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Probability
and Statistics in Complex Systems: Genomics, Networks, and Financial
Engineering, September 1, 2003 - June 30, 2004
Abstracts:
IMA
Workshop:
April
12-16, 2004
Photo
Gallery Material
from Talks
Marco
Avellaneda (Courant Institute of Mathematical
Sciences (CIMS), New York University) avellane@courant.nyu.edu
http://www.math.nyu.edu/faculty/avellane/
A
Market-Induced Mechanism for Stock Pinning on Option Expiration
Dates
Slides:
PinningSlides.pdf
Paper: qf3601.pdf
We
propose a model to describe stock pinning on option expiration
dates. We argue that if the open interest in a particular
contract is unusually large, Delta-hedging in aggregate by
market-makers can impact the stock price and drive it to the
strike price of the option. We derive a stochastic differential
equation for the stock price which has a singular drift that
accounts for the price-impact of Delta-hedging. According
to this model, the stock price has a finite probability of
pinning at a strike. We calculate analytically and numerically
this probability in terms of the volatility of the stock,
the time-to-maturity, the open interest for the option under
consideration and a ``price-elasticity. constant that models
price impact. We also present strong evidence of the validity
of the model, based on historical data from 1996-2004.
David
S. Bates (Tippie College of Business, University
of Iowa and NBER) david-bates@uiowa.edu
http://www.biz.uiowa.edu/faculty/dbates
Maximum
Likelihood Estimation of Latent Affine Processes
Paper:
pdf
This
article develops a direct filtration-based maximum likelihood
methodology for estimating the parameters and realizations
of affine processes with latent state variables. Rather than
working with probability densities, which are not generally
known in continuous-time finance models, a procedure is developed
for recursively updating the associated characteristic functions
of latent variables conditional upon past discrete-time data.
Filtered estimates of latent variable realizations are directly
generated within the procedure, while the likelihood function
of observed data necessary for parameter estimation can be
evaluated numerically by Fourier inversion. An application
to daily stock index returns over 1953-96 reveals substantial
divergences from EMM-based estimates of latent stochastic
volatility and jump risk -- in particular, more substantial
and time-varying jump risk. The relevance for pricing stock
index options is discussed.
Alexandre
d'Aspremont
(Department of Electrical Engineering and Computer Science,
University of California, Berkeley) aspremon@eecs.berkeley.edu
A
Moment Approach to the Static Arbitrage Problem on Baskets
Slides:
pdf
We
consider the problem of computing upper and lower bounds on
the price of a European basket call option, given prices on
other similar baskets. We focus here on an interpretation
of this program as a generalized moment problem, using results
by Berg & Maserick (1984), Putinar & Vasilescu (1999) and
Lasserre (2001) on harmonic analysis on semigroups, the K-moment
problem and its applications to optimization. These allow
us to derive tractable necessary and sufficient conditions
for the absence of static arbitrage between basket straddles,
hence on basket calls and puts.
Rama
CONT (Centre
de Mathematiques Appliquees, Ecole Polytechnique, France)
Rama.Cont@polytechnique.fr
http://www.cmap.polytechnique.fr/~rama/
Recovering
Pricing Models from Option Prices: A Statistical Approach
to an Ill-Posed Inverse Problem
Paper:
pdf
Keywords:
Bayesian methods, evolutionary algorithms, ill--posed inverse
problems, model calibration, particle methods, option pricing.
The
inverse problem of recovering an option pricing model (or
risk--neutral process) from a set of given market prices of
options, known in finance as the model calibration problem,
has been treated in the literature either as an exact inversion
in presence of continuum data or as an optimization problem
involving non--linear least squares or regularized versions
of it. When applied to a given set of market prices, these
methods yield a single set of model parameters calibrated
to the market, whereas in principle (infinitely) many solutions
can exist. The non-uniqueness of the solution is not simply
a mathematical nuisance: it reflects model uncertainty and
should not be neglected.
We describe here a statistical approach to the model calibration
problem, which allows for incomplete data and takes into account
the multiplicity of solutions: we propose a random search
algorithm which converges to a random sample from the set
of calibrated models. Starting from an IID population of candidate
solutions drawn from a prior distribution of the set of model
parameters, the population of parameters is updated through
cycles of independent random moves followed by ``selection"
using the calibration criterion. We examine conditions under
which such an evolving population converges to a set of calibrated
models.
