Probability and Statistics Seminar at Boston University
Seminar on Extreme Values and Financial Risk

Day:	Thursdays and/or Tuesdays	Click here for directions to the Boston University Department of Mathematics. Campus text map, gif map , and a general area map.
Time:	10am-noon
Place:	Room 135, Department of Mathematics, 111 Cummington St., Boston University

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2004-2005 (the seminar will focus on long-range dependence)

We will examine the dependence of extreme events in different financial markets and the impact that it can have on asset allocation.

Recently, Extreme Value Theory (EVT) has been shown to be a promising tool for risk management. Although to date, most work in the application of EVT to risk management has been done in a univariate framework. We will apply and extend tools from multivariate extreme value theory to describe and quantify the dependence of extreme events in financial markets. Throughout, we will make comparisons with the traditional hypothesis of multivariate normal market returns. Multivariate normality, although convenient, may not be an accurate representation of reality. For institutions and investors primarily concerned with guarding against large losses (safety first investors) such a misrepresentation may have serious consequences. We will also examine optimal asset allocation strategies for safety first investors applying tools from extreme value theory.

Other topics, however, may also sometimes be covered particularly when there is an outside speaker.

Last year we talked about "Stable non-Gaussian processes and their connections to non-singular flows." Click here to see last year's lisk of talks.

This seminar has now become a regular feature of Boston University and is also attended by mathematicians, scientists and postdoctoral fellows in the greater Boston area. Announcements will be done by email and through my Web Page:

This fall semester the talks will usually be on Tuesdays (10-12). The first talk will be on September 24, 2002.

Tuesday, September 24, 2002

Extreme values and financial risk 1

Brendan Bradley (Boston University)

This is the first of a series of talks on Extreme values and Financial risk. In this first talk, we review some of the important topics from univariate extreme value theory. Such topics include the extremal types theorem, threshold and point process models of extremes.

Tuesday, October 1, 2002

Extreme values and financial risk 2

Brendan Bradley (Boston University)

We continue our review of univariate extreme value theory. First, we will characterize the maximum domain of attraction of generalized extreme value distributions. This characterization will lead us to a examine threshold methods of extreme value theory. Lastly, we will review the point process characterization of extremes. This will be especially important for multivariate extremes.

Tuesday, October 8, 2002

Stochastic Resonance and Related Problems

Mark Freidlin (University of Maryland)

We consider dynamical systems with fast and slow components. We show that small random perturbations can lead to essential changes in the behavior of the system. For example, the noise can create new stable equilibriums or deterministic oscillations with amplitude and frequency of order 1. These effects are a manifestation of the laws of large deviation. In particular, stochastic resonance, which appears in various applications, can be considered from this point of view. There are many open questions here. For example, the case of perturbations by fractional Brownian motion.

Tuesday, October 22, 2002

Extreme values and financial risk 3

Brendan Bradley (Boston University)

In this talk we review the point process characterization of extremes. We will show that both the block maxima and threshold methods covered in the first two talks are special cases of point process representations. The point process representation will be an important ingredient to multivariate extreme value distributions.

Tuesday, October 29, 2002

Extreme values and financial risk 4

Brendan Bradley (Boston University)

We will start to characterize limit distributions of multivariate extremes. We will do this by first introducing and characterizing max-infinitely divisible distributions.

Tuesday, November 5, 2002

Extreme values and financial risk 5

Brendan Bradley (Boston University)

We will start dealing with multivariate extremes.

Tuesday, November 12, 2002

Multivariate stable distributions: asymptotic analysis, simulation and statistics.

Alexander Nagaev (Nicolaus Copernicus University, Torun, Poland)

Consider a random vector having a stable non-normal distribution, that is, heavy tails.

1) Asymptotic properties:

We will show how classical limit theorems help to make serious progress in the asymptotic analysis of stable distributions.

2) Simulation:

We will discuss an approach to simulation of stable random vectors, which is based on a numerical analysis of the multivariate density functions.

3) Statistics:

Estimators for the stability parameter alpha and the spectral measure, which characterizes the dependence of a stable vector will be discussed.

