Past Mathematical and Computational Finance Internal Seminar

In [Liang, Lyons and Qian(2009): Backward Stochastic Dynamics on a Filtered Probability Space, to appear in the Annals of Probability], the authors demonstrated that BSDEs can be reformulated as functional differential equations, and as an application, they solved BSDEs on general filtered probability spaces. In this paper the authors continue the study of functional differential equations and demonstrate how such approach can be used to solve FBSDEs. By this approach the equations can be solved in one direction altogether rather than in a forward and backward way. The solutions of FBSDEs are then employed to construct the weak solutions to a class of BSDE systems (not necessarily scalar) with quadratic growth, by a nonlinear version of Girsanov's transformation. As the solving procedure is constructive, the authors not only obtain the existence and uniqueness theorem, but also really work out the solutions to such class of BSDE systems with quadratic growth. Finally an optimal portfolio problem in incomplete markets is solved based on the functional differential equation approach and the nonlinear Girsanov's transformation.

The talk is based on the joint work with Lyons and Qian:

http://arxiv4.library.cornell.edu/abs/1011.4499

This talk highlights the role of Gaussian Process models in sequential

data analysis. Issues of active data selection, global optimisation,

sensor selection and prediction in the presence of changepoints are

discussed.

An agent is presented with an N Bandit (Fruit) machines. It is assumed that each machine produces successes or failures according to some fixed, but unknown Bernoulli distribution. If the agent plays for ever, how can he/she choose a strategy that ensures the average successes observed tend to the parameter of the "best" arm?

Alternatively suppose that the agent recieves a reward of a^n at the nth button press for a success, and 0 for a failure; now how can the agent choose a strategy to optimise his/her total expected rewards over all time? These are two examples of classic Bandit Problems.

We analyse the behaviour of two strategies, the Narendra Algorithm and the Gittins Index Strategy. The Narendra Algorithm is a "learning"

strategy, in that it answers the first question in the above paragraph, and we demonstrate this remains true when the sequences of success and failures observed on the machines are no longer i.i.d., but merely satisfy an ergodic condition. The Gittins Index Strategy optimises the reward stream given above. We demonstrate that this strategy does not "learn" and give some new explicit bounds on the Gittins Indices themselves.

We solve the problem of static hedging of (upper) barrier options (we concentrate on up-and-out put, but show how the other cases follow from this one) in models where the underlying is given by a time-homogeneous diffusion process with, possibly, independent stochastic time-change.

The main result of the paper includes analytic expression for the payoff of a (single) European-type contingent claim (which pays a certain function of the underlying value at maturity, without any pathdependence),

such that it has the same price as the barrier option up until hitting the barrier. We then consider some examples, including the Black-Scholes, CEV and zero-correlation SABR models, and investigate an approximation of the exact static hedge with two vanilla (call and put) options.

A behavioral mean-variance portfolio selection problem in continuous time is formulated and studied. Based on the standard mean-variance portfolio selection problem, the cumulative distribution function of the cash flow is distorted by a probability distortion function. Then the problem is no longer a convex optimization problem. This feature distinguishes it from the conventional linear-quadratic (LQ) problems.

The stochastic optimal LQ control theory no longer applies. We take the quantile function of the terminal cash flow as the decision variable.

The corresponding optimal terminal cash flow can be recovered by the optimal quantile function. Then the efficient strategy is the hedging strategy of the optimal terminal cash flow.

The talk concerns experiments that study strategic thinking by eliciting subjects’ initial responses to series of different but related games, while monitoring and analyzing the patterns of subjects’ searches for hidden but freely accessible payoff information along with their decisions.

Using a large panel data set of Swedish pension savers (75,000 investors, daily portfolios 2000-2008) we show that active investors outperform inactive investors and that there is a causal effect of fund switches on performance. The higher performance is earned not by market timing, but by dynamic fund picking (within the same asset class). While activity is positive for the individual investor, there are indications that it generates costs for other investors.

This talk I present a study of BSDEs with non-linear terms of quadratic growth by using Girsanov's theorem. In particular we are able to establish a non-linear version of the Cameron-Martin formula, which can be for example used to obtain gradient estimates for some non-linear parabolic equations.

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We investigate calibrating financial models using a rigorous Bayesian framework. Non-parametric approaches in particular are studied and the local volatility model is used as an example. By incorporating calibration error into our method we design optimal hedges that minimise expected loss statistics based on different Bayesian loss functions determined by an agent's preferences. Comparisons made with the standard hedge strategies show the Bayesian hedges to outperform traditional methods.

Most financial models introduced for the purpose of pricing and hedging derivatives concentrate

on the dynamics of the underlying stocks, or underlying instruments on which the derivatives

are written. However, as certain types of derivatives became liquid, it appeared reasonable to model

their prices directly and use these market models to price or hedge exotic derivatives. This framework

was originally advocated by Heath, Jarrow and Morton for the Treasury bond markets.

We discuss the characterization of arbitrage free dynamic stochastic models for the markets with

infinite number of European Call options as the liquid derivatives. Subject to our assumptions on the

presence of jumps in the underlying, the option prices are represented either through local volatility or

through local L´evy measure. Each of the latter ones is then given dynamics through an Itˆo stochastic

process in infinite dimensional space. The main thrust of our work is to characterize absence of arbitrage

in this framework and address the issue of construction of the arbitrage-free models.

We take a leisurely look at some mathematical models from population genetics and the ways that they can be analysed. Some of the models have a very familiar form - for example diffusion models of population size look a lot like interest rate models. But hopefully there will also be something new.

In this talk, we try to construct a dynamical model for the basket credit products in the credit market under the structural-model framework. We use the particle representation for the firms' asset value and investigate the evolution of the empirical measure of the particle system. By proving the convergence of the empirical measure we can achieve a stochastic PDE which is satisfied by the density of the limit empirical measure and also give an explicit formula for the default proportion at any time t. Furthermore, the dynamics of the underlying firms' asset values can be assumed to be either driven by Brownian motions or more general Levy processes, or even have some interactive effects among the particles. This is a joint work with Dr. Ben Hambly.

This talk will be based on a joint work with Professor Terry Lyons and Mr Gechun Liang (OMI). I will explain a new approach to define and to solve a class of backward dynamic systems including the well known examples of non-linear backward SDE. The new approach does not require any kind of martingale representation or any specific restriction on the probability base in question, and therefore can be applied to a much wider class of backward systems.

Diffusion process models for evolution of neutral genes have a particle dual coalescent process underlying them. Models are reversible with transition functions having a diagonal expansion in orthogonal polynomial eigenfunctions of dimension greater than one, extending classical one-dimensional diffusion models with Beta stationary distribution and Jacobi polynomial expansions to models with Dirichlet or Poisson Dirichlet stationary distributions. Another form of the transition functions is as a mixture depending on the mutant and non-mutant families represented in the leaves of an infinite-leaf coalescent tree.

The one-dimensional Wright-Fisher diffusion process is important in a characterization of a wider class of continuous time reversible Markov processes with Beta stationary distributions originally studied by Bochner (1954) and Gasper (1972). These processes include the subordinated Wright-Fisher diffusion process.