Program for the Summer School on
STOCHASTIC ANALYSIS
AND OPTIMIZATION IN FINANCE
9-21 July, Dimitsana, Arcadia, Greece
Sponsored by the
NATIONAL BANK OF GREECE
Lecturers:
Patrick Cheridito (ETH, Zürich, Switzerland)
Freddy Delbaen (ETH, Zürich, Switzerland)
Takis Konstantopoulos (University of Texas at Austin, USA)
Marc Yor (Université Paris VI, France)
I. Outline of topics
The course is going to cover mathematical models and techniques for the analysis of financial
instruments, pricing, and other applications in financial economics.
Since the beginning of the use of mathematical techniques in finance, Probability Theory and
Stochastic Processes has played an important rôle. Even at the dawn of the 20th century, Bachelier
single-handedly invented the so-called Brownian motion for the use in finance. In fact, he did that
before Einstein used it in Physics.
The use of probability in the financial world is not a surprise to anybody nowadays. What
is probably more important to realize is that the use of continuous-time models (e.g., stochastic
differential equations) necessitated from the globalization of the economy, i.e., from the fact that
markets become larger in scope and size. Indeed, using continuous-time models can often be
justified through limit theorems, through macroscopic approximations, much as what is done in
statistical physics when people try to derive macroscopic equations for the evolution of large-scale
systems, such as gases, fluids, and inter-galactic matter.
Brownian motion is a fundamental stochastic process, and is, in a sense, the identity in the
spaces of processes. Being so fundamental it pops up everywhere, much as a normal random
variable is used often when the only information about a distribution is merely the mean and the
variance.
1
2
So, in this course, we start by explaining how to model financial systems via Brownian motion
and equations driven by Brownian motion. Simultaneously, we are going to cover the basic notions
of mathematical finance, as well as the basic tools from probability theory needed for our course.
However, nowadays, it has been recognized that Brownian motion is often inadequate as a
model, largely because it fails to approximate real phenomena, phenomena observed in practice.
We are interested in two kinds of phenomena.
The first one is rather obvious from graphs of stock prices: quite often one sees huge jumps
(down during a recession or up when, e.g., a startup company becomes public and so its stock price
goes up overnight). Although Brownian motion can capture large fluctuations, it cannot capture
(large) jumps because it is continuous. Fortunately, not all processes sharing some similarities with
Brownian motion are continuous. In fact, if we maintain the fundamental property of independence
(and stationarity) of increments we obtain the so-called Lévy process, named after Paul Lévy who
understood a lot about these processes even before modern tools were available. So, we will introduce and study Lévy processes both from an analytic point of view but also from a probabilistic
one. In fact, several of the stochastic calculus tools carry over to processes and financial problems
can be studied using Lévy processes and related models.
The second phenomenon regards a tendency of persistent statistical dependence between financial data, e.g., stock prices. Processes with independent increments clearly cannot capture this
phenomenon which is commonly referred to as “long-range dependence”. Fortunately again, by
extending another property of Brownian motion, namely self-similarity, we arrive at the so-called
fractional Brownian motion, originally introduced by Kolmogorov for the study of turbulence, and
was, for quite some time, known as Kolmogorov process. (Later, Mandelbrot coined in the name
“fractional Brownian motion” and popularized the model.) We will introduce fractional Brownian
motion, its basic properties, and its stochastic calculus. There is a lot of activity these days on the
subject and we plan to overview those aspects of the model that are relevant to finance.
In more detail, here is the list of subjects and topics that are going to be covered.
1. Introduction to mathematical finance
(a) Basic structure: probability spaces and filtrations
(b) Pricing financial instruments
(c) The principle of no-arbitrage
(d) The fundamental theorem of mathematical finance and the pricing of contingent claims
(e) The interpretation of the fundamental theorem in discrete time
(f) The need for continuous time models and stochastic integrals
(g) The Snell envelope and American options
2. Stochastic calculus
(a) Introduction to martingales
(b) Brownian motion
3
(c) Stochastic integrals
(d) The Itô formula
(e) Stochastic differential equations
(f) The Black-Scholes formula and optimization/control problems
3. Continuous time finance
(a) The Samuelson model
(b) Option pricing (e.g., Asian options, quantile options, European options)
(c) The rôle of stochastic calculus in continuous finance
(d) Completeness and incompleteness of markets
(e) Hedging procedures and newer theories
4. Lévy processes
(a) Poisson processes
(b) Processes with independent increments and the Lévy-Khinchine representation
(c) Important classes of Lévy processes (compound Poisson; stable; subordinators)
(d) Some fluctuation theory
(e) Stochastic calculus with Lévy processes
5. Fractional Brownian motion
(a) Self-similarity and Gaussian processes
(b) Representations of fractional Brownian motion
(c) Non-traditional stochastic calculus for FBM
(d) The possibility of arbitrage
(e) Advanced topics on FBM
II. Bibliography
We list a (necessarily incomplete) list of books that can be used as a reference for the Summer
School. However, many of the topics that will be covered cannot be found in books. The list is
only indicative.
