# VPP Pricing V: Least-Squares Monte-Carlo

For the sake of completeness please find here the code for the evaluation of the virtual power plant (VPP) using a least-squares Monte-Carlo algorithm. The code depends on the latest QuantLib version from the SVN trunk or the upcoming QuantLib 1.2 release. The model and power plant specifications can be found in the previous blog entries. A more general description  of the problem and the algorithms can be found e.g. here [1]. Test forward curves can be taken e.g. from the Kyos example download page.

The regression polynomials are of third order in the spark spread $x$ and the stochastic component of the gas price $y$.

$p_a(x,y)=a_0 + a_1 x + a_2 x^2 +a_3x^3 + a_4y + a_5y^2 + a_6y^3 + a_7xy+a_8x^2y+a_9xy^2$

The regression is carried out for every exercise right (every hour) and every possible VPP state separately. The calibration phase is based on ordinary Monte-Carlo scenarios, whereas the pricing is done using Quasi Monte-Carlo scenarios (Sobol sequence) and a Brownian Bridge (BB).

The following table summarizes the performance of the different pricing algorithms for the example contract and maturity of six month. Target accuracy is around 1% relative error in the NPV. The timings are given for a Core i5@3GHz CPU using four threads or a GTX560@0.8/1.6GHz GPU with 336 cores.

$\footnotesize{ \begin{tabular}{|c|c|c|c|c|c|} \hline Algorithm & Optimisaton & Approximation & Hardware & Runtime & Comment\\ \hline \hline Quasi-MC + BB & dyn. prog. & perfect foresight & GPU & 0.19s & single precision \\ Quasi-MC + BB & dyn. prog. & perfect foresight & CPU & 20.3s & \\ Monte-Carlo & dyn. prog. & perfect foresight & CPU & 286.3s & \\ Finite Difference & dyn. prog.& no & CPU & 487.7s & \\ Least-Squares MC & dyn. prog. & no & CPU & 645.6s & \\ Quasi-MC + BB & linear prog. & perfect foresight & CPU & 4198s & using GLPK \\ \hline \end{tabular} }$

I don’t know the reason for the bad performance of the Gnu Linear Programming Kit for these kind of problems. Some commercial linear optimizer are much faster but they can not compete with dynamic programming for a simple VPP. As soon as e.g. time integral constraints are involved linear programming might become the method of choice.

[1] H. van Dijken, The value of starting up the power plant.