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Financial Mathematics / Mathématiques financières
(Org: Luis Seco)

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ROBERT ALMGREN, University of Toronto, Toronto, Ontario
Continuous-time model for household portfolios [PDF]

We develop a continuous-time model for a Merton-like household portfolio choice problem in which the investor is subject to undiversifiable income risk. A mean-reverting factor predicts excess return of the stock, and wealth must be allocated among investments and consumption. The investor's goal is to maximize the utility of lifetime consumption in the presence of short-sales and borrowing constraints. Using techniques of stochastic optimal control, we derive a non-linear PDE with an internal free boundary. For a reduced problem, we obtain numerical solution for the value function and thus for the optimal portfolio policies. [Joint work with Raymond Cheng]

IAN BUCKLEY, Centre for Quantitative Finance, Imperial College, London, UK
Portfolio optimization for alternative investments [PDF]

(With Gustavo Comezana, Ben Djerroud and Luis Seco.)

A tractable and practical generalization to Markowitz mean-variance style portfolio theory is presented in which portfolios of hedge funds and commodity trading advisors (CTAs) can be handled successfully. Making the assumption that their returns have the finite Gaussian mixture distribution and using the probability of outperforming a target return as the objective function, these assets are optimized in the static setting by solving a non-linear programming problem to find portfolio weights.

ABEL CADENILLAS, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta  T6G 2G1
Optimal stochastic impulse control of the free cash flow [PDF]

We apply the theory of stochastic impulse control to the determination of the optimal policy for cash disbursements and seasonal equity offerings of a financial corporation.

TAHIR CHOULLI, Alberta
Minimal Hellinger martingale measures in incomplete markets [PDF]

In incomplete markets, one of the crucial problem that we face is concerned with the choice of an ``appropiate'' risk-neutral measure to price any payoff. Via Hellinger processes, optimal criterions are proposed. These criterions are charaterized by the explicite forms for the extremal martingale measures as well as the control of markets' information dynamically. Hence the methodology illustrates an interplay between control and information theories. The relationship of the obtained martingale measures and the existing ones is investigated. The existence and comparison results are detailed in the general semimartingale framework.

ALI LAVASSANI, Calgary
[PDF]

ERIC RENAULT, Université de Montréal, CIREQ, CIRANO
Stochastic volatility models [PDF]

(Joint work with F. Comte and L. Coutin).

In this paper, we study a classical extension of the Black and Scholes model for asset prices and option pricing, generally known as the Heston model. In our specification, the volatility is a fractional integral of a Cox, Ingersoll, Ross process (also known as an ``affine'' model): this implies that it is not only stochastic but also admits long memory features. We study the volatility and the integrated volatility processes and prove their long memory properties. We address the issue of option pricing and we study discretizations of the model. Lastly, we provide an estimation strategy and simulation experiments in order to test this methodology.

TOM SALISBURY, Department of Mathematics and Statistics, York University, Toronto, Ontario  M3J 1P3
Liquidity premiums for variable annuities [PDF]

A significant level of US retirement savings are housed in variable annuity accounts. Such accounts typically impose restrictions on how funds can be moved around. What premium should these accounts pay in order to compensate the investor for the resulting lack of liquidity? The problem can be solved by reformulating it in such a way that techniques from the theory of American Options can be applied. This is joint work with Moshe Milevsky, Sid Browne, and Shannon Kennedy.

DAVE SAUNDERS, University of Pittsburgh, Pittsburgh, Pennsylvania  15260, USA
Optimal structuring of asset portfolios for insurance products with minimum guarantee provisions [PDF]

Modern insurance products are becoming increasingly complex, offering various guarantees, surrender options and bonus provisions. Typical products allow investors to participate in the returns of a reference portfolio, subject to some minimum guaranteed floor on the level of returns. The option-like nature of the payout to the investor is evident, and much work has been done on finding appropriate pricing algorithms under various assumptions on the stochastic behaviour of the reference portfolio and market risk factors. Little effort has been devoted to the problem of optimally structuring the reference portfolio. We consider this problem from the point of view of the firm offering the product. The resulting optimization problem is a nonlinear stochastic programming problem. We discuss properties of its solution and different solution algorithms. Examples illustrate how the model can be used to analyze different policy features and offer the optimally structured product for investors and shareholders.

GEORGE STOICA, University of New Brunswick, Saint John, New Brunswick
Market completeness: a return to order [PDF]

We define a trading strategy operator in a two-times stochastic economy and investigate market completeness with respect to the order relation on a linear lattice of functions describing, in a two-times economy, the associated cash flow space. The study is leading us towards alternative definitions for market completeness, in terms of trading strategy operators and approximate uniformly integrable martingales spanning on such linear lattices. In particular, we study the almost everywhere convergence situation on the space of cash flows given by the space of all equivalence classes of real valued random variables.

AGNES TOURIN, Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario  L8S 4L8
Maximizing the probability of being solvent in the presence of transaction costs [PDF]

It is a well known result that, in the presence of transaction costs, the writer of a European option may not be solvent. Here, we present a stochastic control problem which consists in maximizing the probability of being solvent. We compute the optimal probability and the free boundaries characterizing the optimal policies. This is a joint work with Thaleia Zariphopoulou.

TONY WARE, University of Calgary, Calgary, Alberta
Numerical explorations of swing options [PDF]

We describe some partial differential equation models for swing option pricing, and discuss the issue of calibrating those models to the relevant markets. We also describe a finite element scheme for solving the equations and illustrate the characteristics of these options by means of a set of numerical explorations.