MINERVA RESEARCH FOUNDATION LECTURES
SPRING 2009
Supported by a generous grant from the Minerva Research Foundation
http://www.minerva-foundation.org/about.html
Sara Biagini
University of Pisa
Topics in Portfolio Optimization with general
underlying Assets
Lecture 1: Tuesday Jan 20, 9:30AM-11AM
Lecture 2: Wednesday Jan 21, 9:30AM-11AM
Lecture 3: Thursday Jan 22, 9:30AM-11AM
Lecture 4: Friday Jan 23, 9:30AM-11AM
Lecture 5: Tuesday Jan 27, 9:30AM-11AM
1025
School of Social Work
Building
Room 1025, 10th Floor
-
1255 Amsterdam Avenue-between. 121st & 122nd Street
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Aim of the Lectures and Prerequisites:
We provide a theoretical framework for
portfolio optimization with general, possibly non-locally bounded, processes.
Some familiarity is assumed with: 1) the basic concepts of stochastic calculus,
such as predictable processes, (super, local) martingales, semimartingales and
stochastic integration; 2) functional and convex analysis, especially for
Lecture 5. However we will try to be as self-contained as possible.
Lecture 1: The market model and absence of arbitrage.
The increasing complexity of financial instruments requires more general models
for the underlying assets, which can be non-locally bounded. We give some
examples, including Levy models. In such genera-lity, the notion of No Arbitrage
must be replaced by No Free Lunch with Vanishing Risk. In the Fundamental
Theorem of Asset Pricing for non-locally bounded processes, Del-baen and
Schachermayer 1998 showed that the pricing measures in this context are the
sigma-martingale measures instead of (local) martingale measures. The
mathematical concept of sigma-martingale process was introduced by Chou
and Emery in the '70s and it is a generalization of the martingale concept. We
illustrate it in a variety of examples.
Lecture 2: Which set of admissible strategies?
The choice of a good set of admissible strategies is a fundamental and highly
non-trivial issue. Harrison and Kreps noted that a certain type of strategy,
"the doubling" strategy, starting with zero money generates a positive net
return with probability one and within a finite time. Such strategies violate
with their inconsistency the foundations of Mathematical Finance and the
No-Arbitrage Pricing Theory. Therefore, since Harrison and Kreps a wide variety
of constraints has been proposed in order to rule out the doubling strategy. The
class of strategies widely used in the applications, like portfolio selection,
are the uniformly bounded-from-below strategies H which have nice
mathematical properties (an application of the Ansel-Stricker Lemma gives that
these processes are local martingales and supermartingales ) and a clear
financial interpretation (finite credit line during the trading). But if one
wants to account for unbounded stock prices, the set H is not enough, as
it may reduce to the trivial zero strategy: H = {0}. There have
been so far some proposals (e.g. Delbaen and Schachermayer 1998 in the
super-replication price problem, Biagini and Frittelli for utility maximization)
in order to define a good set of strategies in such a way to account for general
asset prices and still preserve the features of the Ansel-Stricker lemma. We
focus on the Biagini-Frittelli definition of admissible set H^W,
consisting of strategies which are bounded from below by a random control
W.
Lecture 3: Utility Maximization (A).
The set of strategies HW performs well in applica-tions, the case study
here analyzed being expected utility maximization from terminal wealth. After
giving some precise mathematical definitions, we point out how the pro-blems of
maximizing utility with restrictions on the wealth (say, the utility U(x)
= log x) and of maximizing unrestricted utility (say U(x)
= 1-e^{-x}) can be unified by the use of Orlicz spaces. We will see that
these spaces, generalizations of the classic L^p spaces, are naturally
induced by the utility function U itself and thus provide a natural
framework for the problem.
Lecture 4: Utility Maximization (B).
We go into the details of the proofs of the optimiza-tion problem, solved via
duality methods. The dual problem has the nice feature of being defined over the
sigma-martingale measures. Also, the optimal investment H^*
satisfies some nice properties. We show how these results can be extended to
cover the problem of utility maximization with random endowment.
Lecture 5: The indifference price as a risk measure.
This new Orlicz formulation enables several key properties of the indifference
price p(B) of a claim B satisfying conditions weaker than
those assumed in the current literature. In particular, the indifference price
functional turns out to be, apart from a sign, a convex risk measure on
the Orlicz space
induced by the utility function
U.
We conclude the lectures by pointing out some open problems.