MPDATA meets Black‐Scholes
We explore a simple yet robust numerical method for integrating PDEs applied to pricing of options (financial instruments). We use the the non-oscillatory forward-in-time second-order MPDATA finite-difference scheme, which originates from geophysical fluid dynamics. The valuation methodology involves casting the Black-Merton-Scholes equation as a transport problem by first transforming it into a homogeneous advection-diffusion PDE via variable substitution, and then expressing the diffusion term as an advective flux using the pseudo-velocity technique. As a result, all terms of the Black-Merton-Sholes equation are consistently represented using a single high-order numerical scheme for the advection operator. So far, we have demonstrated the methodology for European-, American- and Asian-style contracts.
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Paweł Magnuszewski
