HEDGING DEFAULTABLE CLAIMS: MEAN VARIANCE HEDGING AND INDIFFERENCE PRICES

M. Jeanblanc
(joint work with T. Bielecki, M. Rutkowski)

Abstract

We formulate a new paradigm for pricing and hedging financial risks in incomplete markets, rooted in the classical Markowitz mean-variance portfolio selection principle. We consider an investor who is interested in dynamic selection of her portfolio, so that the expected value of her wealth at the end of the pre-selected planning horizon is no less then some floor value, and so that the associated risk, as measured by the variance of the wealth at the end of the planning horizon, is minimized. If the perfect replication is not possible, then the determination of a price that the investor is willing to pay for the opportunity, will become subject to the investor's overall attitude towards trading. In case of our investor, the bid price and the corresponding hedging strategy is to be determined in accordance with the mean-variance paradigm.

We present also a few alternative ways of pricing defaultable claims in the situation when perfect hedging is not possible. We study the indifference pricing approach, that was initiated by Hodges and Neuberger . This method leads us to solving portfolio optimization problems in an incomplete market model, and we shall use the dynamic programming approach. In particular, we compare the indifference prices obtained using strategies adapted to the reference filtration to the indifference prices obtained using strategies based on the enlarged filtration, which encompasses also the observation of the default time. We also solve portfolio optimization problems for the case of the exponential utility. Next, we study a particular indifference price based on the quadratic criterion; it will be referred to as the quadratic hedging price.