Past Mathematical Finance Seminar

29 February 2008
13:15
Walter Schachermayer
Abstract
The Mutual Fund Theorem (MFT) is considered in a general semimartingale financial market S with a finite time horizon T, where agents maximize expected utility of terminal wealth. The main results are: (i) Let N be the wealth process of the numéraire portfolio (i.e. the optimal portfolio for the log utility). If any path-independent option with maturity T written on the numéraire portfolio can be replicated by trading only in N and the risk-free asset, then the (MFT) holds true for general utility functions, and the numéraire portfolio may serve as mutual fund. This generalizes Merton’s classical result on Black-Merton-Scholes markets. Conversely, under a supplementary weak completeness assumption, we show that the validity of the (MFT) for general utility functions implies the replicability property for options on the numéraire portfolio described above. (ii) If for a given class of utility functions (i.e. investors) the (MFT) holds true in all complete Brownian financial markets S, then all investors use the same utility function U, which must be of HARA type. This is a result in the spirit of the classical work by Cass and Stiglitz.
  • Mathematical Finance Seminar
22 February 2008
13:15
Peter Bank
Abstract
When liquidating large portfolios of securities one faces a trade off between adverse market impact of sell orders and the impatience to generate proceeds. We present a Black-Scholes model with an impact factor describing the market's distress arising from previous transactions and show how to solve the ensuing optimization problem via classical calculus of variations. (Joint work with Dirk Becherer, Humboldt Universität zu Berlin)
  • Mathematical Finance Seminar
15 February 2008
13:15
Huyen Pham
Abstract
We consider impulse control problems in finite horizon for diffusions with decision lag and execution delay. The new feature is that our general framework deals with the important case when several consecutive orders may be decided before the effective execution of the first one. This is motivated by financial applications in the trading of illiquid assets such as hedge funds. We show that the value functions for such control problems satisfy a suitable version of dynamic programming principle in finite dimension, which takes into account the past dependence of state process through the pending orders. The corresponding Bellman partial differential equations (PDE) system is derived, and exhibit some peculiarities on the coupled equations, domains and boundary conditions. We prove a unique characterization of the value functions to this nonstandard PDE system by means of viscosity solutions. We then provide an algorithm to find the value functions and the optimal control. This implementable algorithm involves backward and forward iterations on the domains and the value functions, which appear in turn as original arguments in the proofs for the boundary conditions and uniqueness results. Finally, we give several numerical experiments illustrating the impact of execution delay on trading strategies and on option pricing.
  • Mathematical Finance Seminar
8 February 2008
13:15
David Hobson
Abstract
In this talk we will investigate the properties of stochastic volatility models, to discuss to what extent, and with regard to which models, properties of the classical exponential Brownian motion model carry over to a stochastic volatility setting. The properties of the classical model of interest include the fact that the discounted stock price is positive for all $t$ but converges to zero almost surely, the fact that it is a martingale but not a uniformly integrable martingale, and the fact that European option prices (with convex payoff functions) are convex in the initial stock price and increasing in volatility. We give examples of stochastic volatility models where these properties continue to hold, and other examples where they fail. The main tool is a construction of a time-homogeneous autonomous volatility model via a time change.
  • Mathematical Finance Seminar
5 February 2008
13:15
Professor Bernt Oksendal
Abstract
A trader in finance is called an insider if she (or he) knows more about the prices in the market than can be obtained from the market history itself. This is the case if, for example, the trader knows something about the future price/value of a stock. We discuss the following question: What is the optimal portfolio of an insider who wants to maximize her expected profit at a given future time? The problem is that heavy trading by the insider will reveal parts of her inside price information to the market and thereby reduce her information advantage. We will solve this problem by presenting a general anticipative stochastic calculus model for insider trading. Our results generalize equilibrium results due to Kyle (1985) and Back (1992). The presentation is partly based on recent joint work with Knut Aase and Terje Bjuland, both at the Norwegian School of Economics and Business Administration (NHH).
  • Mathematical Finance Seminar
1 February 2008
13:15
Thaleia Zariphopoulou
Abstract
In this paper we derive a stochastic partial di¤erential equation whose solutions are processes relevant to the portfolio choice problem. The mar- ket is incomplete and asset prices are modelled as Ito processes. We provide solutions of the SPDE for various choices of its volatility coe¢ - cient. We also show how to imbed the classical Merton problem into our framework.
  • Mathematical Finance Seminar

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