Numerical Path Integration Applied to Options under the Normal Inverse Gaussian Market Model: Pricing path dependent options on Nord Pool electricity forwards when assuming normal inverse Gaussian dynamics by numerical path integration

Abstract

In citeasnoun{SkaugNaess2007} Path integration (PI) was used on discretely monitored barriers options with a log normal return. Compared to the trinomial three used in citeasnoun{BroadieEtAl1997} it is an order of magnitude faster. However, based on the analytical price of continuously monitored barrier options the same article derived an approximation by adjusting the barriers to compensate for the discrete monitoring. This allows for nearly instant pricing and the PI method is slow in comparison but it is not as accurate and when high precision pricing is needed, PI is the best choice. The situation is different for the normal inverse Gaussian (NIG) market model. No fast analytical approximation exist making the PI method not only a good choice for fast and accurate pricing, but for fast pricing in general cite{Naess2009}. An even faster implementation was provided by citeasnoun{Saebo2009} using interpolation and a non-uniform grid. Here the two implementations are analyzed and compared along with a third choice of grid: interpolation on a uniform grid. To allow for this comparison a large sub section is devoted to discussions regarding the accuracy of different implementations. No framework exist for the accuracy of PI so a heuristic approach is taken. Defining the error of a simulation as the mean distance between the exact density after one step and the density represented with the grid, an indication of how well the grid represents the density is obtained. It does not say anything about the accuracy of the end result and it is not elegant but it provides a rule of thumb for how to chose grids. This is a huge improvement over the approach taken in the literature where grid choices are made with trial and error knowing the price. Trial and error results in faster simulations than using the rule of thumb but testing a range of grids also takes time. In addition, a numerical method is not very useful if it only prices products where the price is already known. Lookback options were first priced with PI in citeasnoun{Saebo2009} and the results are reproduced here together with a few tricks which speeds up the implementation. Regardless, the calculation time is more than ten times higher than for barrier options which makes it about as fast as Monte Carlo (MC) simulations. This makes PI applied to lookback options under the NIG market model a subject of theoretical interest unless further improvements to calculation speed are obtained. Asian options under NIG dynamics are also priced in this thesis, which is a novel application of the PI method. As with the lookback option the implementation is too slow to be of practical interest. If it is possible to reduce the CPU time enough to make PI a good chose for fast accurate pricing is uncertain, but this thesis is a proof of concept .