This thesis consists of four papers, presented in Chapters 2-5, on the topics of derivatives pricing and importance sampling for stochastic processes.
In the first paper a model for the evolution of the forward density of the future value of an asset is proposed. The model is constructed with the aim of being both simple and realistic, and avoid the need for frequent re-calibration. The model is calibrated to liquid options on the S\&P 500 index and an empirical study illustrates that the model provides a good fit to option price data.
In the last three papers of this thesis efficient importance sampling algorithms are designed for computing rare-event probabilities in the setting of stochastic processes. The algorithms are based on subsolutions of partial differential equations of Hamilton-Jacobi type and the construction of appropriate subsolutions is facilitated by a minmax representation involving the \mane potential.
In the second paper, a general framework is provided for the case of one-dimensional diffusions driven by Brownian motion. An analytical formula for the \mane potential is provided and the performance of the algorithm is analyzed in detail for geometric Brownian motion and for the Cox-Ingersoll-Ross process. Depending on the choice of the parameters of the models, the importance sampling algorithm is either proven to be asymptotically optimal or its good performance is demonstrated in numerical investigations.
The third paper extends the results from the previous paper to the setting of high-dimensional stochastic processes. Using the method of characteristics, the partial differential equation for the \mane potential is rewritten as a system of ordinary differential equations which can be efficiently solved. The methodology is used to estimate loss probabilities of large portfolios in the Black-Scholes model and in the stochastic volatility model proposed by Heston. Numerical experiments indicate that the algorithm yields significant variance reduction when compared with standard Monte-Carlo simulation.
In the final paper, an importance sampling algorithm is proposed for computing the probability of voltage collapse in a power system. The power load is modeled by a high-dimensional stochastic process and the sought probability is formulated as an exit problem for the diffusion. A particular challenge is that the boundary of the domain cannot be characterized explicitly. Simulations for two power systems shows that the algorithm can be effectively implemented and provides a viable alternative to existing system risk indices.
The thesis begins with a historical review of mathematical finance, followed by an introduction to importance sampling for stochastic processes.
Stockholm: KTH Royal Institute of Technology, 2015. , viii, 28 p.