Past Mathematical Finance Seminar

5 December 2013
13:00
Curdin Ott
Abstract
<p> <div> <div class="h5"> <div>We consider an option whose payoff corresponds to a “capped American lookback option with floating-strike” and solve the associated pricing problem (an optimal stopping problem) in a financial market whose price process is modeled by an exponential spectrally negative Lévy process. We will present some interesting features of the solution - in fact, it turns out that the continuation region has a feature that resembles a bottleneck and hence the name “Bottleneck option”. We will also come across some well-known optimal stopping problems such as the Russian optimal stopping problem and the American lookback optimal stopping problem</div> </div> </div> </p>
  • Mathematical Finance Seminar
1 October 2013
11:30
Abstract

Relative entropy weighted optimization is convex optimization problem over the space of probability measures. Many convex optimization problems can be rephrased as such a problem. This is particularly useful since this problem type admits a quasi-explicit solution (i.e. as the expectation over a random variable), which immediately provides a Monte-Carlo method for numerically computing the solution of the optimization problem.

In this talk we discuss the background and application of this approach to stochastic optimal control problems, which may be considered as relative entropy weighted problems with Wiener space as probability space, and its connection with the theory of large deviations for Brownian functionals. As a particular application we discuss the minimization of the local time in a given point of Brownian motion with drift.

  • Mathematical Finance Seminar
23 January 2009
14:15
Abstract
We present a theory for  stochastic control problems which, in various ways, are time inconsistent in the sense that they do not admit a Bellman optimality principle. We attach these problems by viewing them within a game theoretic framework, and we look for subgame perfect Nash equilibrium points. <br /> For a general controlled Markov process and a fairly general objective functional we derive an extension of the standard Hamilton-Jacobi-Bellman  equation, in  the form of a system of non-linear equations, for the determination for the equilibrium strategy as well as the equilibrium value function. All  known examples of time inconsistency in the literature are easily seen to be special cases of the present theory. We also prove that for every time inconsistent problem, there exists an associated time consistent problem such that the optimal control and the optimal value function for the consistent problem coincides with the equilibrium control and value function respectively for the time inconsistent problem. We also study some concrete examples.
  • Mathematical Finance Seminar
5 December 2008
14:15
Chris Rogers
Abstract
The theory of risk measurement has been extensively developed over the past ten years or so, but there has been comparatively little effort devoted to using this theory to inform portfolio choice. One theme of this paper is to study how an investor in a conventional log-Brownian market would invest to optimize expected utility of terminal wealth, when subjected to a bound on his risk, as measured by a coherent law-invariant risk measure. Results of Kusuoka lead to remarkably complete expressions for the solution to this problem. The second theme of the paper is to discuss how one would actually manage (not just measure) risk. We study a principal/agent problem, where the principal is required to satisfy some risk constraint. The principal proposes a compensation package to the agent, who then optimises selfishly ignoring the risk constraint. The principal can pick a compensation package that induces the agent to select the principal's optimal choice.
  • Mathematical Finance Seminar
28 November 2008
14:15
Enrico De Giorgi
Abstract
The paper shows that financial market equilibria need not exist if agents possess cumulative prospect theory preferences with piecewise-power value functions. The reason is an infinite short-selling problem. But even when a short-sell constraint is added, non-existence can occur due to discontinuities in agents' demand functions. Existence of equilibria is established when short-sales constraints are imposed and there is also a continuum of agents in the market
  • Mathematical Finance Seminar
21 November 2008
14:15
Fausto Gozzi
Abstract
In this talk we present a work done with M. Di Giacinto (Università di Cassino - Italy) and Salvatore Federico (Scuola Normale - Pisa - Italy). The subject of the work is a continuous time stochastic model of optimal allocation for a defined contribution pension fund with a minimum guarantee. We adopt the point of view of a fund manager maximizing the expected utility from the fund wealth over an infinite horizon. The level of wealth is constrained to stay above a "solvency level". The model is naturally formulated as an optimal control problem of a stochastic delay equation with state constraints and is treated by the dynamic programming approach. We first present the study in the simplified case of no delay where a satisfactory theory can be built proving the existence of regular feedback control strategies and then go to the more general case showing some first results on the value function and on its properties.
  • Mathematical Finance Seminar
14 November 2008
14:15
Ying Hu
Abstract
We begin by the study of the problem of the exponential utility maximization. As opposed to most of the papers dealing with this subject, the investors’ trading strategies we allow underly constraints described by closed, but not necessarily convex, sets. Instead of the well-known convex duality approach, we apply a backward stochastic differential equation (BSDE) approach. This leads to the study of quadratic BSDEs. The second part gives the recent result on the existence and uniqueness of solution to quadratic BSDEs. We give also the connection between these BSDEs and quadratic PDEs. The last part will show that quadratic BSDE is critic. That is, if the BSDE is superquadratic, there exists always some BSDE without solution; and there is infinite many solutions when there is one solution. This phenomenon does not exist for quadratic and superquadratic PDEs.
  • Mathematical Finance Seminar
7 November 2008
14:15
Dilip Madan
Abstract
Stress levels embedded in S&P 500 options are constructed and re-ported. The stress function used is MINMAXV AR: Seven joint laws for the top 50 stocks in the index are considered. The first time changes a Gaussian one factor copula. The remaining six employ correlated Brownian motion independently time changed in each coordinate. Four models use daily returns, either run as Lévy processes or scaled, to the option maturity. The last two employ risk neutral marginals from the V GSSD and CGMY SSD Sato processes. The smallest stress function uses CGMY SSD risk neutral marginals and Lévy correlation. Running the Lévy process yields a lower stress surface than scaling to the option maturity. Static hedging of basket options to a particular level of accept- ability is shown to substantially lower the price at which the basket option may be o¤ered.
  • Mathematical Finance Seminar
31 October 2008
14:15
Nizar Touzi
Abstract
Starting from the problem of perfect hedging under market illiquidity, as introduced by Cetin, Jarrow and Protter, we introduce a class of second order target problems. A dual formulation in the general non-Markov case is obtained by formulating the problem under a convenient reference measure. In contrast with previous works, the controls lie in the classical H2 spaces associated to the reference measure. A dual formulation of the problem in terms of a standard stochastic control problem is derived, and involves control of the diffusion component.
  • Mathematical Finance Seminar
24 October 2008
14:15
Robert Anderson
Abstract
We prove existence of equilibrium in a continuous-time securities market in which the securities are potentially dynamically complete: the number of securities is at least one more than the number of independent sources of uncertainty. We prove that dynamic completeness of the candidate equilibrium price process follows from mild exogenous assumptions on the economic primitives of the model. Our result is universal, rather than generic: dynamic completeness of the candidate equilibrium price process and existence of equilibrium follow from the way information is revealed in a Brownian filtration, and from a mild exogenous nondegeneracy condition on the terminal security dividends. The nondegeneracy condition, which requires that finding one point at which a determinant of a Jacobian matrix of dividends is nonzero, is very easy to check. We find that the equilibrium prices, consumptions, and trading strategies are well-behaved functions of the stochastic process describing the evolution of information. We prove that equilibria of discrete approximations converge to equilibria of the continuous-time economy
  • Mathematical Finance Seminar

Pages