The Quantitative Finance Seminar has been a centerpiece of the Commercial/Industrial
program at the Fields Institute since 1995. Its mandate is to arrange
talks on current research in quantitative finance that will be of interest
to those who work on the border of industry and academia. Wide participation
has been the norm with representation from mathematics, statistics,
computer science, economics, econometrics, finace and operations research.
Topics have included derivatives valuation, credit risk, insurance and
portfolio optimization. Talks occur on the last Wednesday of every month
throughout the academic year and start at 5 pm. Each seminar is organized
around a single theme with two 45-minute talks and a half hour reception.
There is no cost to attend these seminars and everyone is welcome. To
be informed of speakers and titles for upcoming seminars and financial
mathematics activities, please subscribe to the Fields mail
list.

Past seminars 2005-06

### Upcoming Seminars

May 31, 2006

**Tom Coleman**, Dean, Faculty of Mathematics, University of Waterloo

*Minimizing CVaR and VaR for a Portfolio of Derivatives*

Value at Risk (VaR) and Conditional Value at Risk (CVaR) are frequently
used as risk measures in risk management. We analyze the problem of
computing the optimal VaR and CVaR portfolios - we illustrate that if
the portfolios contain derivatives then the resultant optimization problems
are typically ill-posed. We propose corrective measures for this problem
and also look at some of the other computing challenges.

April 26, 2006

**Jean-Pierre Fouque**, University of California Santa Barbara

*Perturbation Methods in Default Modeling*

We show that stochastic volatility incorporated in first passage models
can create reasonable default probabilities over a wide range of

maturities. To achieve that, one has to carefully calibrate the time
scales of volatility, and, to make this approach tractable, we show
that regular and singular perturbations techniques associated to slow
and fast time scales can be used. We then address the multi-name case
and we show that default correlations created by stochastic volatility
give interesting loss distributions. Perturbation techniques are gain
used to compute these distributions and the related tranche prices.

Joint work with Ronnie Sircar (Princeton), Knut SOLNA (UC Irvine),
and Stephen Zhou (PhD student, NC State University).

and

**Sebastian Jaimungal**, Department of Statistics, University
of Toronto

*Indifference Pricing for Equity-Linked Insurance and Reinsurance
Options*

Insurance companies are increasingly facing heavy exposure to capital
market risks, due to the issuance of equity-linked insurance policies.
There is now a growing need for coherent valuation and hedging methodologies
that take into account the interwoven actuarial and financial risks.
In this talk, I will demonstrate how the principle of equivalent utility
provides equity-linked insurance premiums, and explain how to value
double-trigger reinsurance options consistently; such contracts are
crucial risk management vehicles since they provide the insurer with
a means to offload unwanted risks. In addition, I will illustrate
how utility indifference allows for the simultaneous treatment of
counterparty risk. By solving the resulting HJB equations, I determine
that both the premiums and prices satisfy Black-Scholes-like PDEs
with non-linear and non-local risk-aversion correction terms. Numerical
consequences will be explored throughout this talk to clarify the
approach and to aid understanding.

March 29, 2006

**Matheus Grasselli, **McMaster University

* Rational exercise of employee options.*

At the core of the controversy surrounding the accounting status
of employee options lies a lack of agreement on the correct valuation
procedure for them. For this, we propose a discrete-time algorithm
based on pricing techniques for derivatives in incomplete-markets.
The two salient features of the method are that it takes into account
the non-linearity inherent to risk preferences, as well as the possibility
of partial hedge using a correlated instrument, such as a market index.
The immediate effect of non-linearity is that the optimal exercise
policy for the employee consists of partial exercise over en extended
period of time, as opposed to immediate exercise as soon as the underlying
reaches a threshold. The effect of trading on a correlated asset,
on the other hand, counter-balances that of risk aversion and can
be used to greatly increase the value of the option for the employee,
and consequently its cost for the issuing firm.

and

**Dan Rosen**, Fields Institute

*Economic Capital Allocation, Risk Contributions and Diversification
in Credit Portfolios *

Concentration risk is arguably the most important cause of major
problems in banks, according to the Basel committee of Banking Supervision.
The reverse side of the coin, diversification, is one of the key tools
for managing the risk of credit portfolios. A thorough understanding
of diversification/concentration risk is vital for allocating optimally
economic credit capital. This is required for pricing, profitability
assessment and limits, building optimal risk-return portfolios and
strategies, performance measurement and risk based compensation.

This seminar presents a practical overview of the measurement of
diversification and risk capital contributions in credit portfolios
and their application to capital allocation. We stress several key
points. First, marginal risk contributions provide a useful basis
for allocating capital since they are additive and reflect the benefits
of diversification within a portfolio. Second, the choice of the risk
measure can have a substantial impact on capital allocation. In particular,
the quantile level chosen for measuring VaR or expected shortfall
(ES) can also have a significant impact on the relative amount of
capital allocated to portfolio components. Third, diversification
measures and risk contributions can be calculated analytically under
certain models. These methods provide fast calculations and can be
used to understand capital allocation strategies better, but they
may present some practical limitations, as well. Finally, Monte Carlo
methods may be required to compute risk contributions in more realistic
credit models. Computing VaR and ES contributions is challenging,
especially at the extreme quantiles typically used for credit capital
definition.

