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Seminar: Winter 2007
Stanford Financial Mathematics Seminar Schedule
A parameter estimation problem for SPDE's with multiplicative noise
and it application to fixed income markets |
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Igor Cialenco (USC)
We will investigate a parameter estimation
problem for some parabolic SPDEs driven by multiplicative noise.
Some Maximum Likelihood Estimators of the parameter,
based on finite-dimensional approximation of the solution will be
presented and consistency, both in time and space variables,
and asymptotic normality of these estimators will be discussed.
Finally we will show how this problem can be applied in fixed income
market (modeling forward rates).
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| Hedging under L2 convex risk measures |
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Antoine Toussaint (Princeton)
We consider the problem of minimizing the risk of a financial
position (hedging) in an incomplete market. It is well-known that the
industry standard for risk measure, the Value at Risk, does not take into account the natural idea that risk should be minimized through
diversification. This observation led to the theory of convex risk measures by Follmer and Schied. But as a theory on bounded financial positions, it is not ideally suited for the problem of hedging because simple buy-hold strategies may not be bounded. Therefore, we propose as an alternative to extend convex risk measures as functionals on L2. This framework is more suitable for optimal hedging with L2 valued financial markets, where the theory of stochastic integration plays a natural role. We then study the problem of hedging financial contracts with optimal trading strategies to minimize the convex risk measure. In the case of constrained trading strategies we also prove the existence of the optimal hedge.
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| Dupire and Forward Kolmogorov Equations for Two Dimensional Options |
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Olivier Pironneau (University of Paris VI)
Pricing options on multiple underlying or on an underlying modeled with stochastic volatility may involve solving multi-dimensional Black-Scholes like partial differential equations (PDE). Computing several such options at once for various moneyness levels can be a numerical challenge. We investigate here the Kolmogorov equation and Dupire or ``pre-Dupire" equations to solve the problem faster and we validate the approach numerically. The heart of the method is to use the adjoint of the PDE of the option at the discrete level and to use discrete duality identities to obtain Dupire-like relations. The method works on most linear models. Numerical results are given for a European call option on a basket of two assets.
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| A Diffusion Model for Reinsurance |
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Vassilis Papanicolaou (National Technical University of Athens)
We study the problem of the joint risk process undertaken by
an insurer and a reinsurer, in the diffusion approximation. Using techniques
from stochastic analysis and partial differential equations we calculate the
ruin probability of the scheme. We also propose an optimal reinsurance
scheme that minimizes this ruin probability.
(joint work with N. Fragkos, A. Yiannacopoulos, and M. Zazanis)
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