Why Bankers Should Learn Convex Analysis (Part 1) - CARMA Colloquium and SigmaOPT Seminar

Host Institution:

The University of Newcastle

Title of Seminar:

Why Bankers Should Learn Convex Analysis (Part 1)

Speaker's Name:

Qiji Jim Zhu

Speaker's Institution:

Western Michigan University

Time and Date:

11:00am, Thursday 3rd March

Seminar 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 lay out 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.

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