
THE FIELDS INSTITUTE
FOR RESEARCH IN MATHEMATICAL SCIENCES

Actuarial
Science and Mathematical Finance Group Meetings 201112
at the Fields Institute
2:00
p.m., Stewart Library


The Actuarial
Science and Mathematical Finance research group meets on a regular
basis to discuss various problems and methods that arise in Finance
and Actuarial Science. These informal meetings are held at the Fields
Institute for Mathematical Sciences and are open to the public. Talks
range from original research to reviews of classical papers and overviews
of new and interesting mathematical and statistical techniques/frameworks
that arise in the context of Finance and Actuarial Science. This seminar
series is sponsored in part by Mprime through the research project
Finsurance
: Theory, Computation and Applications.
Meetings are normally held on Thursdays from 2pm to 3:30pm in the
Stewart Library, but check calendar for exceptions. If you are interested
in presenting in this series please contact the seminar organizer:
Prof. Sebastian Jaimungal (sebastian [dot] jaimungal [at] utoronto
[dot] ca).
PAST SEMINARS 
March 16
Fields, Room 230
*Please note nonstandard location

Tom Hurd, Department of Mathematics, McMaster University
Modelling financial networks and systemic risk
The study of "contagion" in financial systems,
that is, the spread of defaults through a system of financial
institutions, is very topical these days. In this talk I will
address how mathematical models can help us understand systemic
risk. After reviewing the basic economic picture of the financial
system as a random graph, I will explore some useful "deliberately
simplified models of systemic risk".

February 9

Sebastian Ferrando, Department of Mathematics, Ryerson
University
Trajectory Based Pricing and Arbitrage Opportunities
Assuming as given a trajectory/path space, we define an associated
market model and a notion of pricing interval and also describe
how to obtain arbitrage free results for these models. These
notions and results are purely analytical and do not depend
on a probabilistic assumption. We indicate how one can also
use the results to obtain arbitrage free results for non semimartingale
models. If time permits, we will present a simple, but practical,
model that allow us to compute the pricing interval and to
compare to market data. We report on preliminary numerical
findings related to realizable arbitrage opportunities (as
seen from our model) even when transaction costs are taken
into consideration.

November 21

Elias Shiu, Department of Statistics and Actuarial
Science, University of Iowa
Valuing EquityLinked Death Benefits: Option Pricing Without
Tears
Currently, a major segment of U.S. life insurance business
is the variable annuities, which are investment products with
(exotic) options and insurance guarantees. Many of these options
and guarantees should be priced, hedged, and reserved using
modern optionpricing theory, which involves sophisticated
mathematical tools such as martingales, Brownian motion, stochastic
differential equations, and so on. This talk will show that,
if the guarantees or options are exercisable only at the moment
of death of the policyholder and the underlying asset price
is a geometric Brownian motion, the mathematics simplifies
to an elementary calculus exercise. A key step behind this
method is that the probability density function of the timeuntildeath
random variable can be approximated by a combination of exponential
densities.
This is joint work with Hans U. Gerber of the University
of Lausanne and Hailiang Yang of the University of Hong Kong.
************
Gordon Willmot, Department of Statistics and Actuarial
Science, University of Waterloo
On mixing, compounding, and tail properties of a class
of claim number distributions
The mathematical structure underlying a class of discrete
claim count distributions is examined in some detail. In particular,
the mixed Poisson nature of the class is shown to hold fairly
generally. Using some ideas involving complete monotonicity,
a discussion is provided on the structure of other class members
which are well suited for use in aggregate claims analysis.
The ideas are then extended to the analysis of the corresponding
discrete tail probabilities, which arise in a variety of contexts
including the analysis of the stoploss premium.

Oct. 14

Peter Forsyth, Cheriton School of Computer Science,
University of Waterloo
Comparison between the Mean Variance and the Mean Quadratic
Variation optimal trading strategies
We compare optimal stock liquidation policies in continuous
time in the presence of trading impact using numerical solutions
of Hamilton Jacobi Bellman (HJB)partial differential equations
(PDE). In particular, we compare the timeconsistent meanquadraticvariation
strategy (Almgren and Chriss) with the timeinconsistent (precommitment)
meanvariance strategy. We show that the two different risk
measures lead to very different strategies and liquidation
profiles. In terms of the mean variance efficient frontier,
the original Almgren/Chriss strategy is signficently suboptimal
compared to the (precommitment) meanvariance strategy.
This is joint work with Stephen Tse, Heath Windcliff and
Shannon Kennedy.

July 26

Lane P. Hughston, Imperial College London
General Theory of Geometric Lévy Models for Dynamic
Asset Pricing
The theory of Lévy models for asset pricing simplifies
considerably if we take a pricing kernel approach, which enables
one to bypass market incompleteness issues. The special case
of a geometric Lévy model (GLM) with constant parameters
can be regarded as a natural generalisation of the standard
geometric Brownian motion model used in the BlackScholes
theory. In the onedimensional situation, for any choice of
the underlying Lévy process the associated GLM model
is characterised by four parameters: the initial asset price,
the interest rate, the volatility, and a risk aversion factor.
The pricing kernel is given by the product of a discount factor
and the Esscher martingale associated with the risk aversion
parameter. The model is fixed by the requirement that for
each asset the product of the asset price and the pricing
kernel should be a martingale. In the GBM case, the risk aversion
factor is the socalled market price of risk. In the GLM case,
this interpretation is no longer valid as such, but instead
one finds that the excess rate of return is given by a nonlinear
function of the the volatility and the risk aversion factor.
We show that for positive values of the volatility and the
risk aversion factor the excess rate of return above the interest
rate is positive, and is monotonically increasing in the volatility
and in the risk aversion factor. In the case of foreign exchange,
we know from Siegel's paradox that it should be possible to
construct FX models for which the excess rate of return (above
the interest rate differential) is positive both for the exchange
rate and the inverse exchange rate. We show that this condition
holds for any GLM for which the volatility exceeds the risk
aversion factor. Similar results are shown to hold for multipleasset
markets driven by vectorial Lévy processes, and for
market models based on certain more general classes of Lévy
martingales.
(Work with D. Brody, E. Mackie, F. Mina, and M. Pistorius.)
**************
Andrea Macrina, King’s College London
Randomised Mixture Models for Pricing Kernels
Numerous kinds of uncertainties may affect an economy, e.g.
economic, political, and environmental ones. We model the
aggregate impact of the uncertainties on a financial market
by randomised mixtures of Levy processes. We assume that market
participants observe the randomised mixtures only through
best estimates based on noisy market information. The concept
of incomplete information introduces an element of stochastic
filtering theory in constructing what we term “filtered
martingales”. We use this martingale family and apply
the FlesakerHughston scheme to develop interest rate models.
The proposed approach for pricing kernels is flexible enough
to generate a variety of bond price models of which associated
yield curves may change in level, slope, and shape. The choice
of random mixtures has a significant effect on the model dynamics.
Parameter sensitivity is analysed, and bond option price processes
are derived. We extend the pricing kernel models by considering
a weighted heat kernel approach, and establish the link to
the interest rate models driven by the filtered martingales.
(In collaboration with P. A. Parbhoo)

Past Seminars 201011
Past Semainrs 200910
Past Seminars 200809

