A novel Monte Carlo approach to hybrid local volatility models |
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Authors: | Anthonie W. van der Stoep Lech A. Grzelak Cornelis W. Oosterlee |
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Affiliation: | 1. Pricing Model Validation (PMV), Rabobank, Utrecht, The Netherlands.;2. Centrum Wiskunde &3. Informatica (CWI)—National Research Institute for Mathematics and Computer Science, Amsterdam, The Netherlands.;4. Delft Institute of Applied Mathematics (DIAM), TU Delft, Delft, The Netherlands.;5. Delft Institute of Applied Mathematics (DIAM), TU Delft, Delft, The Netherlands.;6. Quantitative Analytics, ING, Amsterdam, The Netherlands.;7. Centrum Wiskunde & |
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Abstract: | 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. |
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Keywords: | Local volatility Monte Carlo Hybrid Stochastic volatility Stochastic local volatility Stochastic interest rates Stochastic collocation Regression SABR Heston Hull–White |
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