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Risk-minimizing hedging of contingent claims in incomplete models of financial markets

100%

Rutkowski M.

Mathematica Applicanda

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1996

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tom 25

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nr 39

PL

.

EN

The paper is devoted to a specific optimization problem associated with the hedging of contingent claims in continuous-time incomplete models of financial markets. Generally speaking, we place ourselves within the standard framework of the theory of continuous trading, as exposed in Harrison and Pliska [13]. Our aim is twofold. Firstly, we present a relatively concise exposition of the risk-minimizing methodology (due essentially to Follmer and Sondermann [12], Follmer and Schweizer [11] and Schweizer [33]) in a multi-dimensional continuous-time framework. Let us mention here that this approach is based on the specific kind of minimization of the additional cost associated with a hedging strategy at all times before the terminal date T. Secondly, we provide some new results which formalize some concepts introduced in Hofman et a/.[l5], in particular, the general results of the first, part are specialized to the case of multi-dimensional Ito processes. Finally, in Section 6 the general theory is illustrated by means of an example dealing with the risk-minimizing hedging of a stock index option in an incomplete framework. This example is motivated bv the work of Lamberton and Lapeyre [22] who have! solved a related, but simpler, problem of a risk-minimizing hedging under the martingale measure.

2

Optimality of replication in the CRR model with transaction costs

100%

Rutkowski M.

Applicationes Mathematicae

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1998-1999

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tom 25

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nr 1

29-53

EN

Recently, there has been a growing interest in optimization problems associated with the arbitrage pricing of derivative securities in imperfect markets (in particular, in models with transaction costs). In this paper, we examine the valuation and hedging of European claims in the multiplicative binomial model proposed by Cox, Ross and Rubinstein [5] (the CRR model), in the presence of proportional transaction costs. We focus on the optimality of replication; in particular, we provide sufficient conditions for the optimality of the replicating strategy in the case of long and short positions in European options. This work can be seen as a continuation of studies by Bensaid et al. [2] and Edirisinghe et al. [13]. We put, however, more emphasis on the martingale approach to the claims valuation in the presence of transaction costs, focusing on call and put options. The problem of optimality of replication in the CRR model under proportional transaction costs was recently solved in all generality by Stettner[30].

3

Dependent defaults and credit migrations

63%

Bielecki T., Rutkowski M.

Applicationes Mathematicae

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2003

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tom 30

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nr 2

121-145

EN

The paper deals with the modelling of mutually dependent default times of several credit names through the intensity-based approach. We extend to the case of multiple ratings some previous results due to Schmidt (1998), Kusuoka (1999) and Jarrow and Yu (2001). The issue of the arbitrage valuation of simple basket credit derivatives is also briefly examined. We argue that our approach leads, in some cases, to a significant reduction of the dimensionality of the valuation problem at hand.

Ograniczanie wyników

2 Applicationes Mathematicae

1 Mathematica Applicanda

3 Rutkowski M.

1 Bielecki T.

1 2003

1 1998

1 1996

Risk-minimizing hedging of contingent claims in incomplete models of financial markets

Optimality of replication in the CRR model with transaction costs

Dependent defaults and credit migrations