SCIENTIFIC PROGRAMS AND ACTIVITIES

Actuarial Science and Mathematical Finance Group Meetings 2008-09
at the Fields Institute

Volatility and Covariation of Financial Assets: A High-Frequency Analysis
Using high frequency data for the price dynamics of equities we measure the impact that market microstructure noise has on estimates of the: (i) volatility of returns; and (ii) variance-covariance matrix of n assets. We propose a Kalman filter-based methodology that allows us to decompose price series into the true efficient price and the microstructure noise. This approach allows us to employ volatility estimators that achieve very low Root Mean Squared Errors (RMSEs) compared to other estimators that have been proposed to deal with market microstructure noise at high frequencies. Furthermore, this price series decomposition allows us to estimate the variance covariance matrix of n assets in a more efficient way than the methods so far proposed in the literature. We illustrate our results by calculating how microstructure noise affects portfolio decisions and calculations of the equity beta in a CAPM setting.

Birgit Rudloff,ORFE, Princeton University
Optimal Investment Strategies Under Bounded Risk
It is well known that the optimal investment strategy in the classical utility maximization problem can be very risky. As a consequence, in recent research a risk constraint was added to the classical utility maximization problem to control the risky part. We study the utility maximization problem when a convex or a coherent risk measure is used in the risk constraint. As a special case a model with partial information on the drift and an entropic risk constraint will be considered. The optimal terminal wealth and the optimal trading strategies are calculated. Numerical examples illustrate the analytic results. For the general case of an arbitrary convex risk measure, the problem gets more involved. We discuss the limitation of Lagrange Duality, propose Fenchel Duality instead and solve the problem for special cases.

Yukio Muromachi, Tokyo Metropolitan University
Analysis of the concentration risk: decomposing the total risk of a portfolio into the contributions of individual assets.

Analysing the concentration risk in a portfolio is one of the important themes in the fnancial institutions after finishing preparation for the BASEL II. While some measures have been proposed in order to quantify the riskiness of an asset in a portfolio in consideration for the diversifcation e ect, in this talk, we consider the risk contribution (RC) of asset j, RCj, defined by the partial derivative of the total risk of the portfolio (Rp) with respect to the holding amount of asset j (aj), multiplied by aj. In general, the sum of RCs of all assets are equal to the total risk of the portfolio Rp. The RCs has desirable features, however, it is also known that the exact and robust estimation is very difficult, especially by the Monte Carlo method. I talk about an analytical approach called "hybrid method", in which the risk factors are assumed to be conditionally independent, and the approximated values of the RCs is calculated analytically by using saddlepoint approximation.

Alex Levin, Principal Financial Engineer, Algorithmics Inc.
Affine Extensions of the Heston Model with Stochastic Interest Rates

We present affine displaced stochastic volatility extensions of the Heston’93 model for equity indices, FX rates, inflation indices, and correlated with the assets stochastic interest rates described by “additive” version of the multi-factor Hull - White model. The corresponding closed-form “semi-analytical” solutions for the price of European options are derived based on a general Duffie, Pan, and Singleton (2000) “extended transform” approach for solving affine jump-diffusion problems re-written in the form of a “discounted characteristic function” . We consider more general affine diffusion models than described in the Pan and Singleton canonical representation: rectangular volatility matrices with the number of Wiener processes greater than the number of state variables, time-dependent drifts for the interest rates and underlying assets, and time-dependent mean-reversion level for the Heston stochastic variance. This allows for a perfect fit into the observed interest rate term structures, dividend yield (forward rate) term structures and term structure of the variance swap prices (e.g., VIX Index term structure) and effective calibration procedures. Considered model is convenient for pricing and hedging hybrid long-term derivatives and diversified multi-asset (and multi-currency) portfolios including portfolios of insurance companies.

This talk is an extension of the author’s presentation on the same topic at the Fifth Congress of the Bachelier Finance Society, London 2008.

Alexey Kuznetsov, Department of Mathematics and Statistics, York University
Computing distributions of the first passage time, overshoot and some other functionals of a Levy process.

In this talk we will discuss some recent work on computing distributions of various functionals of a Levy process. First, we will present a method for computing the joint density of the first passage time and the overshoot. This method is based on a numerical scheme for solving Wiener-Hopf integral equations coupled with the local information, provided by backward Kolmogorov equation. Second, we will discuss some classical results on Wiener-Hopf factorization method and its numerical implementation for a class of processes with phase-type jumps. Finally, we will introduce a new class of Levy processes, which is qualitatively similar to CGMY family, but for which the Wiener-Hopf factors can be recovered almost explicitly (and very efficiently from the computational point of view). We will also present numerical results, possible applications in Mathematical Finance, and discuss some future directions for research.

Adam Metzler, Department of Applied Mathematics, University of Western Ontario
A Multiname First Passage Model for Credit Risk
In multiname extensions of the seminal Black-Cox model, dependence between corporate defaults is typically introduced by correlating the Brownian motions driving firm values. Despite its significant intuitive appeal, such a framework is simply not capable of describing market data. In this talk we investigate an alternative framework, in which dependence is introduced via stochastic trend and volatility in obligors’ credit qualities. We find that several specifications of the framework are capable of describing market data for synthetic CDO tranches, and compare calibrated parameters from both 2006 and 2008.

Cody Hyndman, Department of Mathematics and Statistics, Concordia University
Forward-backward Stochastic Differential Equations and Term Structure Derivatives

We consider the application of forward-backward stochastic differential equations (FBSDEs) to the problem of pricing and hedging various term structure derivatives. The underlying model assumed for the factors of the economy is a multi-factor affine diffusion. We consider affine term structure models (ATSMs) where the short-rate model is an affine function of the factors process and affine price models (APMs) where the price a risky asset is an exponential affine function of the factors process and the dividend yield is an affine function of the factors. Characterizing the underlying factor dynamics and derivative prices as FBSDEs allows for analytic solutions in certain cases and for the implementation of simulation-based numerical methods for solving FBSDEs.

Angelo Valov, Department of Statistic, University of Toronto
Integral equations arising from the First Passage Time problem via martingale methods
Some of the main tools in attacking the First Passage Time (FPT) problem for Brownian motion are integral equations of Voltera or Fredholm type. In this talk I will discuss a martingale method to construct such equations, generalize existing Voltera equations of the first kind and provide a simple alternative derivation of some known results. Furthermore I will discuss conditions for existence of a unique continuous solution for a subclass of Voltera equations. Finally I will present a partial solution to both the FPT problem and the corresponding inverse problem by introducing a random shift in the Brownian path.

Matt Davison, Canada Research Chair in Quantitative Finance Associate Professor of Applied Mathematics and of Statistical & Actuarial Sciences, The University of Western Ontario.
Applied Stochastic Modelling in Energy Finance

Energy Markets are a frontier areas of Mathematical Finance. They differ from traditional financial mathematics in a number of ways. First, both the spot price processes they engender and the financial derivatives written on these prices tend toward complication. Second, since energy assets are primarily consumption assets, the role of supply demand balance and the physical realities of energy infrastructure play a significant role.

My research, and this seminar, focus on electricity and natural gas markets with a primary focus on electricity. In this talk I review my work with Anderson on hybrid models for electricity prices, and my work with Thompson, Zhao, and Rasmussen on the "real options" problem of valuation and optimal control of energy production and storage assets. I conclude my talk with a discussion of a current research direction, that of extending these ideas to "green" energy assets and, in particular, to the related problem of valuing weather forecasts.