Financial Mathematics and Applied
Probability Seminars 20052006
Unless otherwise indicated, all seminars take place at Lecture Theatre 2C,
King's College London, The Strand, London WC2R 2LS.
Tuesday 11 October, 5:30 pm 
Professor Nizar Touzi
Tanaka Business School, Imperial College London
Optimal investment under capital gains taxes:
an asymptotic expansion result
Abstract: We formulate a model of a continuous time financial market
consisting of a bank account and one risky asset subject to
transaction cost and capital gains taxes.
We consider the problem of maximising expected utility
of future consumption in infinite horizon.

Tuesday 18 October, 5:30 pm

Dr Diane Wilcox
Department of Mathematics, University of Cape Town
On the estimation of crosscorrelations
Abstract:
Covariance matrices have enjoyed a prominent place in finance and
risk management since Markowitz's meanvariance optimisation
method introduced in 1952. As risk measures these objects have
since been superseded by VAR, copulas and coherent risk measures.
Nevertheless, the estimation of crosscorrelation remains a
ubiquitous component in derivatives pricing, portfolio
optimisation and asset allocation problems.
Despite the fundamental simplicity of the problem, covariance
matrix estimation is beset with numerous subtleties which are
addressed with a broad range of methods. We review some techniques
with special attention to the application of Random Matrix Theory
to the analysis of crosscorrelations in South African market
data.

Tuesday 25 October, 5:30 pm 
Dr Erik Ekström
Department of Mathematics, University of Manchester
Properties of option prices in a
jumpdiffusion model
Abstract:
It was shown by Bergman, Grundy and Wiener (1996) that the
value of a convex claim in a diffusion model is increasing
as a function of the volatility of the underlying stock.
A main tool in the proof of this is the preservation of convexity
for such models. In this talk, we provide a sufficient condition
for the preservation of convexity in jumpdiffusion models.
This enables us to derive monotonicity properties of the option
value with respect to different parameters of the model, such as
the volatility, the jump size and the jump intensity. The analysis
is based on investigations of solutions to certain
integrodifferential equations. This is joint work with Johan Tysk.

Tuesday 1 November, 5:30 pm 
Professor William Perraudin
Tanaka Business School, Imperial College London
Real Options with Mixed Strategies
Abstract: Firms which possess real options to cease production may act
strategically if they acquire increased market power when a competitor
quits. This paper analyzes the equilibria that arise in a simple real
options model when firms are engaged in such a war of attrition game. We
show that firms may adopt randomized strategies in which exits are
generated by conditionally Poisson jump processes. We generalize the model
with incomplete information regarding flow costs and show that as the
number of cost types increase the scope for randomized exit decreases.
Surprisingly, in the limit of a continuum of types the firms revert to
their monopoly exit triggers.

Tuesday 8 November, 5:30 pm 
(No Seminar this day)

Tuesday 15 November, 5:30 pm 
Dr Andreas Kyprianou
Department of Actuarial Mathematics and Statistics, HeriotWatt University, Edinburgh
Reflected stable processes
Abstract: We use fluctuation theory and give a new result showing how
to establish the exact distribution of the overshoot over a fixed level of
any strictly stable process when reflected in its infimum (specifically
the case when the Lévy measure is supported on R is included in the
discussion).

Tuesday 22 November, 5:30 pm 
Dr Alexander Cox
Department of Mathematics, University of York
Local Martingales, Bubbles and Option Prices
Abstract: We are interested in option pricing in markets with
bubbles. A bubble is defined to be a price process which, when discounted,
is a local martingale under the riskneutral measure but not a martingale.
In a market with a bubble many standard results from the folklore become
false. Putcall parity fails, the price of an American call exceeds that
of a European call and prices are no longer convex in the underlying. We
show how these results must be modified in the presence of a bubble. It
turns out that the option value depends critically on the definition of
admissible strategy, and that the standard mathematical definition may not
be consistent with the definitions used for trading.

Tuesday 29 November,
2:30 pm room 436

Professor Eugene A. Feinberg
Department of Applied Mathematics & Statistics,
State University of New York at Stony Brook
Optimality of NonRandomized Policies for Certain Stochastic
Systems with Multiple Criteria and Constraints
Abstract: As is wellknown, randomized policies
may outperform nonrandomized
policies for problems with multiple criteria and constraints.
However, in some cases nonrandomized policies may be optimal.
For example, for recurrent finite state and action space Markov
Decision Processes (MDPs) with average rewards per unit time,
the timesharing policies described in the papers by Ross,
Altman, and Shwartz are optimal.
We discuss timesharing and two other situations when
nonrandomized policies are optimal for constrained problems:
continuous time MDPs and discretetime nonatomic MDPs. In each
case we describe the corresponding applications. In particular,
we apply timesharing to radar sensor management. For continuoustime
MDPs, we discuss queueing control and power management, and for nonatomic
MDPs we discuss the link between nonatomic MDPs and the work of Dvoretzky,
Wald, and Wolfowitz on nonrandomized statistical decisions. Moreover, the
results on nonatomic problems imply the optimality of nonrandomized policies
for certain classes of inventory control and financial engineering problems
with multiple objectives and constraints.

