Georgia Institute of Technology

Asymptotics for local and stochastic volatility models

Tue, 12/08/2009 - 3:00pm, Skiles 269

Peter Laurence, Courant Institute of Mathematical Science, N. Y. University         Organizer: Christian Houdré

Pricing Catastrophe Put Options Using Methods in Ruin Theory

Tue, 11/03/2009 - 3:00pm, Skiles 269

Sheldon Lin, Department of Statistics, University of Toronto        Organizer: Liang Peng

The discounted penalty function proposed in the seminal paper Gerber and Shiu (1998) has been widely used to analyze the time of ruin, the surplus immediately before ruin and the deficit at ruin of insurance risk models in ruin theory. However, few of its applications can be found beyond, except that Gerber and Landry (1998) explored its use for the pricing of perpetual American put options. In this talk, I will discuss the use of the discounted penalty function and mathematical tools developed for the function for perpetual American catastrophe put options. Assuming that catastrophe losses follow a mixture of Erlang distributions, I will show that an analytical (semi-closed) expression for the price of perpetual American catastrophe put options can be obtained. I will then discuss the fitting of a mixture of Erlang distributions to catastrophe loss data using an EM algorithm.

Testing the stability of the functional autoregressive processwith an application to credit card transaction volume data

Tue, 10/27/2009 - 3:00pm, Skiles 269

Piotr Kokoszka, Utah State University        Organizer: Liang Peng

The functional autoregressive process has become a useful tool in the analysis of functional time series data. In this model, the observations and the errors are curves, and the role of the autoregressive coefficient is played by an integral operator. To ensure meaningful inference and prediction, it is important to verify that this operator does not change with time. We propose a method for testing its constancy which uses the functional principal component analysis. The test statistic is constructed to have a Kiefer type asymptotic distribution. The asymptotic justification of the procedure is very delicate and touches upon central notions of functional data analysis. The test is implemented using the R package fda. Its finite sample performance is illustrated by an application to credit card transaction data.

Modeling the forward surface of mortality

Tue, 10/20/2009 - 3:00pm, Skiles 269

Daniel Bauer, Georgia State University        Organizer: Liang Peng

In recent literature, different mothods have been proposed on how to define and model stochastic mortality. In most of these approaches, the so-called spot force of mortality is modeled as a stochastic process. In contrast to such spot force models, forward force mortality models infer dynamics on the entire age/term-structure of mortality. This paper considers forward models defined based on best-estimate forecasts of survival probabilities as can be found in so-called best-estimate generation life tables. We provide a detailed analysis of forward mortality models deriven by finite-dimensional Brownian motion. In particular, we address the relationship to other modeling approaches, the consistency problem of parametric forward models, and the existence of finite dimensional realizations for Gaussian forward models. All results are illustrated based on a simple example with an affine specification.

Jumps and Information Flow in Financial Markets

Tue, 10/13/2009 - 3:00pm, Skiles 269

Suzanne Lee, College of Management, Georgia Tech        Organizer: Christian Houdré

We propose a new two stage semi-parametric test and estimation procedure to investigate predictability of stochastic jump arrivals in asset prices. It allows us to search for conditional information that affects the likelihood of jump occurrences up to the intra-day levels so that usual factor analysis for jump dynamics can be achieved. Based on the new theory of inference, we find empirical evidence of jump clustering in U.S. individual equity markets during normal trading hours. We also present other intra-day jump predictors such as analysts recommendation updates and stock news releases.

Pricing Options on Assets with Jump Diffusion and Uncertain Volatility

Tue, 09/22/2009 - 3:00pm, Skiles 269

Gunter Meyer, School of Mathematics, Georgia Tech        Organizer: Liang Peng

When the asset price follows geometric Brownian motion but allows random Poisson jumps (called jump diffusion) then the standard Black Scholes partial differential for the option price becomes a partial-integro differential equation (PIDE). If, in addition, the volatility of the diffusion is assumed to lie between given upper and lower bounds but otherwise not known then sharp upper and lower bounds on the option price can be found from the Black Scholes Barenblatt equation associated with the jump diffusion PIDE. In this talk I will introduce the model equations and then discuss the computational issues which arise when the Black Scholes Barenblatt PIDE for jump diffusion is to be solved numerically.

