THEMATIC PROGRAMS

Thematic Program on Quantitative Finance: Foundations and Applications
January - June, 2010

Graduate Courses held at the Fields Institute
January - June, 2010

All courses will be held at the Fields Institute, Room 230 unless otherwise noted.

 

Starts January 6, 2010 Course 1: Foundations of Mathematical Finance Instructor: Matheus Grasselli (Lecture Notes)

Course starts January 6th and will meet weekly on Wednesdays from 9:30 to 12:15 for roughly 13 weeks until the end of March. Assignment #1 Assignment #2 Assignment #3 Portfolio selection problem Fundamental theorem of asset pricing Semimartingale theory Primal and dual utility optimization problems Risk measure

Guest lectures: (1) Eckhard Platen - University of Technology, Sydney Tuesday, January 19 - 9:30 to 12:15 Wednesday, January 20 - 9:30 to 12:15 TITLE: "The Benchmark Approach" DESCRIPTION: This lecture series introduces a generalized framework for financial market modeling: the benchmark approach. It develops a unified treatment of derivative pricing, portfolio optimization, and risk management without assuming the existence of equivalent risk-neutral probability measures. The benchmark approach compatibly extends beyond the domain of classical asset pricing theories with significant implications for longer dated products, stochastic discount factors, and risk measures. A new Law of the Minimal Price, which generalizes the familiar Law of One Price, provides a revised foundation for derivative pricing. A Diversification Theorem justifies developing a simpler proxy for the full-blown numeraire portfolio. The benchmark approach augments earlier financial modeling frameworks to enable tractable yet realistic market models encompassing equity indices, exchange rates, equities, and the interest rate term structure to be developed based solely upon the real world probability measure. The lecture series carefully explains how the benchmark approach differs from the classical risk-neutral approach. Examples will be presented, using long term and extreme maturity derivatives, to demonstrate the important fact that, in reality, a range of contracts can be less expensively priced and hedged than is suggested by classical theory. The lecture series is based on the book co-authored by Eckhard Platen and David Heath, A Benchmark Approach to Quantitative Finance (Springer Finance, 2006, ISBN 3-540-26212-1). The core ideas from this book will be presented and further expanded upon during the seminar, including: · Basing financial modeling on the key concept of a numeraire portfolio; · Deriving the Law of the Minimal Price; · Approximating the numeraire portfolio via diversification; · Consistent utility maximization and portfolio optimization; · Pricing nonreplicable claims consistently with replicable claims; · Pricing and hedging long term and extreme maturity contracts; (2) Stan Pliska - University of Illinois at Chicago Wednesday, February 17 - 9:30 am to 12:15 pm Wednesday, Febraury 24 - 9:30 am to 12:15 pm To be covered: 1. The History of Options -- From the Middle Ages to Harrison and Kreps 2. The Harrison-Pliska Story (and a little bit more) a. Some continuous time stock price models b. Alternative justification of the Black-Scholes formula c. The preliminary security market model d. Economic considerations e. The general security market model f. Computing the martingale measure g. Pricing contingent claims (European options) h. Complete markets i. American options 3. Portfolio Optimization: The Quest for Useful Mathematics a. Discrete time and Markowitz b. Continuous time and dynamic programming c. Continuous time and the risk neutral approach d. Practical considerations and empirical results e. New developments (3) Marco Frittelli - University of Milan Tuesday, April 20 - 9:30 am to 12:15 pm Wednesday, April 21- 9:30 am to 12:15 pm Lecture 1: Convex Risk Measures Lecture 2: An Orlicz space approach to utility maximization and indifference pricing A considerable part of the vast development in Mathematical Finance over the last two decades was determined by the application of convex analysis. Particular attention will be devoted to the investigation of innovative and advanced methods from stochastic analysis, convex analysis and duality theory that play a fundamental role in the mathematical modelling of finance, and in particular that arise in the context of arbitrage asset pricing, optimization problems and risk measurement. The Lectures will focus on the following three topics: 1) The expected utility maximization problem in continuous time stochastic markets, which can be traced back to the seminal work by Merton, received a renovated impulse in the middle of the eighties, when the so-called convex duality approach to the problem was first developed. During the past twenty years, the theory has constantly improved, and in the last few years the general case of semimartingale stochastic models was tackled with great success. 2) The importance of the analysis of the utility maximization problem is also revealed in the theory of asset pricing in incomplete markets, where the agent's preferences have again to be taken in serious consideration. Indeed, different notion of utility based prices - as the concept of indifference price - have been introduced in the literature, since the middle of the nineties. These concepts determine pricing rules which are often non linear outside the set of marketed claims. Depending on the utility function that is selected, these pricing kernels share many properties with non-linear valuations: we are bordering here the realm of risk measures and capital requirements. 3) Coherent or convex risk measures have been intensively studied in the last ten years with particular emphasis on their dual representation. More recently risk measures have been cons idered in a dynamic context and the theory of non-linear expectations is very appropriate for dealing with the genuinely dynamic aspects of risk measures. In recent papers, risk measures are defined on Orlicz spaces, in order to allow the evaluation of possibly unbounded risk. The main tools for a detailed study of this topics are found in the theory of convex analysis and Frechet lattices. We will also analyse the more recent concept of quasiconvex risk measure in the static and in the dynamic setting.

