Mathematical Finance Internal Seminar
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Thu, 05/05/2011 13:00 |
Konstantinos Zygalakis (OCCAM) |
Mathematical Finance Internal Seminar  |
DH 1st floor SR |
| In this talk we will present results concerning the large scale long time behaviour of particles moving in a periodic (random) velocity field subject to molecular diffusion. The particle can be considered massless (passive tracer) or not (inertial particle). Under appropriate assumptions for the velocity field the large scale long time behavior of the particle is described by a Brownian motion with an effective diffusivity matrix K. We then present some numerical algorithms concerning the calculation of the effective diffusivity in the limit of vanishing molecular diffusion (stochastic geometric integrators). Time permitting we will discuss the case where the driving noise is no longer white but colored and study the effects of this change to the effective diffusivity matrix. | |||
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Thu, 12/05/2011 13:00 |
Ferhana Ahmad |
Mathematical Finance Internal Seminar |
DH 1st floor SR |
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Thu, 19/05/2011 13:00 |
Lukasz Szpruch |
Mathematical Finance Internal Seminar |
DH 1st floor SR |
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Thu, 26/05/2011 13:00 |
Jan Obloj |
Mathematical Finance Internal Seminar |
DH 1st floor SR |
| In this talk I want to ask how to create a coherent mathematical framework for pricing and hedging which starts with the information available in the market and does not assume a given probabilistic setup. This calls for re-definition of notions of arbitrage and trading and, subsequently, for a “probability-free first fundamental theorem of asset pricing". The new setup should also link with a classical approach if our uncertainty about the model vanishes and we are convinced a particular probabilistic structure holds. I explore some recent results but, predominantly, I present the resulting open questions and problems. It is an “internal talk" which does not necessarily present one paper but rather wants to engage into a discussion. Ideas for the talk come in particular from joint works with Alex Cox and Mark Davis. | |||
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Thu, 02/06/2011 13:00 |
Karolina Bujok |
Mathematical Finance Internal Seminar |
DH 1st floor SR |
| We consider a multidimensional structural credit model, where each company follows a jump-diffusion process and is connected with other companies via global factors. We assume that a company can default both expectedly, due to the diffusion part, and unexpectedly, due to the jump part, by a sudden fall in a company's value as a result of a global shock. To price CDOs efficiently, we use ideas, developed by Bush et al. for diffusion processes, where the joint density of the portfolio is approximated by a limit of the empirical measure of asset values in the basket. We extend the method to jump-diffusion settings. In order to check if our model is flexible enough, we calibrate it to CDO spreads from pre-crisis and crisis periods. For both data sets, our model fits the observed spreads well, and what is important, the estimated parameters have economically convincing values. We also study the convergence of our method to basic Monte Carlo and conclude that for a CDO, that typically consists of 125 companies, the method gives close results to basic Monte Carlo." | |||
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Thu, 09/06/2011 13:00 |
Ben Hambly |
Mathematical Finance Internal Seminar |
DH 1st floor SR |
| The aim of this work is to show how to derive the electricity price from models for the underlying construction of the bid-stack. We start with modelling the behaviour of power generators and in particular the bids that they submit for power supply. By modelling the distribution of the bids and the evolution of the underlying price drivers, that is the fuels used for the generation of power, we can construct an spede which models the evolution of the bids. By solving this SPDE and integrating it up we can construct a bid-stack model which evolves in time. If we then specify an exogenous demand process it is possible to recover a model for the electricity price itself. In the case where there is just one fuel type being used there is an explicit formula for the price. If the SDEs for the underlying bid prices are Ornstein-Uhlenbeck processes, then the electricity price will be similar to this in that it will have a mean reverting character. With this price we investigate the prices of spark spreads and swing options. In the case of multiple fuel drivers we obtain a more complex expression for the price as the inversion of the bid stack cannot be used to give an explicit formula. We derive a general form for an SDE for the electricity price. We also show that other variations lead to similar, though still not tractable expressions for the price. | |||
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Thu, 16/06/2011 13:00 |
Christoph Reisinger |
Mathematical Finance Internal Seminar |
DH 1st floor SR |
| The first half of this seminar will discuss the hedging problem faced by a large sports betting agent who has to risk-manage an unwanted position in a bet on the simultaneous outcome of multiple football matches, by trading in moderately liquid simple bets on individual results. The resulting mathematical framework is that of a coupled system of multi-dimensional HJB equations. This leads to the wider question of the numerical approximation of such problems. Dynamic programming with PDEs, while very accurate in low dimensions, becomes practically intractable as the dimensionality increases. Monte Carlo methods, while robust for computing linear expectations in high dimensions, are not per se well suited to dynamic programming. This leaves high-dimensional stochastic control problems to be considered computationally infeasible in general. In the second half of the seminar, we will outline ongoing work in this area by sparse grid techniques and asymptotic expansions, the former exploiting smoothness of the value function, the latter a fast decay in the importance of principal components. We hope to instigate a discussion of other possible approaches including e.g. BSDEs. | |||
