Plenary Session I
Wednesday July 21, 2004
8:45 - 10:30
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From Measure Changes to Time Changes for Asset Pricing
Helyette Geman
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| Security prices are modelled as stochastic processes defined on a filtered probability space (omega, F, Ft, P) where P is the real probability measure. The seminal work by Harrison, Kreps and Pliska led to the beautiful martingale representation for discounted asset prices, and in turn, to the expression of the current price of a contingent claim as an expectation. The change of numeraire coupled with the corresponding change of probability measure allowed us to extend the result to complex options and to establish the key role of the numeraire in the self-financing hedging strategy.
After changing the measure, a natural path to explore was to change the filtration via a time change. A stochastic clock of interest was the transaction clock driven by the intrinsic time of the market. The no-arbitrage assumption implies that asset prices are semimartingales under the real probability measure. Monroe's theorem tells us that semimartingales are time-changed Brownian motions. This result leads to new and
noteworthy classes of processes, in particular, pure jump Levy processes for asset price modelling. In other words, financial theory hinges upon the transformation of martingales into semimartingales originating from changes in time or probability.
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Capital Structure and the Present Value of a Firm's Investment Opportunities:
A Reduced Form Credit Risk Perspective
Robert Jarrow
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| This paper develops a theory of the firm to investigate the impact of a firm's capital structure on the present value of its investment opportunities. Given standard market imperfections, we show that maximizing the firm's equity value is consistent with the need to include a capital charge for an investment specific to a firm's capital structure and in excess of the investment's market determined risk. A reduced form credit risk perspective is taken to enable a continuous time implementation. This continuous time implementation is illustrated within the paper.
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Plenary Session II
Thursday July 22, 2004
3:45 - 5:15
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A Consumption--Investment Problem with Production Possibilities
Masaaki Kijima
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| We investigate a consumption-investment problem in the setting of corporate finance by considering a single agent with production possibilities. He can invest funds into both manufacturing and financial assets to diversify the income. The agent, endowed with an initial fund as well as initial production assets, strives to maximize the total expected utility from consumption over the finite time horizon. We establish for this problem a separation theorem. Namely, it can be solved by a two-stage procedure. The first stage is an independent optimization problem for the manufacturing arm and the second one is a standard Merton consumption-investment (portfolio selection) problem. The input parameter of the latter, namely, the initial budget, is determined by the optimal value of the manufacturing problem. Our analysis uses the Bismut stochastic maximum principle. In the case of deterministic coefficients and absence of random fluctuations the first problem is a classical deterministic problem which can be analyzed by the Pontriagin maximum principle.
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Multivariate Extremes and Market Risk Scenarios
Paul Embrechts
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| The problem we are interested in is finding models for the conditional distribution function of a (typically high-dimensional) vector, conditional on the event that the vector lies in some remote hyperplane. "Remote" is to be interpreted in an asymptotic (probabilistic) sense. In order to make this problem both doable and non-trivial, several restrictions on the type of asymptotics above are to be imposed. As a result we obtain a fairly canonical class of conditional limit distributions which we claim can be considered as models for extreme market risk scenarios. Besides working out the limit laws, we also study the relevant domains of attraction of these limits. We finally compare and contrast our new approach with more classical multivariate extreme value theory and discuss the relative merits for applications to financial risk management.
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Plenary Session III
Friday July 23, 2004
9:00 - 10:30
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Measuring risks and valuing options
Martin Schweizer
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| Risk measurement has become one of the important topics in mathematical finance in the last few years. Recent developments also show close links to the problem of valuing derivatives in (complete or incomplete) financial markets. We try to give an overview of key ideas and concepts in this field.
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Perpetual defaultable callable convertible bonds
L.C.G. Rogers
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| When pricing default-free callable convertible bonds, the firm may choose when to call, and the bond-holders have the choice of whether or not to surrender or convert. In addition, the bond-holders may choose to convert prior to the issue being called. These choices are still available when the bond is risky, but there are other choices; the firm may declare bankruptcy if its asset value falls too far. We study this problem and determine optimal policies for firm and bondholders in a simple model.
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Plenary Session IV
Saturday July 24, 2004
3:30 - 5:00
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Evidence on Actual and Risk Neutral Default Intensities
Darrell Duffie
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| I will present time-series models of actual and risk neutral default intensities, estimated firm-level financial data, macroeconomic variables, and default swap rates. This presentation is based on several collaborative projects. Among the results are evidence of significant time variation in the risk premia for bearing default risk, and of significantly higher risk premia for high-credit-quality rated firms than for low-quality firms, per unit of expected loss. The doubly-stochastic (Cox process) formulation is adopted. Tests for this assumption are provided. A new estimator is provided for the conditional term structure of actual default probabilities, based on firm-level and macro-economic covariates.
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The Variance Gamma Model: Successes, Failures, Extensions and Selected Applications.
Dilip Madan
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| A brief introduction of the Variance Gamma Model and other Lévy process models will be followed by a description of their success in describing return densities, statistically and risk neutrally. The failure of Lévy processes in general to synthesize option prices across maturities leads to the introduction of a number stochastic volatility models. The talk will close with an analysis of optimal derivative positioning, and the use of Lévy processes in the portfolio theory of asset allocation.
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