Thomas R. Hurd

Cubature Kalman Filtering for Continuous-Discrete Systems: Theory and Simulations

In this paper, we extend the cubature Kalman filter (CKF) to deal with nonlinear state-space models of the continuous-discrete kind. To be consistent with the literature, the resulting nonlinear filter is referred to as the continuous-discrete cubature Kalman filter (CD-CKF). We use the Itô-Taylor expansion of order 1.5 to transform the process equation, modeled in the form of stochastic ordinary differential equations, into a set of stochastic difference equations. Building on this transformation and assuming that all conditional densities are Gaussian-distributed, the solution to the Bayesian filter reduces to the problem of how to compute Gaussian-weighted integrals. To numerically compute the integrals, we use the third-degree cubature rule. For a reliable implementation of the CD-CKF in a finite word-length machine, it is structurally modified to propagate the square-roots of the covariance matrices. The reliability and accuracy of the square-root version of the CD-CKF are tested in a case study that involves the use of a radar problem of practical significance; the problem considered herein is challenging in the context of radar in two respects- high dimensionality of the state and increasing degree of nonlinearity. The results, presented herein, indicate that the CD-CKF markedly outperforms existing continuous-discrete filters.

Portfolio choice with jumps: A closed-form solution

We analyze the consumption-portfolio selection problem of an investor facing both Brownian and jump risks. We bring new tools, in the form of orthogonal decompositions, to bear on the problem in order to determine the optimal portfolio in closed form. We show that the optimal policy is for the investor to focus on controlling his exposure to the jump risk, while exploiting differences in the Brownian risk of the asset returns that lies in the orthogonal space.

Saddlepoint approximation method for pricing CDOs

A critical issue in the credit risk industry is the accurate, efficient and robust pricing of collateralized debt obligations (CDO) in a variety of mathematical models. These and many similar basket default products are very complex, due to characteristics of the large number of individual firms upon which they depend. Despite this complexity and because of their versatility, such products have become popular in the market. A central difficulty which arises in most models of CDOs is the efficient computation of conditional default loss distributions. Since exact computation is feasible only in highly symmetric situations, it is necessary to have a variety of acceptable approximation schemes. The present paper explores one general method, the saddlepoint approximation, and shows that it offers an improvement when compared with simpler methods.

Systemic Risk in Banking Networks Without Monte Carlo Simulation

An analytical approach to calculating the expected size of contagion events in models of banking networks is presented. The method is applicable to networks with arbitrary degree distributions, permits cascades to be initiated by the default of one or more banks, and includes liquidity risk effects. Theoretical results are validated by comparison with Monte Carlo simulations, and may be used to assess the stability of a given banking network topology.

Affine Markov chain model of multifirm credit migration

This paper introduces and explores a natural extension of the Chen–Filipovic affine models for credit migration, credit spreads and credit default correlation. The essential addition proposed here is to introduce a Markov chain for the “credit rating” of each firm, which are independent conditioned on a stochastic time change. The stochastic time change is then combined with other stochastic factors, here the interest rate and the recovery rate, into a multidimensional affine process. The resulting general framework has the computational effectiveness of the Chen–Filipovic models, but without certain of their conceptual drawbacks. This paper, as the first of the series, aims to illustrate the potential of the general framework by exploring a minimal implementation which is still capable of combining stochastic interest rates, stochastic recovery rates and the multifirm default process. Already within this minimal version we see very good reproduction of essential features such as credit spread curves, default correlations and multifirm default distributions. 1This research was supported by the Natural Sciences and Engineering Research Council of Canada and the MITACS National Centre of Excellence, Canada.

Contagion! Systemic Risk in Financial Networks

Attempts to define systemic risk are summarized and found to be deficient in various respects. In this introductory chapter, after considering some of the salient features of financial crises in the past, we focus on the key characteristics of banks, their balance sheets and how they are regulated. Bankruptcy! Mr. Micawber, David Copperfield’s debt-ridden sometime mentor, knew first hand the difference between surplus and deficit, between happiness and the debtors’ prison. In Dickens’ fictional universe, and perhaps even in the real world of Victorian England, a small businessman’s unpaid debts were never overlooked but always lead him and his loved ones to the unmitigated misery of the poorhouse. On the other hand, the aristocrats and upper classes, were treated more delicately, and usually given a comfortable escape. For people, firms, and in particular banks, bankruptcy in modern times is more complicated yet it still retains some of the flavour of the olden days. When a bank fails, it often seems that the rich financiers responsible for its collapse and the collateral damage it inflicts walk away from the wreckage with intact bonuses and compensation packages. When a particularly egregious case arises and a scapegoat is needed, then a middle rank banker is identified who takes the bullet for the disaster. A cynic might say that despite the dictates of Basel I, II, III, ...•, bank executives remain free to take excessive risks with their company, receiving a rich fraction of any upside while insulating themselves from any possible disaster they might cause. As we learn afresh during every large scale financial crisis, society at large pays the ultimate costs when banks fail. Spiking unemployment leads to the poverty of 1 Charles Dickens, David Copperfield, Chapter 12, p. 185 (1950). First published 1849–1850.

Portfolio Choice in Markets with Contagion

We consider the problem of optimal investment and consumption in a class of multidimensional jump-diffusion models in which asset prices are subject to mutually exciting jump processes. This captures a type of contagion where each downward jump in an assets price results in increased likelihood of further jumps, both in that asset and in the other assets. We solve in closed-form the dynamic consumption-investment problem of a log-utility investor in such a contagion model, prove a theorem verifying its optimality and discuss features of the solution, including flight-to-quality. The exponential and power utility investors are also considered: in these cases, the optimal strategy can be characterized as a distortion of the strategy of a corresponding non-contagion investor.

Indifference Pricing and Hedging for Volatility Derivatives

Utility based indifference pricing and hedging are now considered to be an economically natural method for valuing contingent claims in incomplete markets. However, acceptance of these concepts by the wide financial community has been hampered by the computational and conceptual difficulty of the approach. This paper focuses on the problem of computing indifference prices for derivative securities in a class of incomplete stochastic volatility models general enough to include important examples. A rigorous development is presented based on identifying the natural martingales in the model, leading to a nonlinear Feynman–Kac representation for the indifference price of contingent claims on volatility. To illustrate the power of this representation, closed form solutions are given for the indifference price of a variance swap in the standard Heston model and in a new “reciprocal Heston” model. These are the first known explicit formulas for the indifference price for a class of derivatives that is important to the finance industry. * Research supported by the Natural Sciences and Engineering Research Council of Canada and MITACS, Mathematics of Information Technology and Complex Systems Canada

Wiener Chaos and the Cox-Ingersoll-Ross model

In this paper we recast the Cox–Ingersoll–Ross (CIR) model of interest rates into the chaotic representation recently introduced by Hughston and Rafailidis. Beginning with the ‘squared Gaussian representation’ of the CIR model, we find a simple expression for the fundamental random variable X∞. By use of techniques from the theory of infinite–dimensional Gaussian integration, we derive an explicit formula for the nth term of the Wiener chaos expansion of the CIR model, for n = 0,1,2,…. We then derive a new expression for the price of a zero coupon bond which reveals a connection between Gaussian measures and Ricatti differential equations.

The Role of Hellinger Processes in Mathematical Finance

This paper illustrates the natural role that Hellinger processes can play in solving problems from ¯nance. We propose an extension of the concept of Hellinger process applicable to entropy distance and f-divergence distances, where f is a convex logarithmic function or a convex power function with general order q, 0 6= q < 1. These concepts lead to a new approach to Mertons optimal portfolio problem and its dual in general L¶evy markets.