In the recent project BENCHOP–the BENCHmarking project in Option Pricing we found that Stochastic and Local Volatility problems were particularly challenging. Here we continue the effort by introducing a set of benchmark problems for this type of problems. Eight different methods targeted for the Stochastic Differential Equation (SDE) formulation and the Partial Differential Equation (PDE) formulation of the problem, as well as Fourier methods making use of the characteristic function, were implemented to solve these problems. Comparisons are made with respect to time to reach a certain error level in the computed solution for the different methods. The implemented Fourier method was superior to all others for the two problems where it was implemented. Generally, methods targeting the PDE formulation of the problem outperformed the methods for the SDE formulation. Among the methods for the PDE formulation the ADI method stood out as the best performing one.@en

In this work, we propose a one time-step Monte Carlo method for the SABR model. We base our approach on an accurate approximation of the cumulative distribution function of the time-integrated variance (conditional on the SABR volatility), using Fourier techniques and a copula. Resulting is a fast simulation algorithm which can be employed to price European options under the SABR dynamics. Our approach can thus be seen as an alternative to Hagan's analytic formula for short maturities that may be employed for model calibration purposes.@en

In this paper we extend the Stochastic Grid Bundling Method (SGBM), a regress-later Monte Carlo scheme for pricing early-exercise options, with an adjoint method to compute in a highly efficient manner the option sensitivities (the “Greeks”)along the Monte Carlo paths, with reasonable accuracy. The path-wise SGBM Greeks computation is based on the conventional path-wise sensitivity analysis, however, for a regress-later technique. The resulting sensitivities at the end of the monitoring period are implicitly rolled over into the sensitivities of the regression coefficients of the previous monitoring date. For this reason, we name the method Rolling Adjoints, which facilitates Smoking Adjoints [M. Giles, P. Glasserman, Smoking adjoints: fast Monte Carlo Greeks, Risk 19 (1)(2006)88–92]to compute conditional sensitivities along the paths for options with early-exercise features.@en

In this paper, we will present a multiple time step Monte Carlo simulation technique for pricing options under the Stochastic Alpha Beta Rho model. The proposed method is an extension of the one time step Monte Carlo method that we proposed in an accompanying paper Leitao et al. [Appl. Math. Comput. 2017, 293, 461–479], for pricing European options in the context of the model calibration. A highly efficient method results, with many very interesting and nontrivial components, like Fourier inversion for the sum of log-normals, stochastic collocation, Gumbel copula, correlation approximation, that are not yet seen in combination within a Monte Carlo simulation. The present multiple time step Monte Carlo method is especially useful for long-term options and for exotic options.@en

In this paper, we present the data-driven COS method, ddCOS. It is a Fourier-based financial option valuation method which assumes the availability of asset data samples: a characteristic function of the underlying asset probability density function is not required. As such, the presented technique represents a generalization of the well-known COS method [1]. The convergence of the proposed method is O(1/n) in line with Monte Carlo methods for pricing financial derivatives. The ddCOS method is then particularly interesting for density recovery and also for the efficient computation of the option's sensitivities Delta and Gamma. These are often used in risk management, and can be obtained at a higher accuracy with ddCOS than with plain Monte Carlo methods.@en

In this paper, we present the data-driven COS method, ddCOS. It is a Fourier-based financial option valuation method which assumes the availability of asset data samples: a characteristic function of the underlying asset probability density function is not required. As such, the presented technique represents a generalization of the well-known COS method [1]. The convergence of the proposed method is O(1/n) in line with Monte Carlo methods for pricing financial derivatives. The ddCOS method is then particularly interesting for density recovery and also for the efficient computation of the option's sensitivities Delta and Gamma. These are often used in risk management, and can be obtained at a higher accuracy with ddCOS than with plain Monte Carlo methods.@en

In this work, a parallel graphics processing units (GPU) version of the Monte Carlo stochastic grid bundling method (SGBM) for pricing multi-dimensional early-exercise options is presented. To extend the method's applicability, the problem dimensions and the number of bundles will be increased drastically. This makes SGBM very expensive in terms of computational costs on conventional hardware systems based on central processing units. A parallelization strategy of the method is developed and the general purpose computing on graphics processing units paradigm is used to reduce the execution time. An improved technique for bundling asset paths, which is more efficient on parallel hardware is introduced. Thanks to the performance of the GPU version of SGBM, a general approach for computing the early-exercise policy is proposed. Comparisons between sequential and GPU parallel versions are presented.@en

Monte Carlo methods are highly appreciated and intensively employed in computational finance in the context of financial derivatives valuation or risk management. The method offers valuable advantages like flexibility, easy interpretation and straightforward implementation. Furthermore, the dimensionality of the financial problem can be increased without reducing the efficiency significantly. The latter feature of Monte Carlo methods is important since it represents a clear advantage over other competing numerical methods. Furthermore, in the case of option valuation problems in multiple dimensions (typically more than five), theMonte Carlo method and its variants become the only possible choices. Basically, theMonte Carlo method is based on the simulation of possible scenarios of an underlying process and by then aggregating their values for a final solution. Pricing derivatives on equity and interest rates, risk assessment or portfolio valuation are some of the representative examples in finance, where Monte Carlo methods perform very satisfactorily. The main drawback attributed to these methods is the rather poor balance between computational cost and accuracy, according to the theoretical rate ofMonte Carlo convergence. Based on the central limit theorem, theMonte Carlo method requires hundred times more scenarios to reduce the error by one order...@en