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The Fields Institute 2007-2008
Seminar Series on Quantitative Finance
sponsored by
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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, finance 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.
| Apr. 30, 2008 |
Leif Andersen, Bank of America Securities
Markov modeling of seasonal commodities
In this talk, we examine low-dimensional Markov models for
practical trading of gas futures and options. We aim for a
realistic (seasonal) correlation structure and for perfect
replication of the strongly seasonal form of gas option volatilities.
We also investigate techniques to incorporate seasonal volatility
smiles into the model, without losing analytical tractability.
For motivation, the talk is sprinkled with empirical data
from the US gas market.
and
Michael Gordy, Federal Reserve Board
Nested Simulation in Portfolio Risk Measurement (with Sandeep
Juneja)
Risk measurement for derivative portfolios almost invariably
calls for nested simulation. In the outer step one draws realizations
of all risk factors up to the horizon, and in the inner step
one re-prices each instrument in the portfolio at the horizon
conditional on the drawn risk factors. Practitioners may perceive
the computational burden of such nested schemes to be unacceptable,
and adopt a variety of second-best pricing techniques to avoid
the inner simulation. In this paper, we question whether such
short cuts are necessary. We show that a relatively small
number of trials in the inner step can yield accurate estimates,
and analyze how a fixed computational budget may be allocated
to the inner and the outer step to minimize the mean square
error of the resultant estimator. Finally, we develop jackknife
and dynamic allocation methods for bias reduction.
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| March 26, 2008 |
Peter Cotton, Julius Finance Corporation
(Audio of talks)
Bowling Alone. Do Copula Models Washout?
Mathematical models for structured credit products enjoy a privileged
and protected existence. The dependence structure amongst corporate
defaults may have many causes and descriptions but delineation
between these is largely defeated by an absence of empirical
data. There are very few realizations of the system over any
reasonable time frame and no guarantee of stationarity. One
might argue that there have been no realizations in fact, whereas
presumably many thousands are required in any rigorous testing
of causality. Consequently, one mans' contagion is another man's
macro-economic co-dependence, and so forth.
We inject some data into the debate - albeit entirely irrelevant
data. We examine a small collection of dependent objects whose
joint distribution can be interrogated by tedious but strangely
satisfying experiments. The objects are pins. There are ten
of them. They display a tendency to fall over in much the same
manner as companies. Their defaults, as we might call them,
have knock-on effects which can sometimes bring down one or
more of their colleagues.
and
Roger Lee, Department of Mathematics, University of
Chicago (Audio of
talks)
Hedging Options on Realized Variance
Contracts on realized variance have become a leading tool
for managing exposure to volatility risk. In particular, variance
swaps pay realized variance, typically defined as the sum
of squared daily returns of the underlying. Moreover, a second
generation of variance contracts has emerged, including variance
_options_ (calls and puts on realized variance) which offer
portfolio managers even greater control over their volatility
risk profile. However, they present greater pricing/hedging
problems to the dealer. We take a robust model-free approach
to these problems. Assuming essentially only the positivity
and continuity of the underlying share price, we hedge continuously-monitored
variance options by dynamically trading the underlying shares,
and statically holding European options. These hedges lead
to upper and lower bounds on variance option values. (Joint
with Peter Carr)
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| Feb. 27, 2008 |
Rama Cont, Columbia University (Audio
of talks)
Calibration of portfolio credit risk models: solution of
an inverse problem via intensity control.
Pricing models for portfolio credit derivatives such as
CDOs involves the construction of a stochastic process for
the losses due to defaults which is compatible with a set
of observations of market spreads for CDO tranches. We propose
an efficient and stable algorithm to solve this inverse problem
by transforming it into a stochastic control probem. We formalize
the problem in terms of minimization of relative entropy with
respect to a prior jump process under calibration constraints
and use convex duality techniques to solve the problem. The
dual problem is shown to be an intensity control problem.
We show that the corresponding nonlinear Hamilton Jacobi system
of differential equations can be represented in terms of a
nonlinear transform of a linear system of ODEs and thus easily
solved. Our method allows to construct a Markovian jump process
for defaults which leads to CDO tranche spreads consistent
with the observations. We illustrate our method ITRAXX index
data: our results reveal strong evidence for the dependence
of loss transitions rates on the past number of defaults,
thus offering quantitative evidence for contagion effects
in the risk-neutral loss process.
Jim Gatheral, Merrill Lynch and Courant Institute
(Audio of talks)
Developments in Volatility Derivatives Pricing.