1. Mathematical finance
• Baxter, M. and A. Rennie. Financial Calculus: An Introduction to Derivative Pricing.
Cambridge Univ. Press, Cambridge, 1996.
• Karatzas, I. and S.E. Shreve. Methods of Mathematical Finance. Springer-Verlag, New
York, 1998.
4
• Lamberton, D. and Lapeyre, B. Introduction to stochastic calculus applied to finance.
Chapman & Hall, London, 1996.
• Steele, J.M. Stochastic Calculus and Financial Applications. Springer Verlag, New
York, 2000.
• Shiryaev, A. Essentials of Stochastic Finance. World Scientific, Singapore, 1999
2. Stochastic calculus
• Durrett R. Brownian Motion and Martingales in Analysis. Wadsworth Inc., 1984
• Karatzas, I. and S.E. Shreve. Brownian Motion and Stochastic Calculus. SpringerVerlag, New York, 1999.
• Øksendal, B. Stochastic Differential Equations: An Introduction with Applications.
Springer Verlag, Berlin, 1998.
• Revuz D. and M. Yor. Continuous Martingales and Brownian Motion. Springer-Verlag,
New York, 1999.
3. Lévy processes
• Bertoin, J. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996.
• Sato, K.-I. Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ.
Press, Cambridge, 1999.
4. Fractional Brownian motion
• Samorodnitsky, G. and M.S. Taqqu. Stable Non-Gaussian Random Processes: Stochastic Models With Infinite Variance. Chapman and Hall, New York, 1994.
5
III. Daily schedule
WEEK 1 (9-14 July)
Monday
8:30-10:30
10:30-11:00
11:00-13:00
13:00-18:00
18:00-20:00
Tuesday
Opening an- Basic
models
nouncements
in finance
————
Introduction Delbaen
to
math.
finance
Delbaen
Wednesday
Stochastic
integrals
and Brownian motion
Konstantopoulos
Thursday
Itô’s formula
Konstantopoulos
mid-morning coffee break
Introduction Martingales Theorems
Option
to stochas- Konstanof finance pricing
tic calculus topoulos
Delbaen
Delbaen
Yor
afternoon recession
Notions of Brownian
Stochastic
Stochastic
stochastic
motion
integration differential
models
Yor
Konstanequations
Konstantopoulos
Yor
topoulos
Friday
Stochastic
differential
equations
in finance
Delbaen
Saturday
Optimization
and control
in finance
Konstantopoulos
Relations
Review and
between
other topics
SDE and in finance
PDE Yor
Advanced
alternative
modeling
with B.M.
Delbaen
6
WEEK 2 (16-21 July)
8:30-10:30
10:30-11:00
11:00-13:00
13:00-18:00
18:00-20:00
Monday
Complete
and
incomplete
markets
Delbaen
Tuesday
Wednesday Thursday
Minimum
Coherent
Arbitrage
Variance
Risk Mea- and FBM
and
sures
Cheridito
Markowitz Delbaen
Theory
Delbaen
mid-morning coffee break
Poisson
Basic the- Fluctuation Stochastic
and Lévy ory of Lévy theory
calculus
processes
processes
Konstanwith Lévy
Yor
Yor
topoulos
processes
Yor
afternoon recession
Introduction Finance
Calculus
Advanced
to
frac- models
with FBM topics
tional
with FBM Cheridito
in
FBM
Brownian
Cheridito
Cheridito
motion
Cheridito
Friday
Capital
Allocation
and Risk
Measures
Delbaen
Saturday
9h30
Closing
Ceremony
Modelling
Informal
with Lévy Discusprocesses
sions.
Yor
Geometry
and Physics
Papadopoulos