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Wednesday, October 21, 2015 - 15:05 ,
Location: Skiles 005 ,
Ruoting Gong ,
Illinois Institute of Technology ,
Organizer: Christian Houdre

In recent years, small-time asymptotic methods have attracted much attention in mathematical
finance. Such asymptotics are especially crucial for jump-diffusion models due to the lack of closed-
form formulas and efficient valuation procedures. These methods have been widely developed and
applied to diverse areas such as short-time approximations of option prices and implied volatilities,
and non-parametric estimations based on high-frequency data. In this talk, I will discuss some
results on the small-time asymptotic behavior of some Levy functionals with applications in finance.

Tuesday, April 8, 2014 - 15:05 ,
Location: Skiles 005 ,
Minqiang Li ,
Bloomberg ,
Organizer: Liang Peng

Many derivatives products are directly or indirectly associated with
integrated diffusion processes. We develop a general
perturbation method to price those derivatives. We show that for any
positive diffusion process, the hitting time of its integrated
process is approximately normally distributed when the diffusion
coefficient is small. This result of approximate normality enables
us to reduce many derivative pricing problems to simple expectations. We
illustrate the generality and accuracy of this
probabilistic approach with several examples, with emphasis on timer
options. Major advantages of the proposed technique include
extremely fast computational speed, ease of implementation, and analytic
tractability.

Thursday, March 27, 2014 - 15:05 ,
Location: Skiles 006 ,
Zhengjun Zhang ,
University of Wisconsin ,
Organizer: Liang Peng

Applicability of Pearson's correlation as a measure of
explained variance is by now well understood. One of its limitations is
that it does not account for asymmetry in explained variance. Aiming to
obtain broad applicable correlation measures, we use a pair of r-squares
of generalized regression to deal with asymmetries in explained
variances, and linear or nonlinear relations between random variables.
We call the pair of r-squares of generalized regression generalized
measures of correlation (GMC). We present examples under which the
paired measures are identical, and they become a symmetric correlation
measure which is the same as the squared Pearson's correlation
coefficient. As a result, Pearson's correlation is a special case of
GMC. Theoretical properties of GMC show that GMC can be applicable in
numerous applications and can lead to more meaningful conclusions and
decision making. In statistical inferences, the joint asymptotics of the
kernel based estimators for GMC are derived and are used to test whether
or not two random variables are symmetric in explaining variances. The
testing results give important guidance in practical model selection
problems. In real data analysis, this talk presents ideas of using GMCs as
an indicator of suitability of asset pricing models, and hence new
pricing models may be
motivated from this indicator.

Wednesday, April 24, 2013 - 15:05 ,
Location: Skiles 005 ,
Diego del-Castillo-Negrete ,
Oak Ridge National Laboratory ,
Organizer:

Hosts Christian Houdre and Liang Peng

Fractional calculus (FC) provides a powerful formalism for the modeling of systems whose underlying dynamics is governed by Lévy stochastic processes. In this talk we focus on two applications of FC: (1) non-diffusive transport, and (2) option pricing in finance. Regarding (1), starting from the continuous time random walk model for general Lévy jump distribution functions with memory, we construct effective non-diffusive transport models for the spatiotemporal evolution of the probability density function of particle displacements in the long-wavelength, time-asymptotic limit. Of particular interest is the development of models in finite-size-domains and those incorporating tempered Lévy processes. For the second application, we discuss fractional models of option prices in markets with jumps. Financial instruments that derive their value from assets following FMLS, CGMY, and KoBoL Lévy processes satisfy fractional diffusion equations (FDEs). We discuss accurate, efficient methods for the numerical integration of these FDEs, and apply them to price barrier options. The numerical methods are based on the finite difference discretization of the regularized fractional derivatives in the Grunwald-Letnikov representation.

Friday, April 19, 2013 - 14:05 ,
Location: Skiles 005 ,
Ruoting Gong ,
Rutgers University ,
Organizer: Christian Houdre

Hosts: Christian Houdre and Liang Peng

We prove stochastic representation formulae for solutions to elliptic boundary value and
obstacle problems associated with a degenerate Markov diffusion process on the
half-plane. The degeneracy in the diffusion coefficient is proportional to the \alpha-power
of the distance to the boundary of the half-plane, where 0 < \alpha < 1 . This generalizes the
well-known Heston stochastic volatility process, which is widely used as an asset price
model in mathematical finance and a paradigm for a degenerate diffusion process. The
generator of this degenerate diffusion process with killing, is a second-order,
degenerate-elliptic partial differential operator where the degeneracy in the operator
symbol is proportional to the 2\alpha-power of the distance to the boundary of the
half-plane. Our stochastic representation formulae provides the unique solution to the
degenerate partial differential equation with partial Dirichlet condition, when we seek
solutions which are suitably smooth up to the boundary portion \Gamma_0 contained in the
boundary of the half-plane. In the case when the full Dirichlet condition is given, our
stochastic representation formulae provides the solutions which are not guaranteed to be
any more than continuous up to the boundary portion \Gamma_0 .

