Mathematical Finance and Probability Seminars (Since covid these events are taking place online.)
Dyson series for the PDEs arising in Mathematical Finance II
Tuesday, April 12, 2011 at 11:15am - 12:15pm
Speaker: Victor Nistor, Penn State University
An important problem in Mathematical Finance is to find a realistic price for the financial instruments traded in the market. For example, the market for Options (or Contingent Claims) has recently surpassed the 600 Trillion mark. The pricing of many of such Options can be reduced to solving a parabolic initial value problem. The best known such parabolic initial value problem is formulated using the famous Black-Scholes equation. A more realistic modeling of the financial market, however, will result if more complicated models are used. These more complicated models, on the other hand, lead to more complicated equations.
Our talks will be devoted to explain a (new) method to approximately solve these more complicated equations that arise in practice. More precisely, in the FIRST TALK we will describe some general results on the analysis of these more general equations. It turns out that there exists an intrinsic geometry associated to the equation, and the properties of this geometry plays an important role in the analysis. We will discuss this and some general analysis results in the framework of "Lie manifolds," a class of manifolds that is useful to model the equations of interest. We will also justify the Dyson series that will be used in the second part of the talk. The SECOND TALK will be devoted to applications to Mathematical Finance. In particular, we will explain how to approximately compute Dyson series using the Campbell-Hausdorff commutator formula and how to apply the results to the computations of option prices, both form small and large times. These results will be tested for different models: Black-Scholes, CEV (constant elasticity of variance), and Heston and other stochastic volatility models. Background material will be included in both talks, and the second talk should be understandable even if you did not attend the first talk.
Speaker: Victor Nistor, Penn State University
Slides: (TBA)
Our talks will be devoted to explain a (new) method to approximately solve these more complicated equations that arise in practice. More precisely, in the FIRST TALK we will describe some general results on the analysis of these more general equations. It turns out that there exists an intrinsic geometry associated to the equation, and the properties of this geometry plays an important role in the analysis. We will discuss this and some general analysis results in the framework of "Lie manifolds," a class of manifolds that is useful to model the equations of interest. We will also justify the Dyson series that will be used in the second part of the talk. The SECOND TALK will be devoted to applications to Mathematical Finance. In particular, we will explain how to approximately compute Dyson series using the Campbell-Hausdorff commutator formula and how to apply the results to the computations of option prices, both form small and large times. These results will be tested for different models: Black-Scholes, CEV (constant elasticity of variance), and Heston and other stochastic volatility models. Background material will be included in both talks, and the second talk should be understandable even if you did not attend the first talk.
Speaker: Victor Nistor, Penn State University
Slides: (TBA)
Location Hill 705