SMS scnews item created by Andrew Mathas at Mon 28 Feb 2011 2240
Type: Seminar
Distribution: World
Expiry: 4 Mar 2011
Calendar1: 3 Mar 2011 1100-1200
CalLoc1: AGR Carslaw 829
CalTitle1: Why Bankers Should Learn Convex Analysis (Parts 1 & 2)
Calendar2: 4 Mar 2011 1100-1200
CalLoc2: AGR Carslaw 829
Auth: mathas@123-243-22-16.static.tpgi.com.au in SMS-auth

# Why Bankers Should Learn Convex Analysis (Parts 1 & 2)

### Qiji Jim Zhu

CARMA Colloquium and SigmaOPT Seminar Why Bankers Should Learn Convex Analysis (Parts 1
& 2)

Speaker: Qiji Jim Zhu, Department of Mathematics, Western Michigan University Location:
Via Access Grid - Room V206, Mathematics Building (Callaghan Campus), The University of
Newcastle Date and Time: Thursday 3rd March 11:00AM (Part 1) & Friday 4th March, 11:00AM
(Part 2)

Abstract

Concave utility functions and convex risk measures play crucial roles in economic and
financial problems.  The use of concave utility function can at least be traced back to
Bernoulli when he posed and solved the St.  Petersburg wager problem.  They have been
the prevailing way to characterize rational market participants for a long period of
time until the 1970’s when Black and Scholes introduced the replicating portfolio
pricing method and Cox and Ross developed the risk neutral measure pricing formula.  For
the past several decades the new paradigm’ became the main stream.  We will show that,
in fact, the new paradigm’ is a special case of the traditional utility maximization
and its dual problem.  Moreover, the convex analysis perspective also highlights that
overlooking sensitivity analysis in the `new paradigm’ is one of the main reason that
leads to the recent financial crisis.  It is perhaps time again for bankers to learn
convex analysis.

The talk will be divided into two parts.

In the first part we layout a discrete model for financial markets.  We explain the
concept of arbitrage and the no arbitrage principle.  This is followed by the important
fundamental theorem of asset pricing in which the no arbitrage condition is
characterized by the existence of martingale (risk neutral) measures.  The proof of this
gives us a first taste of the importance of convex analysis tools.  We then discuss how
to use utility functions and risk measures to characterize the preference of market
agents.

The second part of the talk focuses on the issue of pricing financial derivatives.  We
use simple models to illustrate the idea of the prevailing Black -Scholes replicating
portfolio pricing method and related Cox-Ross risk-neutral pricing method for financial
derivatives.  Then, we show that the replicating portfolio pricing method is a special
case of portfolio optimization and the risk neutral measure is a natural by-product of
solving the dual problem.  Taking the convex analysis perspective of these methods
naturally leads to the consideration of their sensitivity.  It turns out that these
pricing methods are rather sensitive to model perturbations.  This is a key to
understand why these methods have led to financial crisis from time to time.  The above
discussion underscores the importance of using diverse approaches to asset pricing and
trading in financial markets.  One of such method emphasizing the robustness of the
pricing and trading of options will be discussed with tests conducted using real
historical market data.  Again convex analysis plays a crucial role.

To participate in this seminar, book your University's AGR or a university/APAC etc AGR
that you are otherwise able to use.  A listing of Access Grid nodes is available here:
http://www.accessgrid.org/nodes

This seminar notice is available on the AMSI Website (www.amsi.org.au): Events > AGR
Events

Best regards,