Minerva Foundation Lectures
These special lecture series in probability and mathematical finance
are made possible by the generous support of the Minerva
Research Foundation. Time and location vary.
Spring Semester 2011

February 18, 2011
 Johannes MuhleKarbe
(ETH Zurich)
Shadow
Prices in Portfolio Optimization with Transaction
Costs
Poster
Lecture I: 46 pm February 1 in Room 312 Math
Lecture II: 46 pm February 3 in
Room 312 Math
Lecture III: 46 pm February 7 in Room 622 Math
Lecture IV: 46 pm February 8 in Room 312 Math.
Abstract for lectures:
Talk I, Feb 1: Motivation, History, and Existence (based on
http://arxiv.org/abs/0911.4801 and some new results, joint work with Jan
Kallsen, Mark Owen, and Luciano Campi)
Abstract: A ``shadow price'' is a process evolving within the bidask spread
of a market with proportional transaction cots, such that the maximal
expected utility in this frictionless market is the same as in the original
market with transaction costs. In this talk, we introduce this concept and
also outline its origins, which go back to Jouini & Kallal (J. Econom.
Theory, 1995), Kusuoka (Annals Appl. Probab., 1995), and Cvitanic & Karatzas
(Math. Finance, 1996). We also present an elementary existence proof for
finite probability spaces. Moreover, we discuss work in progress on
existence in more general setups.
Talk II, Feb 3: The GrowthOptimal Portfolio under Transaction Costs (based on
http://arxiv.org/abs/1005.5105, joint work with Stefan Gerhold and Walter
Schachermayer)
Abstract: In this talk, we discuss how to use the idea of ``shadow prices''
for computations. More specifically, we determine a shadow price whose
growthoptimal portfolio coincides with the one for proportional transaction
costs in the BlackScholes model. This provides a new simple proof for the
results of Taksar et al. (Math. Oper. Res., 1988). Moreover, it also leads
to asymptotic expansions of the optimal policy and the maximal growth rate
for small transaction costs.
Talk III, Feb 7: Maximizing LogUtility from Consumption under Transaction Costs
(based on http://arxiv.org/abs/1010.4989 resp.
http://arxiv.org/abs/1010.0627, joint work with Jan Kallsen resp. Stefan
Gerhold and Walter Schachermayer)
Abstract: We revisit the problem of maximizing expected logarithmic utility
from consumption over an infinite horizon in the BlackScholes model with
proportional transaction costs, as studied in the seminal paper of Davis and
Norman (Math. Oper. Res., 1990). As in Talk II, we tackle this problem by
determining a shadow price. Moreover, for small transaction costs, we again
determine power series of arbitrary order for the optimal policy and the
value function. This extends work of Janecek and Shreve (Finance Stoch.,
2004), who determined the firstorder terms.
Talk IV, Feb 8: LongRun Optimal Portfolios under Transaction Costs (work in
progress, joint with Stefan Gerhold, Paolo Guasoni, and Walter
Schachermayer)
Abstract: The computations in Talks II and III crucially exploited that the
investor's preferences are modeled by a logarithmic utility function. In
this talk, we describe how to determine shadow prices also for power utility
functions. More specifically, we focus on the longrun optimal portfolio in
the BlackScholes model with proportional transaction costs and provide a
rigorous proof for the results of Dumas and Luciano (J. Finance, 1991).
Moreover, we again explain how to obtain full asymptotic expansions for the
optimal policy and the optimal growth rate.

March 29April 1, 2011
 Darrell Duffie
(Stanford University)
Dark Markets
Poster
Tuesday, March 29
10:30am  12:00pm, SSW 1025
Wednesday, March 30
10:30am  12:00pm, SSW 1025
Friday, April 1
12:00pm  1:00pm, SSW 903
Abstract for lectures:
The financial crisis of 20072009 brought significant concerns and regulatory action regarding the role of overthecounter markets, particularly from the viewpoint of financial instability. Overthe counter markets for derivatives, collateralized debt obligations, and repurchase agreements played particularly important roles in the cri sis and in subsequent legislation in U.S. and Europe. This legisla tion has also focused on increasing competition and transparency. The modeling of OTC markets, however, is still relatively undeveloped in comparison to the available research on central market mechanisms.
Rather than trading through a centralized mechanism such as an
auction, specialist, or limitorder book, overthecounter mar ket
participants negotiate terms privately with other market partici
pants, often pairwise. Overthecounter investors may be largely
un aware of prices that are currently available elsewhere in the
market, or of recent transactions prices. In this sense, OTC markets
are relatively opaque; investors are somewhat in the dark about the
most attractive available terms and about who might offer
them. These lectures addresses how prices, asset allocations, and
information transmission in OTC markets are influenced by this form
of opaque ness. The objective is to provide a brief introduction to
OTC mar kets, including some of the key conceptual issues and
modeling tech niques, and to provide a foundation for reading more
advanced re search in this topic area. The lectures assume a
graduatelevel background in probability theory.
Fall Semester 2010

December 320, 2010
 Boris L. Rozovsky
(Brown University)
Generalized Malliavin Calculus and Stochastic PDEs
Abstract for lectures: [pdf]
 Friday Dec 3, 121 pm in Math 520: Probability Seminar.
 Friday, Dec 3, 2.303.20 pm in Stat 1025: Lecture I. ``Stochastic Quantization and NavierStokes Equation.''
 Friday, Dec 10, 1011:30 am in Math 622: Lecture II. ``Introduction to Malliavin calculus''.
 Friday, Dec 17, 2.304.00 pm in Stat 1025: Lecture III. ``Generalized Malliavin calculus''.
 Monday, Dec 20, 1011.30 am in Stat 1025: Lecture IV. ``Bilinear stochastic PDEs driven by stationary noise''.