We discuss a competitive alternative to stochastic local volatility models, namely the Collocating Volatility (CV) framework, introduced in [L. A. Grzelak (2019) The CLV framework-A fresh look at efficient pricing with smile, International Journal of Computer Mathematics 96 (11), 2209-2228]. The CV framework consists of two elements, a "kernel process"that can be efficiently evaluated and a local volatility function. The latter, based on stochastic collocation-e.g. [I. Babuška, F. Nobile & R. Tempone (2007) A stochastic collocation method for elliptic partial differential equations with random input data, SIAM Journal on Numerical Analysis 45 (3), 1005-1034; B. Ganapathysubramanian & N. Zabaras (2007) Sparse grid collocation schemes for stochastic natural convection problems, Journal of Computational Physics 225 (1), 652-685; J. A. S. Witteveen & G. Iaccarino (2012) Simplex stochastic collocation with random sampling and extrapolation for nonhypercube probability spaces, SIAM Journal on Scientific Computing 34 (2), A814-A838; D. Xiu & J. S. Hesthaven (2005) High-order collocation methods for differential equations with random inputs, SIAM Journal on Scientific Computing 27 (3), 1118-1139]-connects the kernel process to the market and allows the CV framework to be perfectly calibrated to European-type options. In this paper, we consider three different kernel process choices: The Ornstein-Uhlenbeck (OU) and Cox-Ingersoll-Ross (CIR) processes and the Heston model. The kernel process controls the forward smile and allows for an accurate and efficient calibration to exotic options, while the perfect calibration to liquid market quotes is preserved. We confirm this by numerical experiments, in which we calibrate the OU-CV, CIR-CV and Heston-CV frameworks to FX barrier options.

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A general purpose of mathematical models is to accurately mimic some observed phenomena in the real world. In financial engineering, for example, one aim is to reproduce market prices of financial contracts with the help of applied mathematics. In the Foreign Exchange (FX) market, the so-called implied volatility smile plays a key role in the pricing and hedging of financial derivative contracts. This volatility smile is a phenomenon that reflects the prices of European-type options for different strike prices; the implied volatility tends to be higher for options that are deeper In The Money and Out of The Money than options that are approximately At The Money. In order for a pricing model to be accepted in the financial industry, it should at least be able to accurately price back the most simple financial derivative contracts, namely European call and put options. In other words, the model should calibrate well to the implied volatility smile observed in the financial market. The calibration should not only be accurate, but also reasonably fast. Another feature we wish the financial asset model to possess, is an accurate pricing of so-called exotic financial products. Exotic products are not traded on regular exchanges, but over-the-counter, i.e. directly between two parties without the supervision of an exchange. An example is a barrier option, which is a financial contract of which its payoff depends on the possible event that the underlying asset price hits a certain pre-determined level. The model prices of these path-dependent contracts are determined by the transition densities of the relevant underlying asset(s) between future time-points. These transition densities are reflected by the forward volatility smile the model implies; in order for the model to accurately price exotic products, it should yield realistic forward volatilities..@en

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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