Mathematical Finance Applications
We hope to find new corporate sponsors in the mathematical finance and communication industries. Here, such data as a collection of stock prices or the number of jobs at various nodes of a network usually form the observation process, and long-range dependent or heavy-tailed models for observation processes are used for fidelity to reality. Thus, the long-range dependent processes are not Markov. However, as shown in a recent paper by M.L. Kleptsyna, A. Le Breton and M.C. Roubaud, entitled ?x201c;An elementary approach to filtering in systems with fractional Brownian observation noise?x201d;, there are still methods of constructing filters in certain long-range dependent observation settings.
Application of filtering and particle systems to mathematical finance is a growing area. Hence, it seems natural to apply our algorithms to this area in hope of attract an industry partner. In the volatility tracking problem we model and track stochastic volatility together with five unknown parameters using IDEX. In fact, due to the nature of the model we can compute the maximum likelihood parameters from the filter. Despite the large number of parameters to estimate, when applied with real stock data the model still performed very well. We published some of the results in Kouritzin, Remillard and C. Chan (2001) and believe that this technique could benefit an industry partner.
When pricing options using natural random interest rate and stock models, the future price must be brought back to the current time using a random weighting. Several books suggest constructing option prices using straight Monte Carlo methods. However, this random weighting means that a resampled particle system like SERP should outperform straight Monte Carlo pricing. We have applied SERP to a simple (single stock, bank account) model where the bank account interest rate depends on the stock price. A minor modification of SERP is then used to price all pathspace and terminal options simultaneously by the creation of a ``pricing measure''. The results were impressive, requiring far fewer particles than straight Monte Carlo. We look forward to testing SERP on the multi-stock scenario. In November and December, Jarett Hailes and Mike Kouritzin will meet with contacts from the finance sector in an attempt to secure a corporate partner and justify further work in this area.