A maximum principle for controlled stochastic factor model

In the present work, we consider an optimal control for a three-factor stochastic factor model. We assume that one of the factors is not observed and use classical filtering technique to transform the partial observation control problem for stochastic differential equation (SDE) to a full observation control problem for stochastic partial differential equation (SPDE). We then give a sufficient maximum principle for a system of controlled SDEs and degenerate SPDE. We also derive an equivalent stochastic maximum principle. We apply the obtained results to study a pricing and hedging problem of a commodity derivative at a given location, when the convenience yield is not observable.</jats:p

Some contributions to filtering theory with applications in financial modelling

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Two main groups of filtering algorithms are characterised and developed. Their applicability is demonstrated using actuarial and financial time series data. The first group of algorithms involved hidden Markov models (HMM), where the parameters of an asset price model switch between regimes in accordance with the dynamics of a Markov chain. We start with the known HMM filtering set-up and extend the framework to the case where the drift and volatility have independent probabilistic behaviour. In addition, a non-normal noise term is considered and recursive formulae in the online re-estimation of model parameters are derived for the case of students’ t-distributed noise. Change of reference probability is employed in the construction of the filters. Both extensions are then tested on financial and actuarial data. The second group of filtering algorithms deals with sigma point filtering techniques. We propose a method to generate sigma points from symmetric multivariate distributions. The algorithm matches the first three moments exactly and the fourth moment approximately; this minimises the worst case mismatch using a semidefinite programming approach. The sigma point generation procedure is in turn applied to construct algorithms in the latent state estimation of nonlinear time series models; a numerical demonstration of the procedure’s effectiveness is given. Finally, we propose a partially linearised sigma point filter, which is an alternative technique for the optimal state estimation of a wide class of nonlinear time series models. In particular, sigma points are employed for generating samples of possible state values and then a linear programming-based procedure is utilised in the update step of the state simulation. The performance of the filtering technique is then assessed on simulated, highly non-linear multivariate interest rate process and is shown to perform significantly better than the extended Kalman filter in terms of computational time

Valuation of Swing Options in Electricity Commodity Markets

Although electricity is considered to be a commodity, its price behavior is remarkably different from most other commodities or assets on the market. Since power can hardly be stored physically, the storage-based methodology, which is widely used for valuing commodity derivatives, is unsuitable for electricity. Therefore, new approaches are required to understand and reproduce its price dynamics. Concurrently, the demand for derivative instruments has grown and new types of contracts for energy markets have been introduced. Swing options, in particular, have attracted an increasing interest, offering more flexibility and reducing exposure to strong price fluctuations. In this thesis, we propose a mean-reverting model with seasonality and double exponential jumps. It is able to accurately reproduce the behavior and main peculiarities of electricity's spot prices. With this model, we can characterize the swing option value as a solution to a partial integro-differential complementarity problem, which we solve numerically. In the last part of the thesis, we present a more complex type of swing options, in which we also include variable electricity volumes in the contract. This formulation leads to a two-dimensional Hamilton-Jacobi-Bellman (HJB) equation. By applying the method of characteristics, this problem is simplified to a sequence of one dimensional HJB equations, which are solved numerically by using a similar approach as before