Profit-and-Loss of Option Strategies under Quadratic Skew Parametrization

We analyse the profit-and-loss (P&L) of delta-hedging strategies for vanilla options in the presence of the implied volatility skew and derive an approximation for the P&L under the quadratic parametrization of the implied volatility. We apply this approximation to study the P&L of a straddle, a risk-reversal, and a butterfly. Using our results, we derive the break-even realized skew and convexity that equate the average realized P&L of the risk-reversal and the butterfly, respectively, to zero. Furthermore, we analyse the impact of the volatility skew on the delta-hedging of these option strategies. We present some empirical results using implied volatilities of options on the S&P 500 index.