Tuesday 6 December,
5:30 pm

Dr Dorje C. Brody
Blackett laboratory, Imperial College, London
An InformationBased Approach to Asset Pricing
Abstract:
A new framework for asset price dynamics is introduced where the concept
of noisy information about future cashflows is used to derive the
corresponding price processes. In this framework an asset is defined by
its cashflow structure. Each cash flow is modelled by a random variable
that can be expressed as a function of a collection of independent random
variables called market factors. The cash flows and market factors are for
simplicity taken to be continuous random variables here, but more general
situations can be incorporated as well, depending on the context. With
each such market factor we associate a socalled market information
process, the values of which we assume are accessible to market
participants. Each market information process consists of a sum of two
terms, one of which contains true information about the value of the
associated market factor, and the other of which contains noise. The noise
term is modelled by an independent Brownian bridge process that spans the
time interval from the present to the time at which the value of the given
market factor is revealed. The market filtration is assumed to be that
generated by the aggregate of the independent market information
processes. For simplicity we assume that interest rates are deterministic,
and that the risk neutral measure has been specified. The price of an
asset is given by the conditional expectation of the discounted cashflows
in this measure, where the conditional expectation is that arising from
the market filtration constructed as indicated. In the case where the cash
flows are the random dividend payments associated with equities, an
explicit model is thereby obtained for the shareprice process. Dividend
growth is taken into account by introducing appropriate structure on the
market factors associated with the dividends, and various dividend growth
models can be considered. The prices of options on dividendpaying assets
are derived and, remarkably, the form of the price process of a
Europeanstyle call option is of the BlackScholes type. For exponential
and gammadistributed dividend payments a closedform expression for the
shareprice process is obtained, and a semianalytical formula is computed
for the value of a European call option. We consider both the case where
the rate at which information is revealed to the market is constant, as
well as the more general case where the information flow rate varies in
time. The latter case is developed in some detail in this paper. The
framework has another significant feature: it generates a natural family
of stochastic volatility models without the need for specifying on an ad
hoc basis the stochastic dynamics of the volatility. (Work carried out in
collaboration with L. P. Hughston and A. Macrina, Department of
Mathematics, King's College London.)

Tuesday 17 January, 5:30 pm 
Dr Lutz Schloegl
Fixed Income Quantitative Research,
Lehman Brothers, London
Stochastic Recovery Rates and NoArbitrage in the Tranche Markets

Tuesday 24 January, 5:30 pm 
(No seminar this day)

Tuesday 31 January, 5:30 pm 
Dr Wim Schoutens
Department of Mathematics, Catholic University of Leuven
JumpDriven Intensity Models for Credit Risk Modeling
Abstract:
We overview some tractable and popular intensity models in a credit risk
setting.
Next, we introduce intensity models driven by jumps. In these, socalled
OrnsteinUhlenbeck (OU) models,
the intensity is a stationary process driven by a pure jump Levy process
(subordinator). We focus on the
GammaOU and Inverse GaussianOU cases, where closedform formulas for
the default probability are available.
We make a calibration exercise of the models considered on a whole range
of CDS term structures
and compare there fitting abilities. Next, we focus on pricing issues.
As an exercise, we investigate
the model risk by pricing a digital default put.
This is joint work with Jessica Cariboni.

Tuesday 7 February, 5:30 pm 
Dr A. B. Piunovskiy
Department of Mathematical Sciences,
University of Liverpool
Pareto sets for multiple objective Markov Decision Processes
Abstract:
First of all, I intend to remind several properties of polyhedral cones and conegenerated orders which will be used for constructing Pareto sets in multiple objective optimisation problems. Afterwards, I will consider multiple objective
discounted Markov Decision Process. Methods of Convex Analysis and the Dynamic Programming Approach allow to construct the Pareto sets and study their properties. For instance, I will show that in the unichain case, Pareto sets for different initial distributions are topologically equivalent. Finally, I will present an example on the optimal management of a deteriorating system.

