Hedging Large Risks Reduces the Transaction Costs

As soon as one accepts to abandon the zero-risk paradigm of Black-Scholes, very interesting issues concerning risk control arise because diierent deenitions of the risk become unequivalent. Optimal hedges then depend on the quantity one wishes to minimize. We show that a deenition of the risk more sensitive to the extreme events generically leads to a decrease both of the probability of extreme losses and of the sensitivity of the hedge on the price of the underlying (thèGamma'). Therefore, the transaction costs and the impact of hedging on the price dynamics of the underlying are reduced. It is well known that the perfect Black-Scholes hedge only works in the ideal case of a continuous time, log-Brownian evolution of the price of the underlying. Unfortunately, this model is rather remote from reality: the distribution of price changes has`fat tails', which persist even for rather long time lags (see, e.g. 1, 2, 3]). This makes the whole idea of zero-risk strategies and perfect replication shady. An alternative view was proposed in 4, 3, 5], where one accepts from the start that the risk associated with option trading is in general non zero, but can be minimized by adopting an appropriate 1