Proposal: 1
Supervisor: Supratik Roy
Title: Nash competitive equilibria in asset markets
Description: CAPM assumes that all market participants take prices as given. In practice, market data
appear to support this assumption for small investors, but not for large investors like mutual funds
and hedge funds whose transactions have an impact on prices. Hens etal combine price taking and
strategic behavior in a simple asset market model with multiple assets, with the competitive part
based on Arrow and Debreu (1954), and the strategic part based on the strategic market game
(SMG)[Dubey, Shapley and Shubik].
In this approach, a two period model is considered with a finite number of states in the second
period. All decisions are taken in the first period and a finite number of investors are endowed with
wealth that can be spent on first period consumption and on a finite number of assets (bonds and
shares) delivering state contingent payoffs in the second period. Asset payoffs are assumed to be the
only source to finance second period consumption, and both competitively and strategically acting
investors are simultaneously allowed on the asset market. Competitive investors take prices as
given, while strategic investors choices incorporate the market impact of their demand. It is shown
that if two-period fund separation holds, the first-order conditions of the strategic and the
competitive investor only differ by a scalar connected to the existence of a Nash equilibrium in the
SMG. However, when derivatives are considered, strategic behavior is found to differ from
competitive behavior even in those cases where two-period fund separation holds. Constant relative
risk aversion (CRRA) and no-aggregate risk (NAR) are found to be sufficient to ensure two-period
fund separation.
The project involves looking at one or more aspects of the following (a) extending the model to more
than two periods (b) weakening the assumptions of CRRA and/or NAR (c) endogenize wealth by
giving agents endowments in the form of assets. Any other modifications of the various assumptions
or conditions under which the model has been analyzed in the paper, can be suggested for
exploration by the student.
Hens,T, Reimann, S, and Vogt, B. Nash competitive equilibria and two-period fund separation,
Journal of Mathematical Economics, 40 (2004), 321-346. [The paper should be available through UCC
library online]
Proposal: 2
Zero beta model estimation.
Supervisor: Bernard Hanzon
The zero beta model is a CAPM/Asset Pricing Theory type model, which can be found for instance in
[1]. It is used in case one assumes there is no risk-free investment available in the market. In such
case the so-called zero-beta-portfolio takes the place of the risk-free interest rate in the theory. (The
zero-beta approach was first developed by Fischer Black) .
Estimation of zero-beta models via maximum likelihood leads to a non-linear set of equations. These
can be solved by new technique that is being developed in the Edgeworth Centre.
In this project the results of the new technique will be compared to more traditional techniques as
described in [1]. In the first instance this will be done on simulated data. This will allow us to
investigate the behaviour for particular cases in which the two methods may give different answers.
Once we are happy with the outcomes the techniques can be applied to real data.
[1] Campbell, Lo, MacKinlay, The Econometrics of Financial Markets, Princeton University Press
Proposal: 3
Option pricing with Variance Gamma and EPT driven models.
Supervisor: Bernard Hanzon
Classical option pricing is based on models driven by Brownian motion. However empirical studies
show that the Gaussian assumption that is made in such models is incorrect.
Models such as the Variance Gamma model (see e.g. [1]), which are non-Gaussian, are begin put
forward as alternatives. The Variance Gamma model can be seen as a special case of a more general
class of models which we call the EPT-class and which is currently being investigated. Analytical
formulas have been developed for that class. One issue that arises for non-Gaussian models is that
there are no risk-free hedging strategies (market incompleteness). The purpose of the project is to
apply a Monte Carlo simulation approach to investigate the risks in various hedging strategies for
such models. Also Monte Carlo can be used to compute option prices which can be compared to the
theoretical prices that are obtained using the analytical formulas.
[1] Dilip Madan, Peter Carr, Eric Chang, The Variance Gamma process and option pricing, European
Finance Review, 2, pp. 79-105, 1998.
Proposal: 4
Non-gaussian asymptotic distributions for estimation of stochastic state-space models.
Supervisor: Bernard Hanzon
Stochastic state space models are used in many applications including financial mathematics, where
they are used to model interest rates for instance.
Once aspect of the estimation of state space models is that one has to arrive at a correct
specification of the order (state space dimension) of the model. If the order is specified correctly
then the usual estimation methods will have have a distribution which (after appropriate scaling)
tends to a Gaussian distribution for large numbers of observations. However if the order is taken
too large, then much less is known about this distribution except that it will not be Gaussian. In this
project this question will be investigated both using simulation techniques as well as analytical
techniques. A Monte Carlo technique can be used to generate the data and to generate the
corresponding empirical distribution of the estimators. By using various appropriate
parametrizations of the state space model various hypotheses about the asymptotic distribution will
be investigated. We will start with a simple AR(1) model that is estimated while the data are
generated by white noise.
[1] E.J. Hannan, M. Deistler, The Statistical Theory of Linear Systems, Wiley.
Proposal: 5
A study on Ruin Theory with particular applications to reinsurance contracts
Supervisors: Damian Conway and Tom Carroll
Background:
Ruin theory examines the solvency of an insurer in a theoretical setting. The insurer's
surplus or reserve for an insurance portfolio over time is modelled by a stochastic process.
