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 information-based
asset pricing theory of Brody, Hughston and Macrina (BHM), in which so-called
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
information-based 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 convenience-dividend flow;
and (ii) a convenience-dividend process that provides information about current and
past dividend levels. The price of a commodity is given by the risk-neutral 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 Ornstein-Uhlenbeck 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.