

Actuarial Science and Mathematical Finance
Group Meetings 200809
at the Fields Institute


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.
Meetings are normally held on Wednesdays 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 200809
June 10, 2009
2:00 p.m. 
Alvaro Cartea,Universidad Carlos III de Madrid and
Birkbeck University of London
Volatility and Covariation of Financial Assets: A HighFrequency
Analysis
Using high frequency data for the price dynamics of equities
we measure the impact that market microstructure noise has
on estimates of the: (i) volatility of returns; and (ii) variancecovariance
matrix of n assets. We propose a Kalman filterbased methodology
that allows us to decompose price series into the true efficient
price and the microstructure noise. This approach allows us
to employ volatility estimators that achieve very low Root
Mean Squared Errors (RMSEs) compared to other estimators that
have been proposed to deal with market microstructure noise
at high frequencies. Furthermore, this price series decomposition
allows us to estimate the variance covariance matrix of n
assets in a more efficient way than the methods so far proposed
in the literature. We illustrate our results by calculating
how microstructure noise affects portfolio decisions and calculations
of the equity beta in a CAPM setting.

2:00 p.m.
Wed. May 13 
Tim Leung, John Hopkins University
Exponential Hedging in an Incomplete Market with Regime Switching
Standard option pricing models assume continuous trading
of the underlying asset. In many situations, however, the underlying
asset is not traded. Instead, the option buyer or seller trades
a correlated asset as a proxy to the underlying to manage risk
exposure. With this setting, we consider the valuation of European
and American options in a regimeswitching market, where asset
prices follow Markovmodulated dynamics. We adopt a utility
maximization approach, in which the investor optimizes his/her
investment strategy with respect to a timeconsistent exponential
utility function. This leads to the study of a system of coupled
nonlinear PDEs or free boundary problems. We also develop a
finitedifference numerical scheme to solve for the optimal
hedging and exercising strategies under different market regimes.
Finally, we examine the impact of various factors, such as risk
aversion and regime parameters, on option prices and investment
strategies. 
Friday April 17, 2009 
Birgit Rudloff,ORFE, Princeton University
Optimal Investment Strategies Under Bounded Risk
It is well known that the optimal investment strategy
in the classical utility maximization problem can be very
risky. As a consequence, in recent research a risk constraint
was added to the classical utility maximization problem to
control the risky part. We study the utility maximization
problem when a convex or a coherent risk measure is used in
the risk constraint. As a special case a model with partial
information on the drift and an entropic risk constraint will
be considered. The optimal terminal wealth and the optimal
trading strategies are calculated. Numerical examples illustrate
the analytic results. For the general case of an arbitrary
convex risk measure, the problem gets more involved. We discuss
the limitation of Lagrange Duality, propose Fenchel Duality
instead and solve the problem for special cases.

Wednesday, March 25, 2009 
Yukio Muromachi, Tokyo Metropolitan University
Analysis of the concentration risk: decomposing the total
risk of a portfolio into the contributions of individual assets.
Analysing the concentration risk in a portfolio is one of
the important themes in the fnancial institutions after finishing
preparation for the BASEL II. While some measures have been
proposed in order to quantify the riskiness of an asset in
a portfolio in consideration for the diversifcation e ect,
in this talk, we consider the risk contribution (RC) of asset
j, RCj, defined by the partial derivative of the total risk
of the portfolio (Rp) with respect to the holding amount of
asset j (aj), multiplied by aj. In general, the sum of RCs
of all assets are equal to the total risk of the portfolio
Rp. The RCs has desirable features, however, it is also known
that the exact and robust estimation is very difficult, especially
by the Monte Carlo method. I talk about an analytical approach
called "hybrid method", in which the risk factors
are assumed to be conditionally independent, and the approximated
values of the RCs is calculated analytically by using saddlepoint
approximation.

Wednesday, February 18, 2009 
Alex Levin, Principal Financial Engineer, Algorithmics
Inc.
Affine Extensions of the Heston Model with Stochastic Interest
Rates
We present affine displaced stochastic volatility extensions
of the Heston’93 model for equity indices, FX rates,
inflation indices, and correlated with the assets stochastic
interest rates described by “additive” version of
the multifactor Hull  White model. The corresponding closedform
“semianalytical” solutions for the price of European
options are derived based on a general Duffie, Pan, and Singleton
(2000) “extended transform” approach for solving
affine jumpdiffusion problems rewritten in the form of a
“discounted characteristic function” . We consider
more general affine diffusion models than described in the
Pan and Singleton canonical representation: rectangular volatility
matrices with the number of Wiener processes greater than
the number of state variables, timedependent drifts for the
interest rates and underlying assets, and timedependent meanreversion
level for the Heston stochastic variance. This allows for
a perfect fit into the observed interest rate term structures,
dividend yield (forward rate) term structures and term structure
of the variance swap prices (e.g., VIX Index term structure)
and effective calibration procedures. Considered model is
convenient for pricing and hedging hybrid longterm derivatives
and diversified multiasset (and multicurrency) portfolios
including portfolios of insurance companies.
This talk is an extension of the author’s presentation
on the same topic at the Fifth Congress of the Bachelier Finance
Society, London 2008.

