited: Asymptotic Expansions And Singular Perturbations' MF 4/05 Muroi Yoshifumi 'Pricing Contingent Claims with Credit Risk: Asymptotic Expansion Approach' F&S 7/05 Howison Sam ‘A Matched Asymptotic Expansions Approach to Continuity Corrections for Discretely Sampled Options. Part 2: Bermudan Options’ Applied Mathematical Finance Vol. 14, #1 March 2007 Howison Sam, Mario Steinberg ‘A Matched Asymptotic Expansions Approach to Continuity Corrections for Discretely Sampled Options. Part 1: Barrier Options’ Applied Mathematical Finance Vol. 14, #1 March 2007 Howison Sam ‘Matched Asymptotic Expansions in Financial Engineering’ 2005 Oxford Financial Research Center Osajima Yasufumi ‘The Asymptotic Expansion Formula of Implied Volatility for Dynamic SABR Model and FX Hybrid Model’ SSRN 2/07 Osajima Yasufumi ‘The Asymptotic Expansion of Implied Volatility for Long-Term Cross-Currency Hybrid Model’ Bachelier Conference 2006 Uchida Yoshihiko ‘A New Computational Scheme for Computing Greeks by the Asymptotic Expansion’ Bachelier Conference 2006 Knight John, Stephen Satchell 'Asymptotic Expansions for Random Walks with Normal Errors' w.p. U. London 11/92distributions Kunitomo Naoto, Akoihiko Takahashi 'On Validity of the Asymptotic Expansion Approach to Contingent Claim Analysis' 4/2000 contingent claim <Watanabe-Yoshida, Malliavin calculus>Kunitomo Naoto, Akoihiko Takahashi 'The Asymptotic Expansion Approach to the Valuation of Interest Rate Contingent Claims' MF 1/2001 contingent claim <Watanabe-Yoshida, Malliavin calculus> Barndorff-Nielsen Ole, Neil Shephard 'How Accurate is the Asymptotic Approximationto the Distribution of Realized Volatility?' Scan. J. Stat. 6/03 , <volatility,Lévy> 8/01 Sircar Ronnie, Thaleia Zariphopoulou 'Bounds & Asymptotic Approximations for Utility Prices when Volatility is Random' SIAM J. Control & Optim. 2005 <volatility,derivative, stochastic volatility> Sacerdote L., F. Tomassetti 'On Evaluation of Asymptotic Approximations of First-Passage Time Probabilities' (96) [SDE] Advances in App.Prob.

QuoteOriginally posted by: INFIDELThe field you're looking for is Asymptotic approximation of integrals. Google it and you'll get plenty of hits. I don't really know how much it's used in finance -- I'm not a quant yet, am currently job hunting -- but it's almost certainly not used enough.The classic book is the one by Frank Olver...This title is available from the Wilmott Bookshop. Asymptotics and Special Functions