Differential Geometry Techniques in the Black-scholes Option Pricing; Theoretical Results and Approximations

Black-Scholes model for the basket options is used to valuate S & P 500, DAX and other Stock market index options. We explain the lack of closed formulas for the multi-asset European call options, using a differential geometric approach initiated by Labordere and V. Linetski. Our theoretical results can be tested on some extensions of the SABR model, Heston's and Bergomi stochastic volatility models, usually approached using Monte Carlo or numerical partial differential equation simulations.