Monte Carlo Pricing with Local Volatility Grids

Local volatility models are widely used to manage many exotic options in a way consistent with available market prices of vanilla options. Once calibrated, a local volatility grid can be used in numerical methods such as PDE or Monte Carlo to price and hedge exotic options consistently with vanilla market prices. This is important, since these vanilla options will be used to hedge those exotic options. Recent approaches in local volatility calibration from sparse market data have been proposed, including a novel implicit PDE approach in Andreasen-Huge (AH)[1] and a further improvement in Lipton-Sepp (LS)[2]. Both solve the inverse problem of finding the local volatility in Dupire equation. [2] tackles the time discretization problem of the first work. The accuracy and correctness of any local volatility calibration scheme should be gauged by its ability to reprice the vanilla calibration contracts in a Monte Carlo or Black-Scholes type PDE pricing setting, since those methods will be typically used to price exotic options. This not only signals a successful calibration, but also ensures the scheme can be properly employed in a Monte Carlo or PDE setting for pricing or hedging. We report on a simple modification of the algorithm reported in [1] to calibrate local volatility that results in a successful repricing under a Monte Carlo setting, using market data for S&P500 as an example. Our approach is simpler to code and to use in a Monte Carlo setting than the AH method and it improves time discretization error. It also avoids costly calibration and the Laplace transform computations used to solve the Dupire equation, which were recently proposed by LS[2]