Mathematical Finance Seminar
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Fri, 17/10/2008 14:15 |
Ernst Eberlein (Freiburg) |
Mathematical Finance Seminar  |
DH 1st floor SR |
| We discuss the valuation problem for a broad spectrum of derivatives, especially in Levy driven models. The key idea in this approach is to separate from the computational point of view the role of the two ingredients which are the payoff function and the driving process for the underlying quantity. Conditions under which valuation formulae based on Fourier and Laplace transforms hold in a general framework are analyzed. An interesting interplay between the properties of the payoff function and the driving process arises. We also derive the analytically extended characteristic function of the supremum and the infimum processes derived from a Levy process. Putting the different pieces together, we can price lookback and one-touch options in Levy driven models, as well as options on the minimum and maximum of several assets. | |||
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Fri, 24/10/2008 14:15 |
Robert Anderson (Berkeley) |
Mathematical Finance Seminar |
Oxford-Man Institute |
| 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 | |||
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Fri, 31/10/2008 14:15 |
Nizar Touzi (Polytechnique) |
Mathematical Finance Seminar |
DH 1st floor SR |
| 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. | |||
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Fri, 07/11/2008 14:15 |
Dilip Madan (Maryland) |
Mathematical Finance Seminar |
DH 1st floor SR |
| 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. | |||
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Fri, 14/11/2008 14:15 |
Ying Hu (Rennes) |
Mathematical Finance Seminar |
DH 1st floor SR |
| 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. | |||
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Fri, 21/11/2008 14:15 |
Fausto Gozzi (Luiss) |
Mathematical Finance Seminar |
DH 1st floor SR |
| 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. | |||
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Fri, 28/11/2008 14:15 |
Enrico De Giorgi (Lugano) |
Mathematical Finance Seminar |
DH 1st floor SR |
| 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 | |||
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Fri, 05/12/2008 14:15 |
Chris Rogers (Cambridge) |
Mathematical Finance Seminar |
DH 1st floor SR |
| 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. | |||
