June 7 - 18, 2010
We solve the problem of pricing and optimal exercise of American call-type options in markets which do not necessarily admit an equivalent local martingale measure. This resolves an open question proposed by Fernholz and Karatzas [Stochastic Portfolio Theory: A Survey, Handbook of Numerical Analysis, 15:89-168, 2009].
Joint work with Kostas Kardaras and Hao Xing. Available at http://arxiv.org/abs/0908.1082
In economic theory one typically discounts future benefits at a
constant rate. An example of this is the celebrated model of endogeneous
growth, originating with Ramsey (1928), which leads to the so-called golden
rule in macroeconomics. There are now excellent reasons (intergenerational
equity, for instance) to use non-constant discount rates. There is then a
problem of time-inconsistency: a policy which is optimal today will no
longer be so when the time comes to implement it. So optimization is
pointless, and a substitute has to be found for optimal strategies. We will
define such a substitute, namely equilibrium strategies, show how to
characterize them, and investigate what happens to the golden rule. This is
joint work with Ali Lazrak.
There is strong evidence that individuals discount future utilities at non-constant rates. The notion of optimality then disappears, because of time inconsistency (see the Tuesday colloquium) and rational behaviour then centers around equilibrium strategies. I will investigate portfolio management with hyperbolic discounting (the discount rate increases with time), and I will show that this may explain some well-known puzzles of portfolio management. This is joint work with Traian Pirvu.
We propose a model for risk neutral futures price dynamics in the European Unions Emissions Trading Scheme (EU ETS). Historical price dynamics suggests that both allowance prices for different compliance periods and CER prices for different compliance periods are significantly related. To obtain a realistic price dynamics we take into account the specific details of the EU ETS compliance regulations, such as banking and the link to the Clean Development Mechanism (CDM), and exploit arbitrage relationships between futures on EU allowances and Certified Emission Reductions.
Optimal Switching models are concerned with sequential
decision making where the controller has a finite number of policy
regimes. Such models arise naturally in pricing of energy assets,
including tolling agreements for electricity production, natural gas
storage facilities, carbon emission permits, etc. I will discuss the
general mathematical structure of optimal switching models, including
their relation to multiple stopping problems. I will then describe some
work in progress with R. Sircar on exploration control in exhaustible
In the second part of the talk, I will focus on numerical methods and
implementation issues for optimal switching, especially simulation tools
that extend Monte Carlo methods for American options.
We discuss Cournot and Bertrand models of oligopolies, first in the context of static games and then in dynamic models. The static games, involving firms with different costs, lead to questions of how many competitors actively participate in a Nash equilibrium and how many are sidelined or blockaded from entry. The dynamic games lead to systems of nonlinear partial differential equations for which we discuss asymptotic and numerical approximations. Applications include competition between energy producers in the face of exhaustible resources such as oil (Cournot); and markets for substitutable consumer goods (Bertrand).
In recent years Peng prososed a new notion called G-expectation, a type of nonlinear expectation motivated from dynamic risk measures with volatility uncertainty. On the other hand, a martingale under the G-expectation can be viewed as the solution to a "linear" Second Order Backward SDEs, the main subject of the short course which will be given by Nizar Touzi in this workshop. The theory has applications in many areas, e.g. Monte Carlo methods for fully nonlinear PDEs, finanancial problems in models with volatility uncertainty (volatility control, liquidity cost, Gamma constraint). Its main technical feature is the quasi-sure stochastic analysis, which invloves a class of mutually singular probability measures. In this talk we will introduce G-martingales, develop the quais-sure stochastic analysis, and establish the martingale epresentation theorem for G-martingales. This is a joint work with Mete Soner and Nizar Touzi.