Tuesday 14 February, 5:30 pm 
Dr Nick Webber
Warwick Business School
Valuing American options on a lattice
Abstract:
American style options are of considerable importance in the financial markets
but in general no explicit solutions for their value exist. Typically
convergence to the true option value is slow resulting in inaccurate prices. In
this paper we describe a valuation method for American options using a novel
lattice method. Although convergence is not dramatically improved, the
method opens a fresh approach that may have repercussions elsewhere.

Tuesday 21 February, 5:30 pm 
Dr Sam Howison
Mathematical Institute, Oxford
Old wine in new bottles: an asymptotic expansions approach to
discretelysampled Brownian Motion and the BGK correction for barrier and
Bermudan options.

Tuesday 28 February,
4:15 pm 
Professor T. R. Hurd
Deparment of Mathematics and Statistics,
McMaster University, Canada
Indifference pricing of variance swaps and other derivatives
in stochastic volatility models
Abstract: Utility based indifference pricing is now considered to be
the economically natural method for valuing contingent claims in
incomplete markets. However, acceptance of this concept by the wide
financial community has been severely delayed by the computational
and conceptual difficulty of the approach. In this talk I will focus
on the problem of computing indifference prices for derivative
securities in incomplete stochastic volatility markets. As an
alternative to the now standard approach by partial differential
equations, I present a new approach to indifference pricing which
leads to similar results by identifying the natural martingales in
the model. The resulting nonlinear FeynmanKac representations show a
striking connection to formulas from interest rate theory.
Capitalizing on this we are able to provide closed form solutions for
the indifference price of a variance swap in both the standard Heston
model and a new model we call the ``reciprocal Heston" model.
Indifference pricing and hedging for general European style claims written
on integrated variance can be efficiently computed by applying the fast
Fourier transform to the above formulas. To the best of my knowledge,
these are the first known nontrivial closed formulas for the indifference
price of a widely traded class of derivatives.
Joint work with Matheus Grasselli, McMaster, Canada
 Tuesday 28 February,
5:45 pm 
Dr Saul Jacka
Department of Mathematics, University of Warwick
Stochastic Representation and Markets with Costs
Abstract: The relation between stochastic integration, martingale
representation and complete markets is now wellknown. This talk will
discuss (in a discrete time context) some of the analagous results available
to connect markets with transaction costs, stochastic control, coherent risk
measures and forms of stochastic representation

Tuesday 7 March, 5:30 pm 
(No seminar this day)

Tuesday 14 March, 5:30 pm 
Dr Angelos Dassios
Department of Statistics, London School of Economics
Quantiles of Levy processes and applications in
finance
Abstract:
We will present a survey of results on the quantiles of a Brownian
motion with drift as well as a general Levy process. The motivation is to
calculate the price of related financial options. At the end of the talk
some new results on variability orderings between various quantities
associated with path dependent and European options are presented. This
survey is not exhaustive, but intends to provide a flavour of research
carried out in the area.

Tuesday 21 March, 5:30 pm 
(No seminar this day)

Tuesday 28 March, 5:30 pm 
Dr Christopher J. Hunter
BNP Paribas
Aspects of Correlation in the Pricing of Hybrid Derivatives

Tuesday 16 May, 5:30 pm 
Dr Xin Guo
Cornell University
Credit Risk With Incomplete Information: A Unified
Probabilistic Approach
Given the wellknown drawbacks in the two leading
paradigms (the structural models and the reducedform models),
informationbased credit risk study has gained its popularity,
starting from the important paper by Duffie and Lando (2001).
However, there are a number of fundamental issues with this
approach. The first is the mathematical ambiguity about incomplete
information such as the noisy information and the delayed
information; The second is the inconsistency of different
filtration expansion approaches: the minimal filtration expansion
in Duffie and Lando (2001) and Lando (1998), and the progressive
filtration expansion in Elliott, Monique and Yor (2000),
B\'{e}langer, Shreve, and Wong (2004); And the last issue is the
inconsistency of intensity process in finance and mathematics: in
the former it is the instantaneous likelihood of default while in
the latter the RadonNikodyn derivative of a compensator.
In this talk, we shall show how these issues are resolved in a
unified filtration framework. More importantly, this framework
allows us to extend the simple pricing scheme in reducedform
models to structural models, without however the conditional
independence assumptions.

Tuesday 6 June, 5:30 pm 
Dr Martin Baxter
Nomura
Dynamic modelling of singlename credits and CDO tranches
This talk will present a new family of models of the evolution of each
credit within a portfolio basket. The models aim to have dynamics which
are intuitive, which capture the heavy tails of credit distributions;
and which have a correlation structure consistent with CDO market
prices. From a practical point of view, it is also important that
the models are tractable to implement.The model is based on general Levy
processes, which are both
mathematically interesting and well suited to the problem.


