## Seminars and Colloquia by Series

Wednesday, January 25, 2012 - 15:05 , Location: Skiles 006 , Qihe Tang , Department of Statistics and Actuarial Science, University of Iowa , Organizer: Liang Peng

Hosts: Christian Houdre and Liang Peng.

The prevalence of rare events accompanied by disastrous economic and social consequences, the so-called Black-Swan events, makes today's world far different from just decades ago. In this talk, I shall address the issue of modeling the wealth process of an insurer in a stochastic economic environment with dependent insurance and financial risks. The asymptotic behavior of the finite-time ruin probability will be studied. As an application, I shall discuss a portfolio optimization problem. This talk is based on recent joint works with Raluca Vernic and Zhongyi Yuan.
Thursday, October 20, 2011 - 11:05 , Location: Skiles 005 , Don Richards , Penn State, Department of Statistics , Organizer: Christian Houdre

Hosted by Christian Houdre and Liang Peng

At this time, in late September, 2011 the Dow Jones Industrial Average has just suffered its worst week since October, 2008; the Standard & Poor 500 Average just completed its worst week in the past five years; and financial markets worldwide under severe stress. We think it is timely to look at aspects of the role played by "financial engineering" (also known as "mathematical finance" or "quantitative finance") in the genesis of the on-going crisis. In this talk, we examine several structured investment vehicles (SIVs) devised by financial engineers and sold worldwide to many "investors". It will be seen that these SIVs were doomed from inception. In light of these results, we are dismayed by the mathematical models propagated over the past decade by financial engineers'' and experts'' in structured finance, and it heightens our fears about the durability of the on-going worldwide financial crisis.
Wednesday, October 12, 2011 - 15:05 , Location: Skiles 006 , Jin Ma , School of Mathematics, University of Southern California , Organizer: Liang Peng

Hosted by Christian Houdre and Liang Peng

In this work we study the ruin problem for a generalized Cramer-Lundberg reserve model with investments, under the modeling (volatility and claim intensity) uncertainty. We formulate the problem in terms of the newly developed theory on G-Expectation, initiated by S. Peng (2005). More precisely, we recast the problem as to determine the ruin probability under a G-expectation for a reserve process with a G-Compound Poisson type claim process, and perturbed by a G-Brownian motion. We show that the Lundberg bounds for a finite time ruin probability can still be obtained by an exponential $G$-martingale approach, and that the asymptotic behavior of the ruin, as the initial endowment tends to infinity, can be analyzed by the sample path large deviation approach in a G-expectation framework, with respect to the corresponding storage process. This is a joint work with Xin Wang.
Wednesday, October 5, 2011 - 15:05 , Location: Skiles 006 , Bin Chen , Department of Economics, University of Rochester , Organizer: Liang Peng

Hosted by Christian Houdre and Liang Peng

We develop a nonparametric test to check whether the underlying continuous time process is a diffusion, i.e., whether a process can be represented by a stochastic differential equation driven only by a Brownian motion. Our testing procedure utilizes the infinitesimal operator based martingale characterization of diffusion models, under which the null hypothesis is equivalent to a martingale difference property of the transformed processes. Then a generalized spectral derivative test is applied to check the martingale property, where the drift function is estimated via kernel regression and the diffusion function is integrated out by quadratic variation and covariation. Such a testing procedure is feasible and convenient because the infinitesimal operator of the diffusion process, unlike the transition density, has a closed-form expression of the drift and diffusion functions. The proposed test is applicable to both univariate and multivariate continuous time processes and has a N(0,1) limit distribution under the diffusion hypothesis. Simulation studies show that the proposed test has good size and all-around power against non-diffusion alternatives in finite samples. We apply the test to a number of financial time series and find some evidence against the diffusion hypothesis.
Wednesday, September 21, 2011 - 15:05 , Location: Instr. Center 111 , Rong Chen , Department of Statistics, Rutgers University , Organizer: Liang Peng