Over the last two years, we have seen increased trading in
both options on VIX and options on variance swaps. Also, there
has been increased interest in modeling volatility with more
than just one factor (Lorenzo Bergomi's two-factor model for
example); rather than model instantaneous or implied volatility,
the current tendency is to model variance swaps (or curves).
Hans Buehler has derived consistency conditions on variance
curve models in general, concentrating on the so-called Double
Heston model in numerical examples. Bergomi's model on the
other hand has (double) lognormal dynamics.
In this talk, we place the Double Heston and Double Lognormal
models in their historical context, explain Buehler's consistency
condition and explore implications for the pricing of VIX
options. We see that market pricing of VIX options excludes
Heston dynamics but is roughly consistent with lognormal dynamics.
Finally, we extract time series of the two volatility factors
from historical option prices and compare statistical and
risk-neutral parameters, finding that many of the unrealistic
features of one-factor stochastic volatility models are eliminated
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February 6, 2008 - 5:00 p.m.
5:00 p.m. Reception
5:30 p.m |
**The Seminar Series On Quantitative
Finance for tonight, February 6, 2008 has been cancelled due
to weather conditions.**
. Rama Cont, Columbia University
Calibration of portfolio credit risk models: solution of an
inverse problem via intensity control.
Pricing models for portfolio credit derivatives such as
CDOs involves the construction of a stochastic process for
the losses due to defaults which is compatible with a set
of observations of market spreads for CDO tranches. We propose
an efficient and stable algorithm to solve this inverse problem
by transforming it into a stochastic control probem. We formalize
the problem in terms of minimization of relative entropy with
respect to a prior jump process under calibration constraints
and use convex duality techniques to solve the problem. The
dual problem is shown to be an intensity control problem.
We show that the corresponding nonlinear Hamilton Jacobi system
of differential equations can be represented in terms of a
nonlinear transform of a linear system of ODEs and thus easily
solved. Our method allows to construct a Markovian jump process
for defaults which leads to CDO tranche spreads consistent
with the observations. We illustrate our method ITRAXX index
data: our results reveal strong evidence for the dependence
of loss transitions rates on the past number of defaults,
thus offering quantitative evidence for contagion effects
in the risk-neutral loss process.
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| November 28, 2007 |
Michael Walker, University of Toronto (Audio
of talks)
A Calibratable Dynamic Model for CDO's: Application to
Leveraged-Super-Senior Tranche Valuation
The talk begins with an introduction describing the rapid
rise in the use of credit derivatives in the banking and related
industries. CDO's, i.e. collateralized debt obligations (one
of the most important credit derivatives), are also introduced.
Two ideas that allow the construction of easily calibratable
models for the risk-neutral valuation of CDO's, and for the
valuation of related dynamics-sensitive derivatives, are then
described. Finally, illustrative results for the valuation
of an exotic, path-dependent derivative, the leveraged super-senior
tranche, are presented.
Thaleia Zariphopoulou, University of Texas Austin
(Audio of talks)
Stochastic pdes in portfolio choice
A new approach to investment performance measurement is introduced
via the so-called forward performance processes. A related
stochastic pde is derived and analyzed for various cases of
utility coefficients. Applications of the new approach to
risk measures and its connection to the traditional expected
utility maximization problems are discussed.
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| October 31, 2007 |
Martin Schweizer, ETH Zurichand (Audio
of talks)
Arbitrage-free joint models for assets and derivatives
In today's financial markets, many instruments are traded
so liquidly that they can be viewed like primary assets. A typical
example are plain vanilla options which are often used as hedging
instruments when determining prices for other, more exotic derivatives.
Hence one might want to have a model where one can specify dynamics
(SDEs) not only for the underlying stocks, but also for a family
of options written on that stock. However, this is not entirely
straightforward because there are no-arbitrage constraints between
these price processes. A closer look even shows that this problem
is surprisingly tricky.
We present some recent advances on the construction of arbitrage-free
market models for option prices. This includes a presentation
of the problem and of our goals, issues of parametrization
and existence of models, and comparisons to related work in
the literature. While we have some answers, many questions
remain, and hopefully the audience can contribute to better
understanding as well.
This is joint work with Johannes Wissel (ETH Zurich).
David X. Li, Barclays Capital (Audio
of talks)
Dynamical Competing Risk Model for Home Equity Loan Securities:
We present a new approach to model Home Equity Loan (HEL)
securities. We take prepayment and default as two fundamental
variables and model them in a dynamic competing risk model
framework. The valuation of single name ABS CDS, ABX, and
ABS CDO will be discussed.
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