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**Efficient hedging: cost versus shortfall risk.**
*(English)*
Zbl 0956.60074

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 N. El Karoui and M.-C. Quenez, SIAM J. Control Optim. 33, No. 1, 29–66 (1995; Zbl 0831.90010) and 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 to the partial ordering defined by the shortfall risk and the initial capital. These efficient hedges allow the investor to interpolate in a systematic way between the extremes of a perfect hedge (no chance of making a profit) and no hedge (full risk of shortfall, full chance of profit) depending on the accepted level of shortfall risk.

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 to the partial ordering defined by the shortfall risk and the initial capital. These efficient hedges allow the investor to interpolate in a systematic way between the extremes of a perfect hedge (no chance of making a profit) and no hedge (full risk of shortfall, full chance of profit) depending on the accepted level of shortfall risk.

Reviewer: A.V.Swishchuk (Kyïv)

### MSC:

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

91G70 | Statistical methods; risk measures |

62F03 | Parametric hypothesis testing |

62P05 | Applications of statistics to actuarial sciences and financial mathematics |