On the Efficiency of Simplified Weak Taylor Schemes for Monte Carlo Simulation in Finance

The purpose of this paper is to study the efficiency of simplified weak schemes for stochastic differential equations. We present a numerical comparison between weak Taylor schemes and their simplified versions. In the simplified schemes discrete random variables, instead of Gaussian ones, are generated to approximate multiple stochastic integrals. We show that an implementation of simplified schemes based on random bits generators significantly increases the computational speed. The efficiency of the proposed schemes is demonstrated.random bits generators; stochastic differential equations; simplified weak taylor schemes

A Hardware Generator of Multi-Point Distributed Random Numbers for Monte Carlo Simulation

Monte Carlo simulation of weak approximation of stochastic differential equations constitutes an intensive computational task. In applications such as finance, for instance, to achieve "real time" execution, as often required, one needs highly efficient implementations of the multi-point distributed random number generator underlying the simulations. In this paper, a fast and flexible dedicated hardware solution on a field programmable gate array is presented. A comparative performance analysis between a software-only and the poposed hardware solution demonstrated that the hardware solution is bottleneck-free, retains the flexibility of the software solution and significantly increases the computational efficiency. Moreover, simulations in Applications wuch as economics insurance, physics, population dynamics, epidemiology, structural mechanics, checmistry and biotechnology can benefit from the obtained speedups

A hardware generator for multi-point distributed random variables

In Monte Carlo simulation of weak approximation of stochastic differential equations, multi-point distributed random variables play an important role. However, they need highly efficient implementations to meet the "real-time" requirements of applications such as the pricing of complex derivative securities. In this paper a fast and fexible dedicated hardware generator of multi-point distributed random variables on a Field Programmable Gate Array (FPGA) is presented. A comparative performance analysis demonstrates that the hardware solution is bottleneck-free, retains the fexibility of a traditional software implementation and significantly increases the computational fficiency of the overall simulation. © 2005 IEEE

A multi-point distributed random variable accelerator for Monte Carlo simulation in finance

The pricing and hedging of complex derivative securities via Monte Carlo simulations of stochastic differential equations constitutes an intensive computational task. To achieve "real time" execution, as often required by financial institutions, one needs highly efficient implementations of the multi-point distributed random variables underlying the simulations. In this paper a fast and flexible dedicated hardware solution is proposed. A comparative performance analysis demonstrates that the hardware solution is bottleneck-free and flexible, and significantly increases the computational efficiency of the software solution. © 2005 IEEE

GPU accelerated Monte Carlo simulation of Brownian motors dynamics with CUDA

This work presents an updated and extended guide on methods of a proper acceleration of the Monte Carlo integration of stochastic differential equations with the commonly available NVIDIA Graphics Processing Units using the CUDA programming environment. We outline the general aspects of the scientific computing on graphics cards and demonstrate them with two models of a well known phenomenon of the noise induced transport of Brownian motors in periodic structures. As a source of fluctuations in the considered systems we selected the three most commonly occurring noises: the Gaussian white noise, the white Poissonian noise and the dichotomous process also known as a random telegraph signal. The detailed discussion on various aspects of the applied numerical schemes is also presented. The measured speedup can be of the astonishing order of about 3000 when compared to a typical CPU. This number significantly expands the range of problems solvable by use of stochastic simulations, allowing even an interactive research in some cases.Comment: 21 pages, 5 figures; Comput. Phys. Commun., accepted, 201

Recursive marginal quantization: extensions and applications in finance

Quantization techniques have been used in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and the efficient calibration of large derivative books. Recursive marginal quantization of an Euler scheme has recently been proposed as an efficient numerical method for evaluating functionals of solutions of stochastic differential equations. This algorithm is generalized and it is shown that it is possible to perform recursive marginal quantization for two higher-order schemes: the Milstein scheme and a simplified weak-order 2.0 scheme. Furthermore, the recursive marginal quantization algorithm is extended by showing how absorption and reflection at the zero boundary may be incorporated. Numerical evidence is provided of the improved weak-order convergence and computational efficiency for the geometric Brownian motion and constant elasticity of variance models by pricing European, Bermudan and barrier options. The current theoretical error bound is extended to apply to the proposed higher-order methods. When applied to two-factor models, recursive marginal quantization becomes computationally inefficient as the optimization problem usually requires stochastic methods, for example, the randomized Lloyd’s algorithm or Competitive Learning Vector Quantization. To address this, a new algorithm is proposed that allows recursive marginal quantization to be applied to two-factor stochastic volatility models while retaining the efficiency of the original Newton-Raphson gradientdescent technique. The proposed method is illustrated for European options on the Heston and Stein-Stein models and for various exotic options on the popular SABR model. Finally, the recursive marginal quantization algorithm, and improvements, are applied outside the traditional risk-neutral pricing framework by pricing long-dated contracts using the benchmark approach. The growth-optimal portfolio, the central object of the benchmark approach, is modelled using the time-dependent constant elasticity of variance model. Analytic European option prices are derived that generalize the current formulae in the literature. The time-dependent constant elasticity of variance model is then combined with a 3/2 stochastic short rate model to price zerocoupon bonds and zero-coupon bond options, thereby showing the departure from risk-neutral pricing

Monte Carlo Greeks for financial products via approximative transition densities

In this paper we introduce efficient Monte Carlo estimators for the valuation of high-dimensional derivatives and their sensitivities (''Greeks''). These estimators are based on an analytical, usually approximative representation of the underlying density. We study approximative densities obtained by the WKB method. The results are applied in the context of a Libor market model.Comment: 24 page

Pricing a Bermudan option under the constant elasticity of variance model

This dissertation investigates the computational efficiency and accuracy of three methodologies in the pricing of a Bermudan option, under the constant elasticity of variance (CEV) model. The pricing methods considered are the finite difference method, least squares Monte Carlo method and recursive marginal quantization (RMQ) method. Specific emphasis will be on RMQ, as it is the most recent method. A plain vanilla European option is initially priced using the above mentioned methods, and the results obtained are compared to the Black-Scholes option pricing formula to determine their viability as pricing methods. Once the methods have been validated for the European option, a Bermudan option is then priced for these methods. Instead of using the Black-Scholes option pricing formula for comparison of the prices obtained, a high-resolution finite difference scheme is used as a proxy in the absence of an analytical solution. One of the main advantages of the recursive marginal quantization (RMQ) method is that the continuation value of the option is computed at almost no additional computational cost, this with other contributing factors leads to a computationally efficient and accurate method for pricing

Metropolis Sampling

Monte Carlo (MC) sampling methods are widely applied in Bayesian inference, system simulation and optimization problems. The Markov Chain Monte Carlo (MCMC) algorithms are a well-known class of MC methods which generate a Markov chain with the desired invariant distribution. In this document, we focus on the Metropolis-Hastings (MH) sampler, which can be considered as the atom of the MCMC techniques, introducing the basic notions and different properties. We describe in details all the elements involved in the MH algorithm and the most relevant variants. Several improvements and recent extensions proposed in the literature are also briefly discussed, providing a quick but exhaustive overview of the current Metropolis-based sampling's world.Comment: Wiley StatsRef-Statistics Reference Online, 201