95 research outputs found
Information-Based Models for Finance and Insurance
In financial markets, the information that traders have about an asset is reflected in its
price. The arrival of new information then leads to price changes. The ‘information-based
framework’ of Brody, Hughston and Macrina (BHM) isolates the emergence of
information, and examines its role as a driver of price dynamics. This approach has
led to the development of new models that capture a broad range of price behaviour.
This thesis extends the work of BHM by introducing a wider class of processes for the
generation of the market filtration. In the BHM framework, each asset is associated
with a collection of random cash flows. The asset price is the sum of the discounted
expectations of the cash flows. Expectations are taken with respect (i) an appropriate
measure, and (ii) the filtration generated by a set of so-called information processes that
carry noisy or imperfect market information about the cash flows. To model the flow
of information, we introduce a class of processes termed Levy random bridges (LRBs),
generalising the Brownian and gamma information processes of BHM. Conditioned on
its terminal value, an LRB is identical in law to a Levy bridge. We consider in detail
the case where the asset generates a single cash flow XT at a fixed date T. The flow
of information about XT is modelled by an LRB with random terminal value XT.
An explicit expression for the price process is found by working out the discounted
conditional expectation of XT with respect to the natural filtration of the LRB. New
models are constructed using information processes related to the Poisson process, the
Cauchy process, the stable-1/2 subordinator, the variance-gamma process, and the
normal inverse-Gaussian process. These are applied to the valuation of credit-risky
bonds, vanilla and exotic options, and non-life insurance liabilities
Archimedean Survival Processes
Archimedean copulas are popular in the world of multivariate modelling as a
result of their breadth, tractability, and flexibility. A. J. McNeil and J.
Ne\v{s}lehov\'a (2009) showed that the class of Archimedean copulas coincides
with the class of multivariate -norm symmetric distributions. Building
upon their results, we introduce a class of multivariate Markov processes that
we call `Archimedean survival processes' (ASPs). An ASP is defined over a
finite time interval, is equivalent in law to a multivariate gamma process, and
its terminal value has an Archimedean survival copula. There exists a bijection
from the class of ASPs to the class of Archimedean copulas. We provide various
characterisations of ASPs, and a generalisation
Modulated Information Flows in Financial Markets
We model continuous-time information flows generated by a number of
information sources that switch on and off at random times. By modulating a
multi-dimensional L\'evy random bridge over a random point field, our framework
relates the discovery of relevant new information sources to jumps in
conditional expectation martingales. In the canonical Brownian random bridge
case, we show that the underlying measure-valued process follows jump-diffusion
dynamics, where the jumps are governed by information switches. The dynamic
representation gives rise to a set of stochastically-linked Brownian motions on
random time intervals that capture evolving information states, as well as to a
state-dependent stochastic volatility evolution with jumps. The nature of
information flows usually exhibits complex behaviour, however, we maintain
analytic tractability by introducing what we term the effective and
complementary information processes, which dynamically incorporate active and
inactive information, respectively. As an application, we price a financial
vanilla option, which we prove is expressed by a weighted sum of option values
based on the possible state configurations at expiry. This result may be viewed
as an information-based analogue of Merton's option price, but where
jump-diffusion arises endogenously. The proposed information flows also lend
themselves to the quantification of asymmetric informational advantage among
competitive agents, a feature we analyse by notions of information geometry.Comment: 27 pages, 1 figur
Stable-1/2 Bridges and Insurance
We develop a class of non-life reserving models using a stable-1/2 random
bridge to simulate the accumulation of paid claims, allowing for an essentially
arbitrary choice of a priori distribution for the ultimate loss. Taking an
information-based approach to the reserving problem, we derive the process of
the conditional distribution of the ultimate loss. The "best-estimate ultimate
loss process" is given by the conditional expectation of the ultimate loss. We
derive explicit expressions for the best-estimate ultimate loss process, and
for expected recoveries arising from aggregate excess-of-loss reinsurance
treaties. Use of a deterministic time change allows for the matching of any
initial (increasing) development pattern for the paid claims. We show that
these methods are well-suited to the modelling of claims where there is a
non-trivial probability of catastrophic loss. The generalized inverse-Gaussian
(GIG) distribution is shown to be a natural choice for the a priori ultimate
loss distribution. For particular GIG parameter choices, the best-estimate
ultimate loss process can be written as a rational function of the paid-claims
process. We extend the model to include a second paid-claims process, and allow
the two processes to be dependent. The results obtained can be applied to the
modelling of multiple lines of business or multiple origin years. The
multi-dimensional model has the property that the dimensionality of
calculations remains low, regardless of the number of paid-claims processes. An
algorithm is provided for the simulation of the paid-claims processes.Comment: To appear in: Advances in Mathematics of Finance (A. Palczewski and
L. Stettner, editors.), Banach Center Publications, Polish Academy of
Science, Institute of Mathematic
On the stability of self-gravitating accreting flows
Analytic methods show stability of the stationary accretion of test fluids
but they are inconclusive in the case of self-gravitating stationary flows. We
investigate numerically stability of those stationary flows onto compact
objects that are transonic and rich in gas. In all studied examples solutions
appear stable. Numerical investigation suggests also that the analogy between
sonic and event horizons holds for small perturbations of compact support but
fails in the case of finite perturbations.Comment: 10 pages, accepted for publication in PR
Levy Random Bridges and the Modelling of Financial Information
The information-based asset-pricing framework of Brody, Hughston and Macrina
(BHM) is extended to include a wider class of models for market information. In
the BHM framework, each asset is associated with a collection of random cash
flows. The price of the asset is the sum of the discounted conditional
expectations of the cash flows. The conditional expectations are taken with
respect to a filtration generated by a set of "information processes". The
information processes carry imperfect information about the cash flows. To
model the flow of information, we introduce in this paper a class of processes
which we term Levy random bridges (LRBs). This class generalises the Brownian
bridge and gamma bridge information processes considered by BHM. An LRB is
defined over a finite time horizon. Conditioned on its terminal value, an LRB
is identical in law to a Levy bridge. We consider in detail the case where the
asset generates a single cash flow occurring at a fixed date . The
flow of market information about is modelled by an LRB terminating at the
date with the property that the (random) terminal value of the LRB is equal
to . An explicit expression for the price process of such an asset is
found by working out the discounted conditional expectation of with
respect to the natural filtration of the LRB. The prices of European options on
such an asset are calculated
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