Tomasz R. Bielecki

Optimality of zero-inventory policies for unreliable manufacturing systems

We show that there are ranges of parameter values describing an unreliable manufacturing system for which zero-inventory policies are exactly optimal even when there is uncertainty in manufacturing capacity. This result may be initially surprising since it runs counter to the argument that inventories are buffers against uncertainty and therefore one must strive to maintain a strictly positive inventory as long as there is any uncertainty. However, there is a deeper reason why this argument does not hold, and why a zero-inventory policy can be optimal even in the presence of uncertainty. This provable optimality reinforces the case for zero-inventory policies, which is currently made on the separate grounds that it enforces a healthy discipline on the entire manufacturing process.

Hedging of Defaultable Claims

The goal of this chapter is to present a survey of recent developments in the practically important and challenging area of hedging credit risk. In a companion work, Bielecki et al. (2004a), we presented techniques and results related to the valuation of defaultable claims. It should be emphasized that in most existing papers on credit risk, the risk-neutral valuation of defaultable claims is not supported by any other argument than the desire to produce an arbitrage-free model of default-free and defaultable assets. Here, we focus on the possibility of a perfect replication of defaultable claims and, if the latter is not feasible, on various approaches to hedging in an incomplete setting.

Risk Sensitive Asset Management with Transaction Costs

Abstract. This paper develops a continuous time risk-sensitive portfolio optimization model with a general transaction cost structure and where the individual securities or asset categories are explicitly affected by underlying economic factors. The security prices and factors follow diffusion processes with the drift and diffusion coefficients for the securities being functions of the factor levels. We develop methods of risk sensitive impulsive control theory in order to maximize an infinite horizon objective that is natural and features the long run expected growth rate, the asymptotic variance, and a single risk aversion parameter. The optimal trading strategy has a simple characterization in terms of the security prices and the factor levels. Moreover, it can be computed by solving a {\it risk sensitive quasi-variational inequality}. The Kelly criterion case is also studied, and the various results are related to the recent work by Morton and Pliska.

Risk sensitive control of finite state Markov chains in discrete time, with applications to portfolio management

Abstract. In this paper we extend standard dynamic programming results for the risk sensitive optimal control of discrete time Markov chains to a new class of models. The state space is only finite, but now the assumptions about the Markov transition matrix are much less restrictive. Our results are then applied to the financial problem of managing a portfolio of assets which are affected by Markovian microeconomic and macroeconomic factors and where the investor seeks to maximize the portfolios risk adjusted growth rate.

Multiple Ratings Model of Defaultable Term Structure

A new approach to modeling credit risk, to valuation of defaultable debt and to pricing of credit derivatives is developed. Our approach, based on the Heath, Jarrow, and Morton (1992) methodology, uses the available information about the credit spreads combined with the available information about the recovery rates to model the intensities of credit migrations between various credit ratings classes. This results in a conditionally Markovian model of credit risk. We then combine our model of credit risk with a model of interest rate risk in order to derive an arbitrage-free model of defaultable bonds. As expected, the market price processes of interest rate risk and credit risk provide a natural connection between the actual and the martingale probabilities.

Risk sensitive asset allocation

Abstract This paper develops a continuous time modeling approach for making optimal asset allocation decisions. Macroeconomic and financial factors are explicitly modeled as Gaussian stochastic processes which directly affect the mean returns of the assets. We employ methods of risk sensitive control theory, thereby using an infinite horizon objective that is natural and features the long run expected growth rate and the asymptotic variance as two measures of performance, analogous to the mean return and variance, respectively, in the single period Markowitz model. The optimal strategy is a simple function of the factor levels, and, even with constraints on the portfolio proportions, it can be computed by solving a quadratic program. Explicit formulas can be obtained, as is illustrated by an example where the only factor is a Vasicek-type interest rate and where there are two assets: cash and a stock index. The methods are further illustrated by studies of two data sets: U.S. data with two assets and up to three factors, and Australian data with three assets and three factors.

Up and down credit risk

This paper discusses the main modeling approaches that have been developed for handling portfolio credit derivatives, with a focus on the question of hedging. In particular, the so-called top, top down and bottom up approaches are considered. We give some mathematical insights regarding the fact that information, namely the choice of a relevant model filtration, is the major modeling issue. In this regard, we examine the notion of thinning that was recently advocated for the purpose of hedging a multi-name derivative by single-name derivatives. We then illustrate by means of numerical simulations (semi-static hedging experiments) why and when the portfolio loss process may not be a ‘sufficient statistic’ for the purpose of valuation and hedging of portfolio credit risk.

Arbitrage pricing of defaultable game options with applications to convertible bonds

This paper is the first in a series that we devote to studying the problems of valuation and hedging of defaultable game options in general, and convertible corporate bonds in particular. Here, we present mathematical foundations for our overall study. Specifically, we provide several results characterizing the arbitrage price of a defaultable game option in terms of relevant Dynkin games. In addition, we provide important results regarding price decomposition of defaultable options. These general results are then specified to the case of convertible bonds, yielding in particular a decomposition of convertible bonds in an optional and a bond component.

A Markov copulae approach to pricing and hedging of credit index derivatives and ratings triggered step-up bonds

The paper presents selected results from the theory of Markov copulae and some of their applications in finance. ∗This research was partially supported by NSF Grant DMS-0604789 and Moody’s Corporation grant 5-55411. †The authors would like to express their sincere gratitude and appreciation to Matt Woodhams from the GFI Group, for providing us with data relevant to this paper. ‡Corresponding author; Department of Applied Mathematics, Illinois Institute of Technology, 10W 32nd Street, Chicago, IL 60616, USA; e-mail: bielecki@iit.edu §Department of Applied Mathematics, Illinois Institute of Technology, 10W 32nd Street, Chicago, IL 60616, USA; e-mail: vidoseb@iit.edu ¶Department of Applied Mathematics, Illinois Institute of Technology, 10W 32nd Street, Chicago, IL 60616, USA; e-mail: vidofre@iit.edu

Pricing and trading credit default swaps in a hazard process model

In the paper we study dynamics of the arbitrage prices of credit default swaps within a hazard process model of credit risk. We derive these dynamics without postulating that the immersion property is satisfied between some relevant filtrations. These results are then applied so to study the problem of replication of general defaultable claims, including some basket claims, by means of dynamic trading of credit default swaps.