Efficient Programmable Random Variate Generation Accelerator from Sensor Noise

We introduce a method for non-uniform random number generation based on sampling a physical process in a controlled environment. We demonstrate one proof-of-concept implementation of the method that reduces the error of Monte Carlo integration of a univariate Gaussian by 1068 times while doubling the speed of the Monte Carlo simulation. We show that the supply voltage and temperature of the physical process must be controlled to prevent the mean and standard deviation of the random number generator from drifting.Alan Turing Institute award: TU/B/000096 EPSRC grants: EP/N510129/1, EP/R022534/1, EP/V004654/1 and EP/L015889/

A Parallel Algorithm for solving BSDEs - Application to the pricing and hedging of American options

We present a parallel algorithm for solving backward stochastic differential equations (BSDEs in short) which are very useful theoretic tools to deal with many financial problems ranging from option pricing option to risk management. Our algorithm based on Gobet and Labart (2010) exploits the link between BSDEs and non linear partial differential equations (PDEs in short) and hence enables to solve high dimensional non linear PDEs. In this work, we apply it to the pricing and hedging of American options in high dimensional local volatility models, which remains very computationally demanding. We have tested our algorithm up to dimension 10 on a cluster of 512 CPUs and we obtained linear speedups which proves the scalability of our implementationComment: 25 page

Efficient and Accurate Parallel Inversion of the Gamma Distribution

A method for parallel inversion of the gamma distribution is described. This is very desirable for random number generation in Monte Carlo simulations where gamma variates are required. Let $\alpha$ be a fixed but arbitrary positive real number. Explicitly, given a list of uniformly distributed random numbers our algorithm applies the quantile function (inverse CDF) of the gamma distribution with shape parameter $\alpha$ to each element. The result is, therefore, a list of random numbers distributed according to the said distribution. The output of our algorithm has accuracy close to a choice of single- or double-precision machine epsilon. Inversion of the gamma distribution is traditionally accomplished using some form of root finding. This is known to be computationally expensive. Our algorithm departs from this paradigm by using an initialization phase to construct, on the fly, a piecewise Chebyshev polynomial approximation to a transformation function, which can be evaluated very quickly during variate generation. The Chebyshev polynomials are high order, for good accuracy, and generated via recurrence relations derived from nonlinear second order ODEs. A novelty of our approach is that the same change of variable is applied to each uniform random number before evaluating the transformation function. This is particularly amenable to implementation on SIMD architectures, whose performance is sensitive to frequently diverging execution flows due to conditional statements (branch divergence). We show the performance of a CUDA GPU implementation of our algorithm (called Quantus) is within an order of magnitude of the time to compute the normal quantile function

Fast and accurate parallel computation of quantile functions for random number generation

The fast and accurate computation of quantile functions (the inverse of cumulative distribution functions) is very desirable for generating random variates from non-uniform probability distributions. This is because the quantile function of a distribution monotonically maps uniform variates to variates of the said distribution. This simple fact is the basis of the inversion method for generating non-uniform random numbers. The inversion method enjoys many significant advantages, which is why it is regarded as the best choice for random number generation. Quantile functions preserve the underlying properties of the uniform variates, which is beneficial for a number of applications, especially in modern computational finance. For example, copula and quasi-Monte Carlo methods are significantly easier to use with inversion. Inversion is also well suited to variance reduction techniques. However, for a number of key distributions, existing methods for the computational of their quantile functions are too slow in practice. The methods are also unsuited to execution on parallel architectures such as GPUs and FPGAs. These parallel architectures have become very popular, because they allow simulations to be sped up and enlarged. The original contribution of this thesis is a collection of new and practical numerical algorithms for the normal, gamma, non-central χ2 and skew-normal quantile functions. The algorithms were developed with efficient parallel computation in mind. Quantile mechanics—the differential approach to quantile functions—was used with inventive changes of variables and numerical methods to create the algorithms. The algorithms are faster or more accurate than the current state of the art on parallel architectures. The accuracy of GPU implementations of the algorithms have been benchmarked against independent CPU implementations. The results indicate that the quantile mechanics approach is a viable and powerful technique for developing quantile function approximations and algorithms

Parallel Weighted Random Sampling

Data structures for efficient sampling from a set of weighted items are an important building block of many applications. However, few parallel solutions are known. We close many of these gaps both for shared-memory and distributed-memory machines. We give efficient, fast, and practicable algorithms for sampling single items, k items with/without replacement, permutations, subsets, and reservoirs. We also give improved sequential algorithms for alias table construction and for sampling with replacement. Experiments on shared-memory parallel machines with up to 158 threads show near linear speedups both for construction and queries

A Parallel Algorithm for solving BSDEs - Application to the pricing and hedging of American options

We present a parallel algorithm for solving backward stochastic differential equations (BSDEs in short) which are very useful theoretic tools to deal with many financial problems ranging from option pricing option to risk management. Our algorithm based on Gobet and Labart (2010) exploits the link between BSDEs and non linear partial differential equations (PDEs in short) and hence enables to solve high dimensional non linear PDEs. In this work, we apply it to the pricing and hedging of American options in high dimensional local volatility models, which remains very computationally demanding. We have tested our algorithm up to dimension 10 on a cluster of 512 CPUs and we obtained linear speedups which proves the scalability of our implementationbackward stochastic differential equations, parallel computing, Monte- Carlo methods, non linear PDE, American options, local volatility model.