Starts January 6, 2010 Course 2: Interest rates and credit risk Instructor: Tom Hurd (Lectures Notes) Structural Modes of Credit Risk

Course starts on January 6th and will meet weekly on Wednesdays from 1:30 to 4:15 for roughly 13 weeks until the end of March. Assignment #1 - Solutions Assignment #2 - Solutions Assignment #3 - Solutions MIDTERM: (SOLUTIONS) March 3, 2010 1:30 - 3:30 pm 3rd floor Stewart Library Spot rate models Heath-Jarrow-Morton theory Libor market models Structural credit risk models Reduced form credit risk models Multifirm default and correlation modeling Basket credit derivatives

Guest lectures: (1) Tomas Bjork - Stockholm School of Economics 'Finite dimensional realizations of HJM models' Tuesday , January 19, 1:30 to 4:15 (Room 230) Wednesday, January 20 - 1:30 to 4:15 (Room 230) Thursday, January 21 - 9:30 to 12:15 (Room 230) An Introduction to Interest Rate Theory The object of these lectures is to give an introduction to Interest rate theory from the point of view of arbitrage theory. Time permitting we will cover the following areas: Short rate models, affine term structures, inversion of the yield curve, HJM forward rate models, the Musiela parameterization, LIBOR market models, the potential approach to positive interest rates. Prerequisites: The students will be assumed to be familiar with the basics of the martingale approach to arbitrage theory, such as absence of arbitrage, existence of martingale measures, completeness and the uniqueness of the martingale measure. A basic knowledge of stochastic calculus (for Wiener driven processes) is also assumed, including martingale representation theorems and the Girsanov theorem. The lectures will be based on the textbook Bjork, T: "Arbitrage Theory in Continuous Time" 3:rd Ed. Oxford University press. (2009) Overhead slides will be available for the students. (2) Thursday, March 25 - 9:30 to 12:15 Thursday, March 25 - 1:30 to 4:15 Kay Giesecke - Stanford University Portfolio Credit Risk The lectures will cover the mathematical modeling, computation, and estimation of portfolio credit risk. Topics include: portfolio credit derivatives (index and tranche swaps), transform analysis of point processes, exact simulation methods for point processes, time changes for point processes, market conventions and model calibration, actual measure portfolio credit, maximum likelihood methods, model validation via time change. Topics are accompanied by case studies based on market and historical default data. The course will build on the material discussed in previous lectures, in particular interest rate theory.

Starts April 21, 2010 Course 3: Stochastic control, BSDEs, and applications to finance Instructor: Nizar Touzi

Course starts on the week of April 19th and will meet weekly on Wednesdays from 9:30 to 12:15 and then again from 1:30 to 4:15 for about 6.5 weeks until roughly the 1st week of June. Please note that class on Wednesday May 19 will be meeting in the 3rd Floor Stewart Library. The last class on May 26 has been rescheduled to Tuesday June 8 in the 3rd Floor Stewart Library Lecture Notes I- Stochastic control and viscosity solutions 1- Formulation 2- Dynamic Programming principle 3- Verification, application to finance 4- Introduction to viscosity solutions of second order PDEs 5- Stochastic control and viscosity solutions 6- Applications: hedging under portfolio constrants II- Backward stochastic differential equations 1- Existence and uniqueness 2- The Markov case 3- Connection with semilinear PDEs 4- Applications: interacting optimal investors with performance concern III- Probabillistic numerical methods for nonlinear PDEs 1- Introduction to Monte Carlo methods 2- Probabillistic algorithms for American options 3- Numerical scheme for fully nonlinear PDEs 4- Convergence 5- Bounds on the rate of convergence IV- Introduction to Second order backward SDEs 1- G-expectation, application to finance 2- Second order target problems, application to finance 3- Second order BSDEs

Guest lectures: It is a great pleasure to have Bruno Bouchard (University Paris Dauphine), Mete Soner (ETH Zurich), and Agnès Tourin (Fields Research Immersion Fellow), giving advanced lectures as an integral part of the course. Bruno will be giving the afternoon sessions of the 5th and the 12th of May. Mete will be giving the morning session of June 2. Agnès will be giving the afternoon session of June 2. (Lecture Notes)

June 7 - 10, 2010 9:30 am - 1:00 pm Course 4: Advanced Risk Management Methods Instructor: Dan Rosen The topics of the four lectures: 1. Economic and regulatory capital 2. Market and credit risk in a trading book or investment portfolio - Counterparty credit risk – capital and CVA - Incremental risk charge 3. Valuation of illiquid securities and structure finance 4. Capital allocation, risk contributions and risk aggregation