Mathematical Finance Internal Seminar
|
Thu, 13/10/2011 13:00 |
various |
Mathematical Finance Internal Seminar  |
DH 1st floor SR |
|
1pm Kawei Wang
Title: A Model of Behavioral Consumption in Contnuous Time Abstract: Inspired by Jin and Zhou (2008), we try to construct a model of consumption within the framework of Prospect Theory and Cumulative Prospect Theory in continuous time. 1.20 Rasmus Wissmann Title: A Principal Component Analysis-based Approach for High-Dimensional PDEs in Derivative Pricing Abstract: Complex derivatives, such as multi asset and path dependent options, often lead to high-dimensional problems. These are generally hard to tackle with numerical PDE methods, because the computational effort necessary increases exponentially with the number of dimensions. We investigate a Principal Component Analysis-based approach that aims to make the high-dimensional problem tractable by splitting it into a number of low-dimensional ones. This is done via a diagonalization of the PDE according to the eigenvectors of the covariance matrix and a subsequent Taylor-like approximation. This idea was first introduced by Reisinger and Wittum for the basic case of a vanilla option on a basket of stocks [1]. We aim to extend the approach to more complex derivatives and markets as well as to develop higher order versions. In this talk we will present the basic ideas, initial results for the example of a ratchet cap under the LIBOR Market Model and the current plans for further research. [1] C. Reisinger and G. Wittum, Efficient Hierarchical Approximation of High-Dimensional Option Pricing Problems, SIAM Journal of Scientific Computing, 2007:29 1.40 Pedro Vitoria Title: Infinitesimal Mean-Variance and Forward Utility Abstract: Mean-Variance, introduced by Markowitz in his seminal paper of 1952, is a classic criterion in Portfolio Theory that is still predominantly used today in real investment practice. In the academic literature, a number of interesting results have been produced in continuous-time version of this model. In my talk, I will establish a link between the multi-period Mean-Variance model and its continuous-time limit. A key feature of the results is that, under suitable but mild technical conditions, it captures the results of Forward Utility, thus establishing an important link between Mean-Variance and forward utility maximisation. |
|||
|
Thu, 20/10/2011 13:00 |
Simon Cotter (OCCAM) |
Mathematical Finance Internal Seminar |
DH 1st floor SR |
|
When modelling biochemical reactions within cells, it is
vitally important to take into account the effect of intrinsic noise in the
system, due to the small copy numbers of some of the chemical species.
Deterministic systems can give vastly different types of behaviour for the same
parameter sets of reaction rates as their stochastic analogues, giving us an
incorrect view of the bifurcation behaviour.
The stochastic description of this problem gives rise to a multi-dimensional Markov jump process, which can be approximated by a system of stochastic differential equations. Long-time behaviour of the process can be better understood by looking at the steady-state solution of the corresponding Fokker-Planck equation. In this talk we consider a new finite element method which uses simulated trajectories of the Markov-jump process to inform the choice of mesh in order to approximate this invariant distribution. The method has been implemented for systems in 3 dimensions, but we shall also consider systems of higher dimension. |
|||
|
Thu, 27/10/2011 13:00 |
Johannes Ruf (OMI) |
Mathematical Finance Internal Seminar |
DH 1st floor SR |
| : Recently strict local martingales have been used to model exchange rates. In such models, put-call parity does not hold if one assumes minimal superreplicating costs as contingent claim prices. I will illustrate how put-call parity can be restored by changing the definition of a contingent claim price. More precisely, I will discuss a change of numeraire technique when the underlying is only a local martingale. Then, the new measure is not necessarily equivalent to the old measure. If one now defines the price of a contingent claim as the minimal superreplicating costs under both measures, then put-call parity holds. I will discuss properties of this new pricing operator. To illustrate this techniques, I will discuss the class of "Quadratic Normal Volatility" models, which have drawn much attention in the financial industry due to their analytic tractability and flexibility. This talk is based on joint work with Peter Carr and Travis Fisher. | |||
|
Thu, 03/11/2011 13:00 |
Greg Gyurko |
Mathematical Finance Internal Seminar |
DH 1st floor SR |
| Cubature on Wiener space" is a numerical method for the weak approximation of SDEs. After an introduction to this method we present some cases when the method is computationally expensive, and highlight some techniques that improve the tractability. In particular, we adapt the Multilevel Monte-Carlo framework and extend the Milstein-scheme based version of Mike Giles to higher dimensional and higher degree cases. | |||