Through an analogy with systems of interacting particles,
a "propagation of chaos" result allows us to interpret
the result of our algorithm as a random IID sample drawn from
the set of calibrated models, whose heterogeneity can be used
to quantify the degree of ill--posedness of the inverse problem.
Building upon this idea, we propose a minimax measure of model
uncertainty for the price of an exotic option which takes
into account the value of liquidly traded ("vanilla")
options.
Our
algorithm yields a computable example of coherent and convex
measures of risk, which are compatible with observed prices
of benchmark options.
We
test this approach both on simulated data and empirical data
sets of index and foreign exchange options in the context
of diffusion models.
Raphael
Douady
(Riskdata) raphael.douady@riskdata.com
http://www.math.nyu.edu/research/rdouady/
Hedge
Fund Performance and Risk Profile Analysis: Non-Linear Statistics
and Risk Factor Identification
Hedge
Fund positional transparency has raised a number of issues
between investors and Funds of Funds managers on the one hand,
and hedge fund managers on the other hand. We show that, in
general, this controversy is almost irrelevant, and that,
indeed, appropriate statistical techniques allow to extract
from historical return series most of the risk information.
Moreover, a large part of this information cannot be detected,
just knowing the positions of the fund at a given date.
We
will, in particular, focus on the importance of taking into
account the non-linear relationship between hedge fund returns
and market factors. We shall also show the relevance of rolling
statistics of the market in the explanation of return series.
Raphael
Douady
(Riskdata) raphael.douady@riskdata.com
http://www.math.nyu.edu/research/rdouady/
Demonstration
of Riskdata's Fund Risk Profiling tool FOFiX® (poster
session)
We
present how, in practice, works Riskdata's fund analyser in
order to produce the risk profile of Hedge Funds and of Funds
of Hedge Funds, that is, how market factors may impact the
Fund or the FoF returns. During the demo, we shall compare,
on actual fund data series, the various statistical methods
that are presented during the talk, which mainly consists
of back-testing results of these methods.
Nicole
EL KAROUI
(Centre de Mathématiques Appliquées Ecole Polytechnique) elkaroui@cmapx.polytechnique.fr
Max
Plus Decomposition of Supermartingale with Application to
Portfolio Insurance
Slides:Slides: IMA2004.pdf
IMARisk04_3.pdf
Paper: pdf
Joint
work with Asma Meziou.
Hans
Föllmer
(Institut für Mathematik, Humboldt Universität zu Berlin)
foellmer@mathematik.hu-berlin.de
http://wws.mathematik.hu-berlin.de/~foellmer/
Convex
Risk Measures and Robust Optimization Problems
Slides:
pdf
We
discuss the structure of convex risk measures and the solution
of some related robust optimization problems. The talk will
be based on joint work with A. Schied
and on recent results by A. Schied and A. Gundel.
Jean-Pierre
Fouque (Department of Mathematics, North Carolina
State University) fouque@math.ncsu.edu
http://www.math.ncsu.edu/~fouque
Multiscale
Stochastic Volatility
Slides:
pdf
We
consider stochastic volatility diffusion models where volatility
is driven by two factors running on short and long time scales
respectively. Perturbations techniques, singular and regular,
are very efficient to approximate option prices. We show that
five parameters are needed to capture the main effects due
to stochastic volatility. Furthermore we reduce the parametrization
to four effective parameters which can easily be calibrated
to the implied volatility surface. Finally we explain how
to use these parameters to price other exotic derivatives.
Joint work with G. Papanicolaou,
R. Sircar and K.
Solna. Papers available at: www.math.ncsu.edu/~fouque/PubliFM.
Craig
Alan Friedman
(Standard and Poor's and New York University's Courant Institute
of Mathematical Sciences) cqf3296@nyu.edu
A
Financial Approach to Machine Learning with Applications to
Credit Risk
Slides:
html
pdf
ps
ppt
We
review a coherent, financially based approach for measuring
model performance and building probabilistic models that learn
from data. We give information theoretic interpretations of
our model performance measures and provide new generalizations
of entropy and Kullback-Leibler relative entropy. For investors
with utility functions in a three-parameter logarithmic family,
our model building method leads to a regularized relative
entropy minimization. We review applications of this methodology
to two credit problems: estimating the conditional probability
of default, given side information and estimating the conditional
density of recovery rates of defaulted debt, given side information.