A New Nonlinear Long Memory Volatility Process

Tue, 02/10/2009 - 3:00pm, Skiles 269

Rehim Kilic, School of Economics, Georgia Tech        Organizer: Christian Houdré

This paper introduces a new nonlinear long memory volatility process, denoted by Smooth Transition FIGARCH, or ST-FIGARCH, which is designed to account for both long memory and nonlinear dynamics in the conditional variance process. The nonlinearity is introduced via a logistic transition function which is characterized by a transition parameter and a variable. The model can capture smooth jumps in the altitude of the volatility clusters as well as asymmetric response to negative and positive shocks. A Monte Carlo study finds that the ST-FIGARCH model outperforms the standard FIGARCH model when nonlinearity is present, and performs at least as well without nonlinearity. Applications reported in the paper show both nonlinearity and long memory characterize the conditional volatility in exchange rate and stock returns and therefore presence of nonlinearity may not be the source of long memory found in the data.

Quantitative Finance at Bloomberg

Tue, 02/03/2009 - 3:00pm, Skiles 269

Dmitry Kreslavskiy, Bloomberg        Organizer: Christian Houdré

We will give an overview of the company as it relates to the work of a quant. We will discuss projects of interest, typical lifecycle of a project, and involved areas.

To Be Announced

Tue, 01/27/2009 - 11:05am, Skiles 269

Philip Protter, Cornell University        Organizer: Christian Houdré

Wolfgang Doeblin: A Mathematician Rediscovered

Wed, 11/12/2008 - 2:00pm, Skiles 255

Christian Houdré, School of Mathematics, Georgia Tech        Organizer: Christian Houdré

In connection with the class Stochastic Processes in Finance II, we will have a supplementary lecture where a first, 50 minutes long, movie on Doeblin's life will be shown. This will be followed by a second movie, 30 minutes long, where Yor explains on the blackboard Doeblin's contribution to what Shreeve calls the Ito-Doeblin's lemma.

Semiparametric Estimation of ARCH(∞) Model

Wed, 10/29/2008 - 3:00pm, Skiles 269

Lily Wang, Department of Statistics, University of Georgia        Organizer: Liang Peng

We analyze a class of semiparametric ARCH models that nests the simple GARCH(1,1) model but has flexible news impact function. A simple estimation method is proposed based on profiled polynomial spline smoothing. Under regular conditions, the proposed estimator of the dynamic coeffcient is shown to be root-n consistent and asymptotically normal. A fast and efficient algorithm based on fast fourier transform (FFT) has been developed to analyze volatility functions with infinitely many lagged variables within seconds. We compare the performance of our method with the commonly used GARCH(1, 1) model, the GJR model and the method in Linton and Mammen (2005) through simulated data and various interesting time series. For the S&P 500 index returns, we find further statistical evidence of the nonlinear and asymmetric news impact functions.

Quotient Correlation - A New Light of Measuring Variable Associations and Testing Hypotheses of Independence and Tail Independence

Wed, 10/22/2008 - 3:00pm, Skiles 269

ZhengJun Zhang, University of Wisconsin        Organizer: Liang Peng

Various correlation measures have been introduced in statistical inferences and applications. Each of them may be used in measuring association strength of the relationship, or testing independence, between two random variables. The quotient correlation is defined here as an alternative to Pearson's correlation that is more intuitive and flexible in cases where the tail behavior of data is important. It measures nonlinear dependence where the regular correlation coefficient is generally not applicable. One of its most useful features is a test statistic that has high power when testing nonlinear dependence in cases where the Fisher's Z-transformation test may fail to reach a right conclusion. Unlike most asymptotic test statistics, which are either normal or \chi 2, this test statistic has a limiting gamma distribution (henceforth the gamma test statistic). More than the common usages of correlation, the quotient correlation can easily and intuitively be adjusted to values at tails. This adjustment generates two new concepts -- the tail quotient correlation and the tail independence test statistics, which are also gamma statistics. Due to the fact that there is no analogue of the correlation coefficient in extreme value theory, and there does not exist an efficient tail independence test statistic, these two new concepts may open up a new field of study. In addition, an alternative to Spearman's rank correlation: a rank based quotient correlation is also defined. The advantages of using these new concepts are illustrated with simulated data, and real data analysis of internet traffic, tobacco markets, financial markets...