[grads] [Sem-coll] AM Colloquia & Seminars 9/5, 9/6, 9/10

[Joe Millham]

Greetings everyone!

Please join us for the following talks this coming week.  Coffee and snacks
will be served 5-10 minutes prior to each one. 

Colloquia
Thursday, September 6, 2007
4:40 pm E1 122
Prof. Qiang Du of Penn State University

Title: Centroidal Voronoi Tessellations: Concept, Algorithms, and
Applications

Abstract:
A centroidal Voronoi tessellation (CVT) is a Voronoi tessellation of a given
set such that the associated generating points are centroids (centers of
mass) of the corresponding Voronoi regions. It is a concept that has found
interesting applications in diverse areas, well beyond simple geometric
settings. In this talk, we present the basic concept of CVT and its various
generalizations, along with discussions on deterministic and stochastic
numerical algorithms for the construction of CVT. We also give a few
illustrative examples of applications ranging from image and data analysis
to numerical PDEs.

--------------------

Monday, September 10, 2007
4:40 pm E1 106
Prof. Marek Rutkowski of the University of New South Wales

Title: Static Replication of Univariate and Bivariate Claims with
Applications to Realized Variance Swaps.

Abstract:
The main goal of this work is to develop a method of static replication of
multi-variate products, such as realized variance swaps, using more standard
products, such as bivariate binary or barrier European calls and puts. In
Section 1, the concept of a static strategy, based on a continuum of traded
call and/or put options, is formally introduced in terms of a measure-valued
portfolio, and relevant quantities are defined and analyzed. We put emphasis
on the fact that the existence of a wealth process of a static strategy is
not always obvious or easy to prove, due to insuficient information about
options prices.
In Section 2, static replication is applied to univariate claims. We first
examine, following Carr and Madan (2002), replication of path-independent
claims with twice differentiable payoffs. Next, we extend this approach to
payoffs given by differences of two convex functions and subsequently to
payoffs given by functions of finite variation. Since in the last case we
consider claims with a possibly discontinuous payoff, we need to postulate
that all binary call and put options are traded (for a similar observation,
see Carr and Chou (2002)).
In Section 3, the static replication of bivariate claims is introduced and
examined. Initially, bivariate functions of finite variation are defined and
it is consequently shown how to replicate claims with payoffs given by
bivariate functions of finite variation. The work concludes by an
application of general results to static hedging of a realized variance
swap.


-------------------------------------

Applied Mathematics Department Mathematical Finance Seminar

Wednesday, September 5, 2007
4:40 pm E1 106
Prof. Marek Rutkowski of the University of New South Wales

Title: Implied Volatility: Basic Properties and Behavior Close to Expiry

Abstract:
We present a complete description of the small time to expiry asymptotics of
the Black-Scholes implied volatility under the condition of no arbitrage and
a very weak technical condition. The presented formulae are in terms of the
option and stock prices, strike price and time to expiry. In the process of
obtaining these results, we derive small time to expiry asymptotics of the
Black-Scholes price of a European call option as well as a number of
interesting properties of the time-scaled implied volatility. Our results
show that close to expiry the at-the-money (ATM) implied volatility behaves
very differently to the implied volatility of options that are not ATM. The
final section of the note demonstrates that implied volatility may fail to
converge, to a finite or infinite limit, as time to expiry goes to zero even
in the simple case of the Black-Scholes model with time-dependent
volatility. The talk will also give an overview of some other issues and
results related to the existence and properties of implied volatility in a
(local) martingale model of the stock price.


Thank you and see you there!

Joe Millham
Administrative Assistant
Applied Mathematics Department
312.567.8984
jmillham at iit.edu 

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