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I'm reading Peter Carr's "A Practitioner’s Guide to Mathematical Finance". When talking about the math used in mathematical finance, he mentions Lie groups, differential geometry, string theory. Can anyone explain (either in an understandable or informal way is ok), and with examples if possible, how these 3 can be applied in MF?

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    $\begingroup$ The book "Analysis, Geometry, and Modelling in Finance" by Pierre Henry - Labordere will give you a good introduction. $\endgroup$ – Gordon May 25 '15 at 14:24
  • $\begingroup$ Thanks! It seems to introduce detailed connections with many types of geometry (not only in differential geometry), but maybe not with string theory and Lie groups. $\endgroup$ – SiXUlm May 25 '15 at 14:34
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MF is linked with physics mostly because it solves the same PDEs (Black-Scholes equation is a certain type of Schrödinger equation for instance). As for the specific links you mentioned :

  • Lie Algebra : Magnus expansion (to build fast approximation of time dependent ODEs like those arising in credit risk)
  • Differential geometry : link with Varadhan approximation (used for instance in Avellaneda's SABR)
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  • $\begingroup$ Thanks! And do you have any suggestion for string theory? $\endgroup$ – SiXUlm May 26 '15 at 11:11
  • $\begingroup$ Not an expert at all in physics. For what it's worth path integrals are used in string theory and finance. $\endgroup$ – vanna May 26 '15 at 11:17

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