Title:

Informationbased commodity pricing and the theory of signal processing with Levy information

In mathematical finance, increasing attention is being paid to (a) the construction of explicit models for the flow of market information, and (b) the use of such models as a basis for asset pricing. One notable approach in this spirit is the informationbased asset pricing theory of Brody, Hughston and Macrina (BHM), in which socalled information processes are introduced and ingeniously integrated into the general theory of asset pricing. Building on the BHM theory, this thesis presents a number of new developments in this area. I begin with a brief review of the BHM framework, leading to a discussion of the simplest asset pricing models. Then the first main topic of the thesis, which is based in part on Brody, Hughston & Yang (2013b), is developed, which concerns asset pricing with continuous cash flows in the presence of noisy information. In particular, an informationbased model for the pricing of storable commodities and associated derivatives thereof is introduced. The model employs the concept of market information about future supply and demand as a basis for valuation. Physical ownership of a commodity is regarded as providing the beneficiary with a continuous "convenience dividend", equivalent to a continuous cash flow. The market filtration is assumed to be generated jointly by: (i) an information process concerning the future conveniencedividend flow; and (ii) a conveniencedividend process that provides information about current and past dividend levels. The price of a commodity is given by the riskneutral expectation of the cumulative future convenience dividends, suitably discounted, conditional on the information provided by the market filtration. In the situation where the convenience dividend is modelled by an OrnsteinUhlenbeck process, the prices of options on commodities, both when the underlying is a spot price and when the underlying is a futures price, can be derived in closed form. The dynamical equation of the price process is worked out, leading to an identification of the associated innovations process. The resulting model is sufficiently tractable to allow for simulation studies of the resulting commodity price trajectories. The second main topic of the thesis, which is based in part on Brody, Hughston & Yang (2013a), concerns a generalisation of concept of information process to the situation where the noise is modelled by a general Levy process. There are many practical circumstances in which signal or noise, or both, exhibit discontinuities. This part of the thesis develops a rather general theory of signal processing involving Levy noise, with the view to the introduction of a broad and tractable family of information processes suitable for modelling situations involving discontinuous signals, discontinuous noise, and discontinuous information. In this context, each information process is associated with a certain "noise type", and an information process of a given noise type is distinguished by the message that it carries. More specifically, each information process is associated in a precise way to a Levy process, which I call the fiducial process. The fiducial process is the information process that results in the case of a null message, and can be regarded as a "pure noise" process of the given noise type. Information processes can be classified by the characteristics of the associated fiducial processes. To keep the discussion simple, I mainly consider the case where the message is represented by a single random variable. I construct the optimal filter in the form of a map that takes the a priori distribution of the message to an a posteriori distribution that depends on the information made available. A number of examples are presented and worked out in detail. The results vary remarkably in detail and character for the different types of information processes considered, and yet there is an overall unity in the scheme that allows for the construction of numerous explicit and interesting examples. The results presented in the second part of the thesis therefore have the potential to pave the way toward a variety of new applications, including applications to problems in finance.
