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

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 riskneutral 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 fullblown 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 riskneutral 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 coauthored by
Eckhard Platen and David Heath, A Benchmark Approach to
Quantitative Finance (Springer Finance, 2006, ISBN 3540262121).
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 HarrisonPliska Story (and
a little bit more)
a. Some continuous time stock price models
b. Alternative justification of the BlackScholes 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 socalled 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 nonlinear
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 nonlinear
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.


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
 HeathJarrowMorton 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.


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 Gexpectation, 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

Taking the Institute's Courses for Credit
As graduate students at any of the Institute's University Partners,
you may discuss the possibility of obtaining a credit for one or
more courses in this lecture series with your home university graduate
officer and the course instructor. Assigned reading and related
projects may be arranged for the benefit of students requiring these
courses for credit.

