Connections Between Mathematical Finance and Physics-Inspired Mathematics

Summary

The document asks how Lie groups, differential geometry, and string theory relate to mathematical finance, but the available answer gives only a brief connection. It notes that mathematical finance often involves partial differential equations also studied in physics, citing the Black–Scholes equation as a type of Schrödinger equation. This analogy helps explain why ideas and mathematical tools can cross between the fields.

For differential geometry, the answer points to Varadhan approximation and its use in work on the SABR model. It does not explain the approximation, derive a pricing formula, or give examples involving Lie groups or string theory. As a result, the material is a limited signpost rather than a full account of the three topics. Readers should treat the physics analogy as mathematical context, not evidence that financial models inherit the assumptions or interpretations of physical theories. The source provides no empirical evaluation or comparison of methods.

Key ideas

• Some mathematical finance problems use partial differential equations that also appear in physics.
• The Black–Scholes equation is described as belonging to a type of equation related to the Schrödinger equation.
• Differential geometry is connected to Varadhan approximation and applications involving the SABR model.
• The answer does not develop the roles of Lie groups or string theory, so its coverage is incomplete.

Tags

# Why Lie groups, differential geometry and string theory relate to MF?


# Why Lie groups, differential geometry and string theory relate to MF?












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?

## Answer by vanna (score 6, accepted)

https://quant.stackexchange.com/a/17976

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 :



- Differential geometry : link with Varadhan approximation (used for instance in Avellaneda's SABR)

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