A Novel Monte Carlo Approach to Hybrid Local Volatility Models

We present in a Monte Carlo simulation framework a novel approach for the evaluation of hybrid local volatility (Dupire 1994, Derman and Kani 1998) models. In particular, we consider the stochastic local volatility model - see e.g. Lipton et al. (2014), Piterbarg (2007), Tataru and Fisher (2010), Lipton (2002) - and the local volatility model incorporating stochastic interest rates - see e.g. Atlan (2006), Piterbarg (2006), Deelstra and Rayée (2012), Ren et al. (2007). For both model classes a particular (conditional) expectation needs to be evaluated, which cannot be extracted from the market and is expensive to compute. We establish accurate and 'cheap to evaluate' approximations for the expectations by means of the stochastic collocation method (Babuska et al. 2007, Xiu and Hesthaven 2005, Beck et al. 2012, Nobile et al. 2008, Sankaran and Marsden 2011) which was recently applied in the financial context (Grzelak et al. 2014, Grzelak and Oosterlee 2017), combined with standard regression techniques. Monte Carlo pricing experiments confirm that our method is highly accurate and fast