February 22, 2006

**Hyejin Ku**, Department of Math and Statistics, York University

Liquidity Risk with Coherent Risk Measures

We consider questions related to the regulation of liquidity risk.
Basically, the firm should be able to unwind its current position
without too much loss of its wealth if it were required to do so.
Liquidity risk is important in deciding whether a firm's position
is "acceptable" or not. We develop a method to incorporate
liquidity risk into risk measurement. We consider a portfolio to be
acceptable if it can (by trading) be turned into an ?acceptable"
cash-only position having positive future cash flows at some fixed
date, and present an example of modeling liquidity.

and

**Ajay Subramanian**, Assistant Professor of Risk Management and
Insurance

J. Mack Robinson College of Business, Georgia State University

*Asymmetric Beliefs, Agency Conflicts, and Venture Capital Investment
*

We develop a dynamic principal-agent model to examine the interplay
among risk, imperfect information, agency conflicts, and asymmetric
beliefs on the characteristics of venture capital (VC) relationships---the
economic value that they generate, the durations of relationships,
the structures of long-term contracts between VCs and entrepreneurs
(ENs), and the manner in which VC investment is staged over time.
We show that the presence of asymmetric beliefs about project quality
has a substantial beneficial impact on project value and the expected
payoff to the VC implying that VCs have significant incentives to
encourage entrepreneur optimism. We analytically characterize the
effects of the project's characteristics---its systematic and technical
risk, and the degree of asymmetry in beliefs about its quality---on
the path of staged investments by the VC and the structure of the
long-term contract between the VC and the EN. Consistent with empirical
evidence, we predict that varying project characteristics lead to
significant heterogeneity in contractual structures and investment
schedules. The systematic and technical risks of projects have opposite
effects on the durations and economic values of VC relationships.
The duration, project value, and the expected payoff to the VC decrease
with the project's systematic risk but increase with its technical
risk, which leads to the striking implication that the value of the
project and the expected payoff to the VC are actually enhanced when
there is greater noise in the perception of project quality. Broadly,
our study not only demonstrates that the interactions among agency
conflicts, imperfect information, and asymmetric beliefs have a major
impact on the VC-EN relationship, but also precisely describes the
manner in which they affect this relationship.

Authors are Yahel Giat, Steven T. Hackman, and Ajay Subramanian .

January 25, 2006 -- 5:00 pm.

**Roger Stein**, Moodys

* Better Predictions of Income Volatility Using a Structural Default
Model*

We propose a novel approach to predicting future volatility of company
earnings, in this case EBITDA. Our approach combines predictions of
a firm’s probability of default with insights from a structural
model of default. The source of the probabilities of default can be
econometric, structural, reduced-form or other models or agency ratings,
provided the source has high predictive power. We use these probabilities
to imply EBITDA volatility using a stylized, liquidity-based model
of firm default similar in some ways to that originally proposed by
Wilcox (1971). The method does not require market information and
our out-of-sample testing suggests that our approach is more accurate
in estimating future volatility than the historical volatility of
EBIDTA. Importantly, the method also produced reasonable estimates
of volatility when historical data is quite limited, for instance
when no historical financial data are available for the firm. In addition
in comparison with historical volatility estimates the implied volatility
estimates appear provide incremental information useful in identifying
those firms that are more likely to experience EBITDA. Beyond implied
volatility, we explore extensions of the approach for estimating implied
liquidity requirements and target growth rates for firms, given a
starting capital structure and variable cash flow stream.

and

**David Lando**, Copenhagen Business School

*Decomposing Swap Spreads*

We analyze a six-factor model for Treasury bonds, corporate bonds,
and swap rates and decompose swap spreads into three components: A
convenience yield from holding Treasuries, a credit risk element from
the underlying LIBOR rate, and a factor specific to the swap market.
In the later part of our sample, the swap-specific factor is strongly
correlated with hedging activity in the MBS market. The model further
sheds light on the relationship between AA hazard rates and the spread
between LIBOR rates and GC repo rates and on the level of the riskless
rate compared to swap and Treasury rates.

(Joint work with Peter Feldhütter)

November 23, 2005 -- 5:00 p.m.

**Steven Kou,** Columbia University

*Credit Spreads, Optimal Capital Structure, and Implied Volatility
with Endogenous Default and Jump Risk*

We propose a two-sided jump model for credit risk by extending the
Leland-Toft endogenous default model based on the geometric Brownian
motion. The model shows that jump risk and endogenous default can
have significant impacts on credit spreads, optimal capital structure,
and implied volatility of equity options: (1) The jump and endogenous
default can produce a variety of non-zero credit spreads, including
upward, humped, and downward shapes; interesting enough, the model
can even produce, consistent with empirical findings, upward credit
spreads for speculative grade bonds. (2) The jump risk leads to much
lower optimal debt/equity ratio; in fact, with jump risk, highly risky
firms tend to have very little debt. (3) The two-sided jumps lead
to a variety of shapes for the implied volatility of equity options,
even for long maturity options; and although in general credit spreads
and implied volatility tend to move in the same direction under exogenous
default models, but this may not be true in presence of endogenous
default and jumps. In terms of mathematical contribution, we give
a proof of a version of the ``smooth fitting'' principle for the jump
model, justifying a conjecture first suggested by Leland and Toft
under the Brownian model.

and

**Mary Hardy**, University of Waterloo

* Stock Return Models for Long Term Embedded Options'*

Insurance companies in Canada and the USA have found that adding
out-of-the-money guarantees to mutual fund type investments creates
a product which is highly popular. The risk management of these contracts
is challenging. A crucial part of the problem is finding a real world
model for the mutual fund returns that adequately captures the tails.