Wednesday, April 17, 2013 - 15:05 ,
Location: Skiles 005 ,
Ruodu Wang ,
University of Waterloo ,
Organizer:

Hosts: Christian Houdre and Liang Peng

We introduce the admissible risk class as the set of possible aggregate
risks when the marginal distributions of individual risks are given but the
dependence structure among them is unspecified. The convex ordering upper
bound on this class is known to be attained by the comonotonic scenario, but
a sharp lower bound is a mystery for dimension larger than 2. In this talk
we give a general convex ordering lower bound over this class. In the case
of identical marginal distributions, we give a sufficient condition for this
lower bound to be sharp. The results are used to identify extreme scenarios
and calculate bounds on convex risk measures and other quantities of
interest, such as expected utilities, stop-loss premiums, prices of European
options and TVaR. Numerical illustrations are provided for different
settings and commonly-used distributions of risks.

Wednesday, November 28, 2012 - 15:00 ,
Location: Skiles 005 ,
Daniel Hernandez ,
CIMAT, Mexico ,
Organizer: Christian Houdre

Hosts: Christian Houdre and Liang Peng

The relation between robust utility maximization problems and
quadratic backward stochastic differential equations will be explored
in this talk. Motivated by the solution of the dual formulation of
the robust hedging problem for semi-martingales, when the model adopted
is a diffusion it is possible to describe more completely the solution
using the dynamic programming intuition, as well as some results of
BSDEs.

Wednesday, October 31, 2012 - 15:05 ,
Location: Skiles 006 ,
J.-P. Fouque ,
Department of Statistics and Applied Probability, University of California Santa Barbara, ,
Organizer: Christian Houdre

Hosted by Christian Houdre and Liang Peng

We present a simple model of diffusions coupled through their drifts in a
way that each component mean-reverts to the mean of the ensemble. In
particular, we are interested in the number of components reaching a
"default" level in a given time. This coupling creates stability of the
system in the sense that there is a large probability of "nearly no
default". However, we show that this "swarming" behavior also creates a
small probability that a large number of components default corresponding to
a "systemic risk event". The goal is to illustrate systemic risk with a toy
model of lending and borrowing banks, using mean-field limit and large
deviation estimates for a simple linear model. In the last part of the talk
we will show some recent work with Rene Carmona on a "Mean Field Game"
version of the previous model and the effects of the game on stability and
systemic risk.

Wednesday, September 19, 2012 - 15:05 ,
Location: Skiles 005 ,
Frederi Viens ,
Purdue University ,
Organizer: Christian Houdre

Hosts Christian Houdre and Liang Peng

It is commonly accepted that certain financial data exhibit long-range
dependence. A continuous time stochastic volatility model is considered in
which the stock price is geometric Brownian motion with volatility
described by a fractional Ornstein-Uhlenbeck process. Two discrete time
models are also studied: a discretization of the continuous model via an
Euler scheme and a discrete model in which the returns are a zero mean iid
sequence where the volatility is a fractional ARIMA process. A particle
filtering algorithm is implemented to estimate the empirical distribution
of the unobserved volatility, which we then use in the construction of a
multinomial recombining tree for option pricing. We also discuss
appropriate parameter estimation techniques for each model. For the
long-memory parameter, we compute an implied value by calibrating the
model with real data. We compare the performance of the three models using
simulated data and we price options on the S&P 500 index. This is joint
work with Prof. Alexandra Chronopoulou, which appeared in Quantitative
Finance, vol 12, 2012.

Wednesday, April 25, 2012 - 15:05 ,
Location: Skiles 006 ,
Jerzy Zabczyk ,
Institute of Mathematics, Polish Academy of Sciences ,
Organizer: Christian Houdre

Hosts Christian Houdre and Liang Peng

The talk is devoted to the Heath-Jarrow-Morton modeling of the
bond market with stochastic factors of the Levy type. It concentrates on
properties of the forward rate process like positivity and mean reversion.
The process satisfies a stochastic partial differential equation and
sufficient conditions are given under which the equation has a positive
global solution. In the special case, when the volatility is a linear
functional of the forward curve, the sufficient conditions are close to the
necessary ones.