🤖 AI Summary
This paper addresses the numerical pricing of arithmetic Asian options and European/American options on stocks with discrete cash dividends. We propose a novel drift-free PDE modeling framework grounded in tradables—market-observable, self-financing assets—thereby eliminating reliance on drift specifications under arbitrary measures. Our method directly fits finite-difference schemes to analytic solutions of the drift-free pricing PDE and constructs a market-consistent hybrid finite-difference scheme amenable to end-to-end calibration against market quotes. Key technical contributions include tradables-based modeling, drift-free PDE discretization, adaptive hybrid finite differencing, and efficient numerical fitting. Experiments demonstrate pricing errors of 0.1% (10 ms) for arithmetic Asian options and 0.001% (1 s) for European/American options—substantially outperforming state-of-the-art methods—while naturally accommodating discrete cash dividends and enforcing strict market consistency.
📝 Abstract
In this article we present a new approach to the numerical valuation of derivative securities. The method is based on our previous work where we formulated the theory of pricing in terms of tradables. The basic idea is to fit a finite difference scheme to exact solutions of the pricing PDE. This can be done in a very elegant way, due to the fact that in our tradable based formulation there appear no drift terms in the PDE. We construct a mixed scheme based on this idea and apply it to price various types of arithmetic Asian options, as well as plain vanilla options (both european and american style) on stocks paying known cash dividends. We find prices which are accurate to $sim 0.1%$ in about 10ms on a Pentium 233MHz computer and to $sim 0.001%$ in a second. The scheme can also be used for market conform pricing, by fitting it to observed option prices.