Wednesday, January 21, 2009 
Alexey Kuznetsov, Department of Mathematics and Statistics,
York University
Computing distributions of the first passage time, overshoot
and some other functionals of a Levy process.
In this talk we will discuss some recent work on computing
distributions of various functionals of a Levy process. First,
we will present a method for computing the joint density of
the first passage time and the overshoot. This method is based
on a numerical scheme for solving WienerHopf integral equations
coupled with the local information, provided by backward Kolmogorov
equation. Second, we will discuss some classical results on
WienerHopf factorization method and its numerical implementation
for a class of processes with phasetype jumps. Finally, we
will introduce a new class of Levy processes, which is qualitatively
similar to CGMY family, but for which the WienerHopf factors
can be recovered almost explicitly (and very efficiently from
the computational point of view). We will also present numerical
results, possible applications in Mathematical Finance, and
discuss some future directions for research.

Wednesday, January 14, 2009
11:00 a.m. *Please note nonstandard time* 
Adam Metzler, Department of Applied Mathematics, University
of Western Ontario
A Multiname First Passage Model for Credit Risk
In multiname extensions of the seminal BlackCox model,
dependence between corporate defaults is typically introduced
by correlating the Brownian motions driving firm values. Despite
its significant intuitive appeal, such a framework is simply
not capable of describing market data. In this talk we investigate
an alternative framework, in which dependence is introduced
via stochastic trend and volatility in obligors’ credit
qualities. We find that several specifications of the framework
are capable of describing market data for synthetic CDO tranches,
and compare calibrated parameters from both 2006 and 2008.

Wednesday, November 5, 2008 2:00 pm 
Cody Hyndman, Department of Mathematics and Statistics,
Concordia University
Forwardbackward Stochastic Differential Equations and
Term Structure Derivatives
We consider the application of forwardbackward stochastic
differential equations (FBSDEs) to the problem of pricing
and hedging various term structure derivatives. The underlying
model assumed for the factors of the economy is a multifactor
affine diffusion. We consider affine term structure models
(ATSMs) where the shortrate model is an affine function of
the factors process and affine price models (APMs) where the
price a risky asset is an exponential affine function of the
factors process and the dividend yield is an affine function
of the factors. Characterizing the underlying factor dynamics
and derivative prices as FBSDEs allows for analytic solutions
in certain cases and for the implementation of simulationbased
numerical methods for solving FBSDEs.

Wednesday, October 22, 2008 2:00
pm 
Angelo Valov, Department of Statistic, University
of Toronto
Integral equations arising from the First Passage Time
problem via martingale methods
Some of the main tools in attacking the First Passage
Time (FPT) problem for Brownian motion are integral equations
of Voltera or Fredholm type. In this talk I will discuss a
martingale method to construct such equations, generalize
existing Voltera equations of the first kind and provide a
simple alternative derivation of some known results. Furthermore
I will discuss conditions for existence of a unique continuous
solution for a subclass of Voltera equations. Finally I will
present a partial solution to both the FPT problem and the
corresponding inverse problem by introducing a random shift
in the Brownian path.

Thursday, September 25, 2008  2:00 P.M.
Sidney Smith Hall SS2098 (enter through room SS2096). 
Matt Davison, Canada Research Chair in Quantitative
Finance Associate Professor of Applied Mathematics and of
Statistical & Actuarial Sciences, The University of Western
Ontario.
Applied Stochastic Modelling in Energy Finance
Energy Markets are a frontier areas of Mathematical Finance.
They differ from traditional financial mathematics in a number
of ways. First, both the spot price processes they engender
and the financial derivatives written on these prices tend
toward complication. Second, since energy assets are primarily
consumption assets, the role of supply demand balance and
the physical realities of energy infrastructure play a significant
role.
My research, and this seminar, focus on electricity and natural
gas markets with a primary focus on electricity. In this talk
I review my work with Anderson on hybrid models for electricity
prices, and my work with Thompson, Zhao, and Rasmussen on
the "real options" problem of valuation and optimal
control of energy production and storage assets. I conclude
my talk with a discussion of a current research direction,
that of extending these ideas to "green" energy
assets and, in particular, to the related problem of valuing
weather forecasts.


