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 Risk, 1994, 7, 18–20], Int. J. Theor. Appl. Finance, 1998, 1, 61–110] models. In particular, we consider the stochastic local volatility model—see e.g. Lipton et al. Quant. Finance, 2014, 14, 1899–1922], Piterbarg Risk, 2007, April, 84–89], Tataru and Fisher Quantitative Development Group, Bloomberg Version 1, 2010], Lipton Risk, 2002, 15, 61–66]—and the local volatility model incorporating stochastic interest rates—see e.g. Atlan ArXiV preprint math/0604316, 2006], Piterbarg Risk, 2006, 19, 66–71], Deelstra and Rayée Appl. Math. Finance, 2012, 1–23], Ren et al. Risk, 2007, 20, 138–143]. 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 SIAM J. Numer. Anal., 2007, 45, 1005–1034], SIAM J. Sci. Comput., 2005, 27, 1118–1139], Math. Models Methods Appl. Sci., 2012, 22, 1–33], SIAM J. Numer. Anal., 2008, 46, 2309–2345], J. Biomech. Eng., 2011, 133, 031001], which was recently applied in the financial context Available at SSRN 2529691, 2014], J. Comput. Finance, 2016, 20, 1–19], combined with standard regression techniques. Monte Carlo pricing experiments confirm that our method is highly accurate and fast.