Hosted by Christian Houdre and Liang Peng

Risk neutral density is extensively used in option pricing and risk management in finance. It is often implied using observed option prices through a complex nonlinear relationship. In this study, we model the dynamic structure of risk neutral density through time, investigate modeling approach, estimation method and prediction performances. State space models, Kalman filter and sequential Monte Carlo methods are used. Simulation and real data examples are presented.
Friday, February 18, 2011 - 15:05 , Location: Skiles 002 , Roger Cooke , Resources for the Future , Organizer: Liang Peng

Hosted by Christian Houdre and Liang Peng

"Tail risk" refers to an 'unholy trinity' Fat Tails, Micro Correlations, and Tail Dependence, that confound traditional risk analysis and are very much under-appreciated. The talk illustrates this with some punchy data. Of great interest is the question: when does aggregation amplify tail dependence? I'll show some data and new results. Tail obesity is not well defined mathematically, we have at least three definitions, leptokurtic, regularly varying and subexponential. A measure of tail obesity for finite data sets is proposed, and some theoretical properties explored.
Friday, February 11, 2011 - 15:05 , Location: Skiles 002 , Gennady Samorodnitsky , School of Operations Research and Information Engineering, Cornell University , Organizer: Liang Peng

Hosted by Christian Houdre and Liang Peng

We prove large deviation results for Minkowski sums S_n of iid random compact sets, both convex and non-convex, where we assume that the summands have a regularly varying distribution and either finite or infinite expectation. The results confirm the heavy-tailed large deviation heuristics: "large'' values of the sum are essentially due to the "largest'' summand.
Friday, October 22, 2010 - 15:05 , Location: Skiles 002 , Ruoting Gong , School of Mathematics, Georgia Tech , Organizer: Christian Houdre

Hosted by Christian Houdre and Liang Peng.

We consider a stochastic volatility model with Levy jumps for a log-return process Z = (Z_t )_{t\ge 0}of the form Z = U + X , where U = (U_t)_{t\ge 0}is a classical stochastic volatility model and X = (X_t)_{t\ge 0} is an independent Levy process with absolutely continuous Levy measure \nu. Small-time expansion, of arbitrary polynomial order in time t, are obtained for the tails P(Z_t\ge z), z > 0 , and for the call-option prices E( e^{z+ Z_t| - 1), z \ne 0, assuming smoothness conditions on the Levy density away from the origin and a small-time large deviation principle on U. The asymptotic behavior of the corresponding implied volatility is also given. Our approach allows for a unified treatment of general payoff functions of the form \phi(x)1_{x\ge z} for smooth function \phi and z > 0. As a consequence of our tail expansions, the polynomial expansions in t of the transition densities f_t are obtained under rather mild conditions.
Friday, September 24, 2010 - 15:00 , Location: Skiles 002 , J.E. Figueroa-Lopez , Purdue University , Organizer: Christian Houdre
The first order small-time approximation of the marginal distribution of a L\'evy process has been known for long-time. In this talk, I present higher order expansions polynomial in time for the distributions of a L\'evy process. As a secondary objective, I illustrate the application of our expansions in the estimation of financial models with jumps as well as in the study of the small-term asymptotic behavior of the implied volatility for this class of financial models. This talk presents joint work with C. Houdr\'e and M. Forde. Associated reading (available in the web site of the speaker): (1) Small-time expansions for the transition distribution of Levy processes. J.E. Figueroa-L\'opez and C. Houdré. Stochastic Processes and their Applications 119 pp. 3862-3889, 2009. (2) Nonparametric estimation of time-changed Levy models under high-frequency data. J.E. Figueroa-L\'opez. Advances in Applied Probability vol. 41, number 4, pp. 1161-1188, 2009. (3) The small-maturity smile for exponential Levy model. J.E. Figueroa-L\'opez and M. Forde. Preprint.
Tuesday, December 8, 2009 - 15:00 , Location: Skiles 269 , Peter Laurence , Courant Institute of Mathematical Science, New York University , Organizer: Christian Houdre
We focus on time inhomogeneous local volatility models, the cornerstone of projection methods of higher dimensional models, and show how to use the heat kernel expansion to obtain new and, in some sense optimal, expansions of the implied volatility in the time to maturity variable. This is joint work with Jim Gatheral, Elton Hsu, Cheng Ouyang and Tai-Ho Wang.