Tuesday, November 19, 2002

Extreme values and financial risk 6

Brendan Bradley (Boston University)

We will finish characterizing max infinitely divisible distributions. We then introduce the class of max stable distributions and show that this class of distributions is of interest because it coincides with the class of multivariate extreme value distributions.

Tuesday, December 3, 2002

Extreme values and financial risk 7

Brendan Bradley (Boston University)

We continue our discussion on multivariate extreme value distributions. This class of distributions coincides with the class of max-stable distributions and hence is characterized by an exponent measure. We show that this exponent measure, after transformation of the marginals, has a nice homogeneity property associated with it. We use this property to completely characterize multivariate extreme value distributions.

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SPRING SEMESTER 2002

TUESDAY, January 28, 2003:

Samuelson's Fallacy of Large Numbers and Optional Stopping

Erol Pekoz (Boston University)

Accepting a sequence of independent positive mean bets that are individually unacceptable is what Samuelson called a fallacy of large numbers. Recently Ross (1999) characterized utility functions where this occurs rationally, and gave examples of utility functions where any finite number of good bets should never be accepted. Here we show how things change if you are allowed the option to quit early: subject to some mild conditions, you should essentially always accept a sufficiently long finite sequence of good bets. Interestingly, the strategy of quitting when you get ahead does not perform well, but quitting when you get behind does. This sheds some light on more possible behavioral reasons for Samuelson's fallacy, as well as strategies for handling a series of sequentially observed good investments.

TUESDAY, February 4, 2003:

Splitting/Coalescence Phenomena in Skew Brownian Motion

Haya Kaspi (Technion, Israel)

In this talk we shall consider a stochastic flow in which individual particles follow skew Brownian motions, with each one of these processes driven by the same Brownian motion and starting at a different points in space and time. Due to the lack of the simultaneous strong uniqueness for the whole system of stochastic differential equations, the flow contains lenses, i.e., pairs of skew Brownian motions which start at the same time at the same point, bifurcate, and then coalesce in a finite time. We shall discuss both qualitative and quantitative (distributional) results on the geometry of the flow and lenses.

The talk is based on a joint work with Krzysztof Burdzy.

TUESDAY, February 11, 2003:

Generalized chaos decomposition for (non-commutative) Levy processes

Michael Anshelevich (University of California at Riverside)

Nualart and Schoutens have recently proven a generalized chaos decomposition property for Levy processes. Using a representation of Levy processes on a Fock space, I will give a simple proof of this property. In the second half of the talk, I will describe a more general setup: non-commutative Levy processes defined on a q-deformed full Fock space. The proof of the chaos decomposition property carries over to this case with no modifications. I will also describe certain analogs of the Appell polynomials related to these processes. No "non-commutative" background will be assumed.

TUESDAY, February 25, 2003:

Properties and identification of the multiscale fractional Brownian mo tion with biomechanical applications

Pierre Bertrand (University Blaise Pascal, Clermont-Ferrand II, France)

In some applications, for instance finance, biomechanics or internet traffic, it appears that it could be relevant to model data with a generalization of a fractional Brownian motion for which the Hurst parameter H is depending on the frequency.

We describe the multiscale fractional Brownian motions which present a parameter H as a piece-wise constant function of the frequency. We provide the main properties of these processes and propose a statistical method based on wavelet analysis to detect the frequency changes, estimate the different parameters and test the goodness of fit.

Biomechanical data are studied with these new tools, that leads to interesting conclusions.

TUESDAY, March 4, 2003:

Typical dynamics for volume preserving homeomorphisms

V. S. Prasad (University of Massachusetts at Lowell)

Many physical systems which evolve in time often preserve a natural volume on the state space. After Birkhoff's proof of the ergodic theorem it became important not only to give examples of ergodic transformations, but also to show that (paraphrasing Birkhoff) they were "typical". While they were Junior Fellows at Harvard in the 1930's, J.C. Oxtoby and S.M. Ulam worked on this conjecture of G.D. Birkhoff that ergodicity is the "general case" for volume preserving homeomorphisms of the cube (or a compact manifold).

In this talk I will survey their work and later developments extending the result of Oxtoby and Ulam in different directions. I hope to describe a simple self contained proof of their result from basic principles. No knowledge of ergodic theory will be presumed.