In this talk I will briefly describe how actuaries approach the risk
management of these contracts, and then will present some of the many
models proposed for equity returns. It emerges that very small tweaks
in the equity model can make a significant difference to the resulting
regulatory capital. Using (and abusing) a bootstrap approach we show
how to determine which of these models can be justified using the
historic data.

October 26, 2005 -- 5:00 p.m.

**Nizar Touzi**, University Paris I-Pantheon-Sorbonne

*Modelling continuous-time financial markets with capital gains
taxes*

We formulate a model of continuous time financial market consisting
of a bank account with constant interest rate and one risky asset
subject to transaction costs and capital gains taxes. The taxation
rule is linear so that it allows for tax credits when capital losses
are experienced. We consider the problem of maximizing expected utility
from future consumption in infinite horizon. We first derive lower
and upper bounds on the value function in erms of the corresponding
value function in the tax free and frictionless model. In particular,
these bounds allow to obtain an explicit first order expansion of
our value function for small interest rate and tax rate coefficients.
We next provide a characterization of the value function in terms
of the associated dynamic programming equation, and we suggest a numerical
approximation scheme based on finite differences and the Howard

algorithm. The numerical results show that the first order Taylor
expansion is very accurate for reasonable market data.

and

Marco Frittelli, Università degli Studi di Firenze

*A Unifying Framework for Utility Maximization Problems with Unbounded
Semimartingales*During the past twenty years, the theory of expected
utility maximization in continuous-time stochastic incomplete markets
has

constantly improved, but a case has been left apart: exactly the situation
examined in this talk where the semi-martingale X, describing the
price evolution of a finite number of assets, can be possibly unbounded.
This is a non-trivial extension, from a mathematical but also from
a financial point of view.

In fact, in highly risky markets (i.e. with unbounded losses in the
trading: think of X as a Compound Poisson process on a finite horizon,
with unbounded jumps) the traditional approach to the problem leads
to trivial maximization: the optimal choice for the agent would be
investing the initial endowment entirely in the risk free asset. However,
it could happen that some of the investors are willing to take a greater
risk: mathematically speaking, they accept trading strategies that
may lead to unbounded losses. This risk-taking attitude gives them
the concrete possibility of increasing their expected utility from
terminal wealth.

In a unified framework, we consider the utility maximization problem
for utility functions that can be finite valued on the whole real
line or only on the positive semi axes and we select a generalized
class of trading strategies which allows for unbounded stochastic
integrals. By duality methods we prove existence of the optimal solution
to both the dual and the primal problems.

As it is widely known, the utility maximization problem is linked
to derivative pricing through the so-called indifference pricing technique.
Such a technique is far from being a theoretical speculation, since
it is currently used by financial institutions to price new and/or
illiquid derivatives. The results here presented allow tackling this
problem in the general case of a non-necessarily locally bounded semi-martingale
price process.

September 28, 2005 -- 5:00 p.m.

**Dmitry Kramkov**, Carnegie Mellon

*Sensitivity analysis of utility based prices and risk-tolerance
wealth processes*

In the general framework of a semimartingale financial model and
a utility function U defined on the positive real line we compute
the first order expansion of marginal utility based prices with respect
to a ``small'' number of random endowments. We show that this linear
approximation has some important qualitative properties if and only
if there is a risk-tolerance wealth process. In particular, they hold
true in the following polar cases:

(i) for any utility function U if and only if the set of state price
densities has a greatest element from the point of view of second
order stochastic dominance

(ii) for any financial model if and only if U is a power utility function
(U is an exponential utility function if it is defined on the whole
real line).

The presentation is based on a joint paper with Mihai Sirbu.

and

**Mark Reesor**, University of Western Ontario

*A Debt Strategy Simulation Framework and Interest-rate Model Risk*

Debt strategy is the manner in which governments (or agencies
and corporations) issue bonds to cover their funding requirements.
The issuer has some control over the relative amounts of bond issuance
across the maturity spectrum, making this problem analogous to the
typical portfolio selection problem. Furthermore, there are additional
constraints that are unique to the problem of managing a large portfolio
of public debt. We discuss how this bond issuance problem can be formulated
as a (constrained) stochastic optimal control problem, along with
a simulation framework that allows for its analysis. Clearly a model
for interest rates is one of the main components of such a framework.
Using a simple example, we investigate the issue of model risk in
the debt strategy analysis.

This is joint work with Shudan Liu, a PhD student in the Dept of Applied
Math, UWO.

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