|
Thu, 10/11/2011 13:00 |
Hanqing Jin |
Mathematical Finance Internal Seminar |
DH 1st floor SR |
| In this work, we study equilibrium solutions for a LQ control problem with state-dependent terms in the objective, which destroy the time-consisitence of a pre-commited optimal solution. We get a sufficient condition for equilibrium by a system of stochastic differential equations. When the coefficients in the problem are all deterministic, we find an explicit equilibrium for general LQ control problem. For the mean-variance portfolio selection in a complete financial market, we also get an explicit equilibrium with random coefficient of the financial. | |||
|
Thu, 17/11/2011 13:00 |
Vladimir Cherny |
Mathematical Finance Internal Seminar |
DH 1st floor SR |
| We consider a portfolio optimisation problem on infinite horizon when the investment policy satisfies the drawdown constraint, which is the wealth process of an investor is always above a threshold given as a function of the past maximum of the wealth process. The preferences are given by a utility function and investor aims to maximise an asymptotic growth rate of her expected utility of wealth. This problem was firstly considered by Grossman and Zhou [3] and solved for a Black-Scholes market and linear drawdown constraint. The main contribution of the paper is an equivalence result: the constrained problem with utility U and drawdown function w has the same value function as the unconstrained problem with utility UoF, where function F is given explicitly in terms of w. This work was inspired by ideas from [2], whose results are a special case of our work. We show that the connection between constrained and unconstrained problems holds for a much more general setup than their paper, i.e. a general semimartingale market, larger class of utility functions and drawdown function which is not necessarily linear. The paper greatly simplifies previous approaches using the tools of Azema-Yor processes developed in [1]. In fact we show that the optimal wealth process for constrained problem can be found as an explicit Azema-Yor transformation of the optimal wealth process for the unconstrained problem. We further provide examples with explicit solution for complete and incomplete markets. [1] Carraro, L., Karoui, N. E., and Obloj, J. On Azema-Yor processes, their optimal properties and the Bachelier-Drawdown equation, to appear in Annals of Probability, 2011. [2] Cvitanic, J., and Karatzas, I. On portfolio optimization under drawdown constraints. IMA Volumes in Mathematics and Its Applications 65(3), 1994, 35-45 [3] Grossman, S. J., and Zhou, Z. Optimal investment strategies for controlling drawdowns. Mathematical Finance 3(3), 1993, 241-276 | |||
|
Tue, 22/11/2011 14:15 |
Vicky Henderson |
Mathematical Finance Internal Seminar |
Oxford-Man Institute |
| NB: EXTRA SEMINAR THIS WEEK Executives compensated with stock options generally receive grants periodically and so on any given date, may have a portfolio of options of differing strikes and maturities on their company’s stock. Non-transferability and trading restrictions in the company stock result in the executive facing unhedgeable risk. We employ exponential utility indifference pricing to analyse the optimal exercise thresholds for each option, option values and cost of the options to shareholders. Portfolio interaction effects mean that each of these differ, depending on the composition of the remainder of the portfolio. In particular, the cost to shareholders of an option portfolio is lowered relative to its cost computed on a per-option basis. The model can explain a number of empirical observations - which options are attractive to exercise first, how exercise changes following a new grant, and early exercise. Joint work with Jia Sun and Elizabeth Whalley (WBS). | |||
|
Thu, 24/11/2011 13:00 |
Alan Whitley |
Mathematical Finance Internal Seminar |
DH 1st floor SR |
| We analyse the effect of a natural change to the time variable on the convergence of the Crank-Nicholson scheme when applied to the solution of the heat equation with Dirac delta function initial conditions. In the original variables, the scheme is known to diverge as the time step is reduced with the ratio (lambda) of the time step to space step held constant - the value of lambda controls how fast the divergence occurs. After introducing the square root of time variable we prove that the numerical scheme for the transformed PDE now always converges and that lambda controls the order of convergence, quadratic convergence being achieved for lambda below a critical value. Numerical results indicate that the time change used with an appropriate value of lambda also results in quadratic convergence for the calculation of gamma for a European call option without the need for Rannacher start-up steps. Finally, some results and analysis are presented for the effect of the time change on the calculation of the option value and greeks for the American put calculated by the penalty method with Crank-Nicholson time-stepping. | |||
