In Return of the Lumberjacks, I have outlined that Andreas Binder has introduced Adaptive Integration at a workshop that turned out as our entry into the quant finance world.
Adaptive Integration has two characteristic ingredients:
- Use high-order integration schemes for the calculation of the integrals arising from the application of Geen's functions to the PDEs of computational finance
- Use adaptive gridding schemes for the underlying as well as for time to make sure that fine grids are introduced only where necessary
This works not only in the generalized Black-Scholes world, but was easily transferred to the valuation of (one factor) interest rate deal types.
Green's functions are only available in intervals (subdomains). So, with domain decomposition and recomposition we apply an asymptotic mathematics approach for the total solution of the non trivial, realistic PDEs.
Adaptive Integration is still the optimal method for the valuation of a variety of moderately complex instruments. Blazingly fast, accurate and robust.