Hélyette
Geman
(DESS 203 "Security Markets, Commodity Markets and Risk Management"
University Paris Dauphine & ESSEC) geman@dauphine.fr
Pure
Jump Lévy Processes for Asset Price Modelling
Articles: Pure
Jump Lévy Processes for Asset Price Modelling.pdf
Stochastic
Volatility for Lévy Processes.pdf
The
goal of the paper is to show that some types of Lévy processes
such as the hyperbolic motion and the CGMY are particularly
suitable for asset price modelling and option pricing. We
wish to review some fundamental mathematic properties of Lévy
distributions, such as the one of infinite divisibility, and
how they translate observed features of asset price returns.
We explain how these processes are related to Brownian motion,
the central process in finance, through stochastic time changes
which can in turn be interpreted as a measure of the economic
activity. Lastly, we focus on two particular classes of pure
jump Lévy processes, the generalized hyperbolic model and
the CGMY models, and report on the goodness of fit obtained
both on stock prices and option prices.
Peter
W. Glynn
(Department of Management Science and Engineering, Stanford
University) glynn@stanford.edu
Parameter
Estimation Methods for Discretely Observed Markov Processes
Slides:
pdf
When
Markov processes are continuous observed, it is generally
possible to write down the likelihood explicitly. Given the
statistical efficiency of maximum likelihood-based methods,
the corresponding maximum likelihood estimators are generally
then the method of choice for parameter estimation. However,
in many settings, the processes of interest are not continuously
observed. The difficulty of computing the corresponding transition
density that enters the likelihood then creates a tension
between what is statistically efficient and what is computationally
tractable. In particular, one may need to consider non-likelihood
based methods for computing parameter estimates. In this talk,
we will discuss some of the mathematical and computational
issues that arise at this interface between computation and
statistics.
Yevgeny
Goncharov (Department of Mathematics, University
of Michigan, Ann Arbor ) yevgeny@umich.edu
New
Approaches to Valuation of CMO's (poster
session)
Popularity
of Collateralized Mortgage Obligations declined in recent
years due losses experienced by CMO investors incurred by
refinancing waves. Inability to properly hedge CMO's can be
partially attributed to complexity of their valuation. Cash
flow in CMO's can be very complex and this Gordian knot is
currently cut with simulations of prepayment that demand a
lot of computational time. I present two new ideas to remove
this necessity. Price representations are given as Feynman-Kac
expectations, and this allows me to calculate the price via
a PDE. In one case the PDE is formulated with the terminal
condition given on a manifold rather then on a hyperplane
"t=maturity."
Victor
Isakov (Department of Mathematics and Statistics,
Wichita State University, Wichita, KS 67260-0033, USA) victor.isakov@wichita.edu
The Inverse Option Pricing Problem
Lecture:
pdf
ps
We consider the problem of recovery of the time independent
volatility from the current market data. By using the Dupire
equation we reduce this problem to an inverse problem for a
parabolic equation with the final overdetermination. We review
available uniqueness and stability results for this inverse
problem and two numerical algorithms, based on use of the fundamental
solution and on a linearization. We discuss results of their
numerical tests and further possibilities and challenges.
Kiseop
Lee
(Department of Mathematics, University of Louisville kiseop.lee@louisville.edu
and Seongjoo Song
(Department of Statistics, Purdue University, ssong@stat.purdue.edu))
Insider's
Hedging in a Jump Diffusion Model (poster
session)
Slides:
pdf
ps
We
formulate the optimal hedging problem when the underlying stock
price has jumps, especially for insiders who have more information
than general public. The jumps in the underlying price process
depends on another diffusion process, which models a sequence
of firm specific information. This diffusion process is observed
only by insiders. Nevertheless, the market is incomplete to
insiders as well as to general public. We use the local risk
minimization method to find a closed form of an optimal hedging
strategy. We also provide a numerical example of the value process
of an option based on the local risk minimization approach in
this setting.
Wei
Li
(Department of Finance, Henry B. Tippie College of Business,
The University of Iowa, Iowa City, IA 52242-1000) wei-li-2@uiowa.edu
Joint
work with Ashish Tiwari
ashish-tiwari@uiowa.edu
On
Performance Chasing, Mutual Fund Tournaments, and Managerial
Incentives (poster session)
Paper:
pdf
Why
do mutual fund investors chase past winner funds despite the
absence of performance persistence among such funds? In this
paper we adopt a tournament framework to analyze the incentives
of two fund managers, with unequal performance at an interim
stage, who compete for investor cash flows. Our model is characterized
by an absence of differential ability among competing fund managers.
