Lecture I: 4--6 pm February 1 in Room 312 Math
Lecture II: 4--6 pm February 3 in
Room 312 Math
Lecture III: 4--6 pm February 7 in Room 622 Math
Lecture IV: 4--6 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 bid-ask 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 Growth-Optimal Portfolio under Transaction Costs (based on
http://arxiv.org/abs/1005.5105, joint work with Stefan Gerhold and Walter
Abstract: In this talk, we discuss how to use the idea of ``shadow prices''
for computations. More specifically, we determine a shadow price whose
growth-optimal portfolio coincides with the one for proportional transaction
costs in the Black-Scholes 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 Log-Utility 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 Black-Scholes 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 first-order terms.
Talk IV, Feb 8: Long-Run Optimal Portfolios under Transaction Costs (work in
progress, joint with Stefan Gerhold, Paolo Guasoni, and Walter
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 long-run optimal portfolio in
the Black-Scholes 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.
The financial crisis of 2007-2009 brought significant concerns and regulatory action regarding the role of over-the-counter markets, particularly from the viewpoint of financial instability. Over-the- 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 limit-order book, over-the-counter mar- ket
participants negotiate terms privately with other market partici-
pants, often pair-wise. Over-the-counter 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
graduate-level background in probability theory.