As time and audience permits I will describe extensions of the Oxtoby Ulam result to non compact settings where the volume measure is not necessarily finite not necessarily finite.

TUESDAY, March 18, 2003:

Random coefficient autoregression, regime switching and long memory I

Stilian Stoev (Boston University)

We will discuss recent results of R. Leipus and D. Surgailis on a time series model which exhibits heavy-tails. Consider the following modification of the classical AR(1) model (*) X(t+1) = a(t) X(t) + Z(t), t = ...,-1, 0, 1,... where {Z(t)} is a finite variance white noise process and 0 < a(t) < 1 is also random and independent of the noise. The process {a(t)} is suitably chosen as a renewal-reward process with heavy-tailed inter-renewal times, so that the equation (*) has a covariance-stationary solution {X(t)}. The surprising result of the paper is that the rescaled version of this finite-variance process {X(t)}, converges in the sense of finite-dimensional distributions, to an infinite variance Levy process {Z_r(t)}. Namely, (**) n^(-1/r) ( X(1) + X(2) + ... + X([nt]) ) ==> Z_r(t), r<2, as n converges to infinity. We will present the construction the authors use and sketch their proof.

TUESDAY, March 25, 2003:

Random coefficient autoregression, regime switching and long memory II

Stilian Stoev (Boston University)

Continuation of last week's talk.

TUESDAY, April 1, 2003:

Local contagion in financial markets

Brendan Bradley (Boston University)

We use an alternative definition of contagion of financial markets and proposes a relevant test. The test is based on a measure of local correlation and does not require the specification of crisis and normal periods. As such, it avoids difficulties associated with testing for correlation breakdown, such as hand picking subsets of the data, and it provides a better understanding of the degree of dependence between financial markets. Using our test, we find evidence of contagion between developed and U.S. equity markets and evidence of flight to quality from the U.S. equity market to the U.S. government bond market.

TUESDAY, April 8, 2003:

Extremal Dependence and Financial Asset Allocation

Brendan Bradley (Boston University)

We investigate the portfolio construction problem for risk-averse investors seeking to minimize quantile based measures of risk. Using dependence measures from extreme value theory, we find that most international equity markets are asymptotically independent. We also find that the few cases of asymptotic dependence occur mostly in markets which are in close geographic proximity. We then examine how extremal dependence affects the asset allocation problem. Following the structure variable approach, we focus on the portfolio and model its tail in a manner consistent with extreme value theory. We then develop a methodology for asset allocation where the goal is to guard against catastrophic losses. The methodology is tested through simulations and applied to portfolios made up of two or more international equity markets. We analyze in detail three typical types of markets, one where the assets are asymptotically independent and the ratio of marginal risks is not constant, the second where the assets are asymptotically independent but the ratio of marginal risks are approximately constant and the third where the assets are asymptotically dependent and the ratio of marginal risks is not constant. The results are compared with the optimal portfolio under the assumption of normally distributed returns. Surprisingly, we find that the assumption of normality incurs only a modest amount of extra risk for all but the largest losses.

TUESDAY, April 15, 2003:

Multivariate approach to financial asset allocation

Brendan Bradley (Boston University)

We finish our discussion on Extremal Dependence and Asset Allocation. We then apply a multivariate approach to the asset allocation problem for investors seeking to minimize the probability of large losses. It involves modelling the tails of joint distributions using techniques motivated by extreme value theory. We compare results with a corresponding univariate approach using simulated and financial data. The multivariate approach is significantly more computationally expensive. We find that the estimated risks using the multivariate and univariate approaches are in close agreement for a wide range of losses and allocations. We therefore suggest that the univariate approach be used for the typical level of loss that an investor may want to guard against.

TUESDAY, April 22, 2003:

The Authorship Attribution of a Text: A Review I

Mikhail Malioutov (Northeastern University)

I will review documentary, literary, style and statistical cryptography methods for the attribution of text authorship. As examples, I will discuss the well-known controversy concerning who wrote the works ascribed to Shakespeare, the disputed Federalist papers, etc.

TUESDAY, April 29, 2003:

The Authorship Attribution of a Text: A Review II

Mikhail Malioutov (Northeastern University)

The first talk was devoted to the history of the subject. This second talk will focus on the statistical analysis.

Go to the top of the list for time and place.