We show that in equilibrium (a) it is optimal for the fund manager
who is trailing behind at the interim stage (i.e., the interim
loser) to increase the idiosyncratic risk of her portfolio,
and (b) risk-averse investors anticipate the incentives facing
losing fund managers and rationally chase winners. Our analysis
yields a number of testable predictions. In particular, we show
that the increase in the idiosyncratic risk of the interim loser
manager~Rs portfolio is directly related to the magnitude of
the performance gap at the interim stage, and to the strength
of the investor (cash flow) response to the relative performance
rankings of the funds (i.e., the strength of the tournament
effect). Furthermore, we show that the ex-ante utility of long-term
fund investors is decreasing in the strength of the tournament
effect. Our results have implications for several aspects of
fund design including the optimal fund entry/exit policy, and
choice of organizational form (i.e., closed-end vs. open-end).
Keywords:
Nash Equilibrium, Portfolio risk, Mutual Fund Tournaments, Delegated
Portfolio Management, Performance Chasing, Managerial Incentives.
Francesco
Rapisarda
(Product and Business Development Group, Banca IMI - Milan (Italy))
francesco.rapisarda@bancaimi.it
http://it.geocities.com/rapix/frames.html
Smiles
and Baskets: Multidimensional Dynamics for Basket Options
Slides: pdf
Papers: MultivariateSmile.pdf
UncertainVolModel.pdf
It is known that the Black Scholes model does not price all
European options quoted on a given market in a consistent way.
In reality the implied volatility generally shows a dependence
on both the option maturity and strike. The aim of this talk
is to incorporate the effect of this dependence in the pricing
and hedging of structured securities, with particular interest
on securities dependent on many assets that each show a volatility
smile/skew.
We
start from the formulation of an embryonic stochastic volatility
model and of its projection onto the local volatility manifold.
Both models have the advantage of being as tractable as Black
and Scholes', with a combination of simplicity and tractability
that make them extremely appealing to practitioners. We then
show that these models can be extended in an intuitive way from
the univariate to the multivariate setting. In particular in
the local volatility version, the resulting theory allows to
sample from an entirely new type of dynamics that still enjoys
an internal consistency with the observed volatility surfaces
for the individual securities, but with strong computational
implications on the calculation of prices of European options
on baskets of securities.
Rituparna
Sen (Department of Statistics, University of
Chicago) rsen@galton.uchicago.edu
Modeling
the Stock Price Process as a Continuous-time Discrete Jump Process
(poster session)
Slides: pdf
ps
An
important aspect of the stock price process, which has often
been ignored in the financial literature, is that prices on
organized exchanges are restricted to lie on a grid. We consider
pure jump models for the stock price process which integrate
the randomness of jump times with the discreteness of the jump
size. The convergence, estimation, discrete time approximation,
and uniform integrability conditions for this model are studied.
The effect of stochastic volatility is studied in this setting.
A Bayesian filtering technique is proposed as a tool for risk
neutral valuation and hedging. This emphasizes the need for
using statistical information for valuation of derivative securities,
rather than relying on implied quantities.
Michael
Stutzer (Burridge Center for Securities Analysis
and Valuation, Leeds School of Business) michael.stutzer@colorado.edu
Ockham's
Razor Critique of Investor Objective Functions: Neither Samuelson
nor Rabin and Thaler Are Right
Slides:
pdf
Influential
early articles by Paul Samuelson advocated use of expected concave
utility of wealth criteria in T-repeated betting and investment
problems. He and other founders of modern decision theory viewed
their work as normative prescriptions for choice under uncertainty;
not just as predictive theories of pre-existing behavior. Results
of Rabin (Econometrica, 2000), exposited and applied in Rabin
and Thaler (Journal of Economic Perspectives, 2001, pp. 219-232),
directly challenged both the prescriptive and predictive usefulness
of any expected concave utility criterion in these settings.
As a predictive alternative, they advocated the use of loss
averse preferences as a substitute for expected concave utility.
While
different systems of preference axioms have been found that
respectively imply the use of expected utility and loss aversion
criteria, they are not normatively convincing. Moreover, neither
loss aversion criteria, nor several other alternatives to expected
utility, do anything to solve a problem that plagues both the
prescriptive and predictive use of expected utility: prescriptive
results critically depend on practically unobservable, adjustable
preference parameters, and hence ad-hoc techniques for attempting
to indirectly identify them.
