Department Mathematik



Mathematisches Kolloquium

Am Samstag, 22. November 2008, um 11 Uhr s.t. spricht

Prof. Dr. Francesca Biagini
(LMU München)

im Hörsaal A027 über das Thema

Local Risk-Minimization for Defaultable Markets

Zusammenfassung: In this paper we discuss the problem of pricing and hedging defaultable claims, i.e. options that can lose partially or totally their value if a default event occurs. We consider a simple market model with two non-defaultable primitive assets and a occurence of a default by using a portfolio consisting only if the primitive assets, sense to applz some of the methods used for pricing and hedging derivatives in incomplete markets. In particular we focus here on the local risk-minimization approach and consider a general case where the dynamics of the risky assets may be influenced by the occurring of a default event and also the default time itself may depend on the assets prices behavior. In this general setting we are able to provide the Föllmer-Schweizer decomposition of a defaultable claim with random recovery rate. In particular we focus on two cases where we compute explicitly the pseudo-locally risk-minimizing strategy and the optimal cost. First we consider the situation where the default time depends on the behavior of the risky asset price, but not vice versa. In the second case we assume that drift and volatility of underlying discounted asset are affected by the default time and we show how our result fits in the approach of local risk-minimization for markets affected by incomplete information.

Our results are then of general interest for computing hedging strategies in incomplete markets in presence of an additional source of randomness, that is "orthogonal" to the asset price dynamics, but not necessarely independent of them and vice versa. In particular local risk-minimization naturally appears as suitable hedging method for the new financial instruments recently introduced to hedge against systematic mortality risk in life insurance contracts. This talk is based on a joint work with Alessandra Cretarola.