Efficient hedging: cost versus shortfall risk
type: Item
instance of: scholarly article
title: Efficient hedging: cost versus shortfall risk
zbMATH Open document ID: 0956.60074
author: Hans Föllmer
author: Peter Leukert
published in: Finance and Stochastics
publication date: 2000-11-01T00:00:00Z
full work available at URL: http://hdl.handle.net/10419/61739
review text: In a complete financial market a given contingent claim can be replicated by a self-financing trading strategy, and the cost of replication defines the price of the claim. In incomplete financial markets one can still stay on the safe side by using a ``superhedging'' strategy [see \textit{N. El Karoui} and \textit{M.-C. Quenez}, SIAM J. Control Optim. 33, No. 1, 29--66 (1995; Zbl 0831.90010) and \textit{I. Karatzas}, Lectures on the mathematics of finance. Providence, RI: Am. Math. Soc. (1996; Zbl 0878.90010)]. But from a practical point of view the cost of superhedging is often too high. Also perfect (super-)hedging takes away the opportunity of making a profit together with the risk of a loss. Suppose that the investor is unwilling to put up the initial amount of capital required by a perfect (super-)hedge and is ready to accept some risk. What is the optimal ``partial hedge'' which can be achieved with a given smaller amount of capital? In order to make this question precise we need a criterion expressing the investor attitude towards the shortfall risk. The authors [Finance Stoch. 3, No. 3, 251--273 (1999; Zbl 0977.91019 )] introduced strategies of ``quantile hedging'' which maximize the probability that a hedge is successful. In that case the investor applies a dynamic version of the static Value at Risk (VaR) concept. Just as the static VaR approach, the dynamic concept of quantile hedging does not take into account the size of the shortfall but only the probability of its occurrence. The authors describe the investor attitude towards the shortfall in terms of a loss function. Convexity of the function corresponds to risk aversion. The shortfall risk is defined as the expectation of the shortfall weighted by the loss function. The problem is to minimize this shortfall risk given some capital constraint. Instead they could prescribe a bound on the shortfall risk and minimize the cost. In other words, they are looking for hedges which are efficient with respect …
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 91G70
Mathematics Subject Classification ID: 62F03
Mathematics Subject Classification ID: 62P05
zbMATH DE Number: 1524263
zbMATH Keywords: hedging
zbMATH Keywords: shortfall risk
zbMATH Keywords: efficient hedges
zbMATH Keywords: risk management
zbMATH Keywords: lower partial moments
zbMATH Keywords: convex duality
zbMATH Keywords: stochastic volatility
MaRDI profile type: Publication
OpenAlex ID: W1973946491
DOI: 10.1007/S007800050008
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description: scientific article; zbMATH DE number 1524263