As
a simpler alternative criterion, this paper proposes the probability
of outperforming an observable benchmark the agent wants to
beat. This criterion does not suffer from the possible ills
of some other probabilistic criteria that were (influentially)
critiqued by Samuelson. Large deviations theory is used to show
that for suitably large T, this criterion is equivalent to maximizing
an expected CRRA (power) habit-formation utility, but with a
coefficient of risk aversion that varies endogenously with the
alternative evaluated. This eliminates the adjustable curvature
parameter used in other expected and non-expected utility (e.g.
loss aversion) preference theories, in accord with the scientific
principle of parsimonious parameterization called Ockham's Razor.
Peter
Tankov
(Centre de Mathématiques Appliquees, Ecole Polytechnique,
Palaiseau, France.) tankov@cmapx.polytechnique.fr
Non-Parametric
Calibration of Jump-Diffusion Option-Pricing Models
Paper:
pdf
Slides: pdf
ps
Joint
work with Rama Cont.
We
present a non-parametric method for calibrating jump-diffusion
and, more generally exponential Lévy models to a finite
set of observed option prices. We show that the usual formulations
of the inverse problem via nonlinear least squares are ill-posed
and propose a regularization method based on relative entropy:
we reformulate our calibration problem into a problem of finding
a risk neutral exponential Lévy model that minimizes
a certain weighted sum of the pricing error and the relative
entropy of the pricing measure with respect to a chosen prior
model. We discuss the numerical implementation of our method
using a gradient based optimization algorithm and show both
theoretically and via simulation tests on various examples that
the entropy penalty resolves the numerical instability of the
calibration problem. Finally, we apply our method to data sets
of index options and discuss the empirical results obtained.
Thaleia
Zariphopoulou
(Department of Mathematics, University of Texas at Austin) zariphop@math.utexas.edu
Optimal
Investments in Markets with Stochastic Opportunity Sets
Slides:
zariphopoulou1.pdf
zariphopoulou2.pdf
A
class of optimal investment and consumption models in incomplete
market environments will be analyzed. The focus will be on a
universal characterization of the optimal portfolios (myopic
and excess risky demand) in terms of hedging strategies of supporting
pseudoclaims. These claims are written on the market price of
risk and are priced by indifference. Recent results on indifference
prices will be used for the sensitivity and robustness analysis
of the optimal investments. Issues related to model specification,
and to the interplay between market incompleteness and risk
preferences, will be also discussed.
Yong
Zeng (Department of Mathematics and Statistics,
University of Missouri at Kansas City) zeng@mendota.umkc.edu
http://mendota.umkc.edu/
A
General Equilibrium Model of the Term Structure of Interest
Rates under Regime-switching Risk (poster
session)
Slides:
pdf
ps
This
work incorporates the systematic risk of regime shifts into
a general equilibrium model of the term structure of interest
rates. The model shows that there is a new source of time-variation
in bond term premiums in the presence of regime shifts. This
new component is a regime-switching risk premium that depends
on the covariations between discrete changes in marginal utility
and bond prices across different regimes. A closed-form solution
for the term structure of interest rates is obtained under an
affine model using log-linear approximation. The model is estimated
by Efficient Method of Moments. The regime-switching risk is
found to be statistically significant and mostly affect the
long-end of the yield curve. This is a joint work with Shu
Wu at the University of Kansas.
JEL
Classification: G12, E43
Key
Words: The Term Structure, General Equilibrium, Markov
Regime Shifts
Gady
Zohar
(Faculty of IE & Management Technion - Israel Institute of Technology)
gady@tx.technion.ac.il
Excess
Yields in Bond Hedging (poster
session)
Paper: pdf
Joint
work with Haim Reisman.
We
explore a dynamic term structure factor model that implicitly
allows for arbitrage opportunities, and we estimate it on Treasury
data. Using this model we construct instantaneously risk free
portfolios, and we write down a formula for their possibly non-zero
excess returns. Our model anticipates that such excess returns
may be quite large. When testing the performance of these portfolios
we find that their returns in practice perfectly match what
our model predicts. An important implication of our approach
is that hedging against factor risk may involve substantial
excess gains or losses, that can be determined by our model.
Photo
Gallery Material
from Talks
Probability
and Statistics in Complex Systems: Genomics, Networks, and Financial
Engineering, September 1, 2003 - June 30, 2004
|
