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Local volatility calibration during turbulent periods

We propose a methodology to calibrate the local volatility function under a continuous time setting. For this purpose, we used the Markov chain approximation method built on the well-established idea of local consistency. The chain was designed to approximate jump–diffusions coupled with a local volatility function. We found that this method outperforms traditional numerical algorithms that require time discretization. Furthermore, we showed that a local volatility jump–diffusion model outperformed the in- and out-of-sample pricing that the market practitioners benchmark, namely the Practitioners Black–Scholes, in turbulent periods during which at-the-money implied volatilities have risen substantially. Hedging experiments show a moderate portfolio risk under the local volatility jump–diffusion case. As in previous literature concerning local volatility estimation, we represent the local volatility function using a space-time cubic spline.