Delta-hedging vega risk?

In this article we compare the Profit and Loss arising from the delta-neutral dynamic hedging of options, using two possible values for the delta of the option. The first one is the Black– Scholes implied delta, while the second one is the local delta, namely the delta of the option in a generalized Black–Scholes model with a local volatility, recalibrated to the market smile every day. We explain why in negatively skewed markets the local delta should provide a better hedge than the implied delta during slow rallies or fast sell-offs, and a worse hedge, though to a lesser extent, during fast rallies or slow sell-offs. Since slow rallies and fast sell-offs are more likely to occur than fast rallies or slow sell-offs in negatively skewed markets (provided we have physical as well as implied negative skewness), we conclude that on average the local delta provides a better hedge than the implied delta in negatively skewed markets. We obtain the same conclusion in the case of positively skewed markets. We illustrate these results by using both simulated and real time-series of equity-index data, that have had a large negative implied skew since the stock market crash of October 1987. Moreover we check numerically that the conclusions we draw are true when transaction costs are taken into account. In the last section we discuss the case of barrier options.