Book Chapter

Jump Bond Markets Some Steps towards General Models in Applications to Hedging and Utility Problems

Xiong, Dewen, Kohlmann, Michael

2011

Overview

In finance, a bond is a debt security in which the authorized issuer owes the holder a debt and, depending on the terms of the bond, is obliged to pay interest (the coupon) and/or to repay the principal a a later date, termed maturity. A bond is a formal contract to repay borrowed money with interest at fixed intervals. Thus a bond is like a loan: The issuer is the borrower, the holder is the lender, and the coupon is the interest. Bonds provide the borrower with external funds to finance long-term investments, or, in the case of government bonds, to finance current expenditure. Bonds must be repaid a fixed intervals over a period of time. Bonds and stocks are both securities, but the major difference between the two is that stockholders ave an equity stake in the company (i.e. they are owners), whereas bondholders have a creditor stake in the company (i.e. they are lenders). Another difference is that bonds have a defined term or maturity after which the bond is redeemed, whereas may be outstanding indefinitely. The simplest case of a bond is a bank account with fixed interest rate rt: $$ dP(t,\\,T)\\, = \\,r_t P(t,\\,T)dt,\\,P(T,\\,T)\\, = \\,1. $$ A short look at the chart of a banks interest rates over some years however shows that the interest rate is by no means fixed so that we should assume that it is a random variable. This leads to the well known classical bond models (in alphabetical order of the authors) by Black-Derman, Chen, Cox-Ingersoll-Ross (CIR), Heath-Jarrow-Morton (HJM), Ho-Lee, Hull-White, Vasicek (V), among many others, where those with an abbreviation after the names are the best known to my observation. However, while models for stock markets are meanwhile quite satisfactory, the models for bonds/interest rates have diverse deficiencies, so that reality is not really well described. Also the models appear mathematically more difficult and technical than the classical stock market models: While researchers and practitioners are concerned with the fine-tuning in stock models, the basics on a general model for bond markets are still discussed without a commonly agreed result. The first attempt to describe a general model is found in the seminal paper by Björk et al(1997)7 and we are often referring to this article. Here we will discuss two mainstreams of problems, namely mean variance hedging and utility optimization (exponential utility indifference) in a general jump bond market. The purpose of this paper then is to introduce some new techniques, especially techniques from the theory of backward stochastic differential equations (BSDEs) in mean variance hedging, and to contribute to a general model. In the first part we will consider a model based on n maturities to apply recent results from MVH in jump stock markets. Carmona et al(2004)10 impressively describe the shortcomings of the models based on a finite number of maturities. To make things short: In the corresponding continuous HJM model the market is infinite dimensional with only a finite number of random sources so that this market always is complete which is contrary to the observations. Further shortcomings caused by the infiniteness of the market are described in Cont (2004)17. There are several attempts to overcome these difficulties. Carmona et al(2004)10 introduce an infinite dimensional Brownian motion and so use Malli-avin calculus methods to treat hedging problems with hedging strategies derived from a Clark-Ocone formula. Baran et al.2 consider generalized strategies in an infinite dimensional HJM-model. A similar approach is taken by DeDonno18. However these generalized strategies are not very useful to solve hedging problems more explicitly. In the second part of this paper we will here propose and extend an infinitely dimensional market where we consider measure valued strategies. Of course also these strategies have certain drawbacks when we come to the economical interpretation. And the main drawback is the fact that we always have to work in martingale markets to consider (𝕄, Q0)-normalized martingales as approximate wealth processes. For many problems this is sufficient but e.g. to describe superhedging we needed the notion of semimartingale. This, however, still is a long-standing open problem already described by Schwartz(1994)43: Il n'est pas facile de savoir exactement de qu'on doit appeler une semimartingale valeurs dans un espace vectoriel topologique. So, the contents of this paper should be seen as one perhaps promising looking step towards a general model for which we then consider exponential hedging problems. The paper is organized as follows: In the first part we construct a market of bonds with jumps driven by a general marked point process as well as by an ℝn-valued Wiener process based on Björk et al(1997)7, in which there exists at least one equivalent martingale measure Q0. Then we consider the mean-variance hedging of a contingent claim $H \\in L^2(\\mathcal{F}_{T_0})$ w.r.t. self-financing portfolios based on the given maturities T1,… ,Tn with T0 < T1 < … < Tn ≤ T*. We introduce the concept of variance-optimal martingale (VOM) and describe the VOM by a backward semimartingale equation (BSE). By making use of the concept of $\\mathcal{E}^*$-martingales introduced by Choulli et al.(1998)13, we obtain another BSE which has a unique solution. We derive an explicit solution of the optimal strategy and the optimal cost of the mean-variance hedging by the solutions of these two BSEs. In the second section we consider the optimal exponential utility in a bond market with jumps basing on a model similar to Björk, Kabanov and Runggaldier(1997)7 which is arbitrage-free. Similar to the normalized integral with respect to the cylindrical martingale first introduced in Mikulevicius and Rozovskii(1998)39, we introduce the (𝕄, Q0)-normalized martingale and local (𝕄, Q0)-normalized martingale. For a given maturity T0 ∈ [0, T*], we describe the minimal entropy martingale (MEM) based on [T0, T*] by a backward semi-martingale equation (BSE) w.r.t. the (𝕄, Q0)-normalized martingale. Then we give an explicit form of the optimal approximate wealth to the optimal exp utility problem by making use of the solution of the BSE. Finally, we describe the dynamics of the exp utility indifference valuation of a bounded contingent claim $H \\in L^{\\infty}(\\mathcal{F}_{T_0})$ by another BSE under the minimal entropy martingale measure in the incomplete market. The present paper strongly relies on unpublished works 45–48 of the authors and a seminar talk of the first author at the Nomura Institute of Oxford University in October 2009. Full proofs of results left out in this report are found in the cited preprints.

• Publisher: WORLD SCIENTIFIC
• Subject: 60G44 / 60H30 / 90A09 / Part II. Finance and Stochastics
• ISBN: 9814355704, 9789814355711, 9814458481, 9814355712, 9789814355704, 9789814458481