Abigail Mellor - Asymptotic analysis and efficient numerical solution of the Fokker-Planck Equations (with application to derivatives pricing)
| Dates: | 3 May 2023 |
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| Times: | 12:00 - 13:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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Abigail Mellor (University of Manchester) will speak in the Probability seminar. (in-person)
The Fokker-Planck Equations (FPEs) are ubiquitous in numerous disciplines, not least in financial mathematics, where a popular technique for pricing derivatives is enabled through the derivation of the density function via the FPE governing its evolution. Accuracy and efficiency are both keys in derivatives pricing, but satisfying both is a challenge due to the delicate mathematical structure often encountered. Our work currently focuses on pricing derivatives with one and two assets following generalised Ornstein-Uhlenbeck processes. In such scenarios, the elasticity of variance parameter(s) of the process(es) may induce a degenerate region at the origin (for instance, when this parameter is fractional). Here, the solution structure can be intricate, and standard computational methods fail as they violate the necessary ‘particle-conserving’ condition. We present rigorous asymptotic analyses to shed light on the aforementioned intricate/singular mathematical structures which render standard numerical techniques unfeasible. By incorporating these structures into our numerical schemes, while ensuring that the particle-conservation condition is fully satisfied, we obtain robust numerical solutions (as confirmed by extensive numerical-scheme experimentation) with modest computational facilities.
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Maurice Priestley Room G.108
Alan Turing Building
Manchester