Objectives:
1) Build Collective Risk Models to analyse an insurer’s annual surplus position at each yearend.
2) Review and describe the main elements of classical Ruin Theory paying particular
attention to Lundberg’s Inequality for ruin probabilities.
3) Combine the theory of Collective Risk Models and Ruin Theory to estimate ruin
probabilities / expected time to ruin for an insurer.
4) Consider the effect that different reinsurance treaties will have on the insurer’s
probability of ruin and expected profitability levels and decide on the optimal levels of
reinsurance which should be used on various reinsurance contracts.
5) Develop simple iterative stochastic models of an insurance company (in Microsoft Excel
or R) and use simulation techniques to compare the empirical ruin probabilities with the
theoretical values of ruin.
6) Briefly review and discuss Solvency 2 legislation in the context of this project.
References:
Boland P.J. (2007). ‘Statistical and Probabilistic Methods in Actuarial Science’. Chapman and
Hall.
Lin X and Willmot G (1998). "Ruin Theory with Excess of Loss Reinsurance and
Reinstatements” (http://www.utstat.utoronto.ca/~sheldon/ACT451/risk0373RuinTheory.pdf)
‘Solvency 2’ – UK Financial Services Authority
(http://www.fsa.gov.uk/pages/About/What/International/solvency/index.shtml)
Proposal: 6
Actuarial data vs. Financial data: comparative analysis of models and estimation techniques
Supervisor; Dr Gregory Temnov
Traditionally, methods used to describe evolution of financial indices are different from the ones
applied for modelling insurance data.
However, in present days, dealing with highly volatile markets motivates adapting actuarial models
for the purpose of realistic modelling of financial indices.
It is suggested to analyze selected samples of financial data and to perform a comparative analysis
with available actuarial data. One of the issues to study is whether the elements of Extreme value
theory usually applied to actuarial data are suitable to use for the analysis of financial indices.
Suitable numerical estimates may be used for this problem, depending on the convenience of
implementation, in particular with respect to heavy-tailed distributions - see e.g. [1], [2] for general
references.
As the implementation of the numerical estimates should be performed with respect to sufficient
accuracy and computational speed, the use of programming languages or/and mathematical
packages (e.g., Matlab, R) is essential. It is expected that certain computational (programming)
efforts should be made.
To summarize, the general aim of the project is to investigate the matter of proper choice of suitable
distributions for fitting financial and actuarial data, which is essential for risk management problems
in the context of financial modelling, so that the project can be viewed as a link between financial
and actuarial research areas.
[1] S. Coles (2001). An Introduction to Statistical Modeling of Extreme Values. Springer [2] P.
Embrechts, C. Klüppelberg and T. Mikosch (1997). Modelling extremal events for insurance and
finance, Springer - Business & Economics.
Proposal: 7
Title: Using finite field computations in algebraic optimization.
Supervisor: B. Hanzon
Description: Many optimization problems in financial mathematics and related areas are algebraic in
nature (for instance if the criterion function is a multivariate polynomial or a multivariate rational
function). In order to exploit this one can try to apply algebraic techniques. A drawback is that this
typically requires exact computations leading to large integer coefficients. If one could work modulo
a prime number p, the computations would simplify enormously, if p is not too large. In this project
we want to investigate for a number of cases how such finite field computations could be applied
and compare the computational speed of the resulting methods with the standard ("infinite field")
approach. A study of literature in this area is part of the project. We may have to look into literature
in adjacent areas such as coding and cryptography as finite field computations are used quite
extensively in those areas. If time permits we can work out one or more examples from financial
mathematics or related fields to show how the methods work.
Proposal: 8
Title: C++ Finite Element Algorithms and C++ Software Implementation for Option Pricing Models.
Supervisor: Dr. Jim Grannell
Summary: Finite difference methods are of limited scope – essentially confined to problems over
domains with very simple boundaries. Finite element methods do not suffer from this limitation. The
project involves guided learning about finite methods for solving partial differential equations (and
possibly partial differential inequalities) arising in option pricing and implementing an algorithm for
solving one problem in C++.
Proposal: 9
Title: A Study on Longevity Risk
Supervisor: Damian Conway
Student:
Ciara Marie Coady
A structural shift in demographics is taking place as people are living longer and
longer. While an increasing number of people look forward to retirement insurers,
pension funds and governments, with hundreds of billions of dollars in definedbenefit pension scheme liabilities is clear, face a much heightened risk of larger
payouts.
Actuaries use the term Longevity Risk to describe the potential risks attached to
the increasing life expectancy of pensioners and policyholders, which can
eventually translate in higher than expected pay-out-ratios for many pension funds
and insurance companies.
In this project we will:
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Consider ways to estimate the longevity risk in mortality projections.
Consider ways to model the mortality and longevity risks so as to
allow a quantitative interpretation of such risks on the insurer
profitability and solvency levels.
Consider how mortality and longevity risks could be hedged through a
portfolio of investments of a pension fund or by the use of mortality
bonds.
Consider also the pricing of such bonds.
Complete an analysis of the Solvency II standard model approach to
longevity risk.
References:
http://www.soa.org/files/pdf/research-long-risk-quant-rpt.pdf
http://0www.sciencedirect.com.ditlib.dit.ie/science/article/pii/S0167668711000771