Adaptive radial basis functions for option pricing

In this thesis, we have developed meshless adaptive radial basis functions (RBFs) method for the pricing of financial contracts by solving the Black-Scholes partial differential equation (PDE). In the 1-D problem, we priced the financial contracts of a European call option, Greeks (Delta, Gamma and Vega), an American put option and a barrier up and out call option with this method. In the BENCHOP project with Challenge Parameter Set (Parameter Set 2) [97], we have shown that our adaptive method is highly accurate and with less computational cost in comparison with the finite difference method for the European call option and barrier up and out call option. And also we have presented the numerical result of the equally spaced RBF method for both Parameter Set 1 and 2. In our numerical simulations with Parameter Set 2, we note that our adaptive method is more accurate and faster than the equally spaced RBF method. For example for the barrier up and out call option, the equally spaced method (MQ) with 3000 uniform nodes has the maximum error of 1.30e-02 at three evaluation points, but our adaptive method (101 nodes) has maximum error of 9.98e-05 at the same three points. This is about 100 times better than the equally spaced method with about 30 times less CPU time. Since our adaptive strategy is accurate and efficient, we substantially increase the accuracy with fewer number of nodes. We also developed an adaptive algorithm for the 2 assets Black-Scholes problem, in this algorithm we used the rectangular Voronoi points for the refinement, and the thin plate spline is used for the local approximation in order to assess the error. The numerical results of pricing a Margrabe call option are presented for both adaptive and non-adaptive methods. The adaptive method is more accurate and requires fewer nodes when compared to the equally spaced RBF method.