Brownian Bridge Construction Research Articles

Optimising Poisson bridge constructions for variance reduction methods

Abstract In this paper we discuss different Monte Carlo (MC) approaches to generate unit-rate Poisson processes and provide an analysis of Poisson bridge constructions, which form the discrete analogue of the well-known Brownian bridge construction for a Wiener process. One of the main advantages of these Poisson bridge constructions is that they, like the Brownian bridge, can be effectively combined with variance reduction techniques. In particular, we show here, in practice and proof, how we can achieve orders of magnitude efficiency improvement over standard MC approaches when generating unit-rate Poisson processes via a synthesis of antithetic sampling and Poisson bridge constructions. At the same time we provide practical guidance as to how to implement and tune Poisson bridge methods to achieve, in a mean sense, (near) optimal performance.

Comparing Option Pricing Methods in q

We use kdb+ and the q language to compare the use of Monte Carlo (MC) and Quasi-Monte Carlo (QMC) methods for pricing options. Low-discrepancy Sobol’ sequences are used to price European and Asian options, using both incremental discretization and Brownian-bridge construction. Results are compared to the deterministic Black–Scholes price for each option type. Analysis was carried out using the time-series database, kdb+, from Kx. Kdb+ is a hybrid on-disk and in-memory columnar database, optimized for the ingestion, storage, and analysis of huge amounts of structured data. Kx software is widely used in the financial industry, for streaming, real-time, and historical analysis of market data. Our code makes use of the efficient and concise nature of the q language, to mirror the results of Kucherenko and Shah [1].

Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model

The rough Bergomi (rBergomi) model, introduced recently in Bayer et al. [Pricing under rough volatility. Quant. Finance, 2016, 16(6), 887–904], is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet remarkably fits empirical implied volatility surfaces. In the absence of analytical European option pricing methods for the model, and due to the non-Markovian nature of the fractional driver, the prevalent option is to use the Monte Carlo (MC) simulation for pricing. Despite recent advances in the MC method in this context, pricing under the rBergomi model is still a time-consuming task. To overcome this issue, we have designed a novel, hierarchical approach, based on: (i) adaptive sparse grids quadrature (ASGQ), and (ii) quasi-Monte Carlo (QMC). Both techniques are coupled with a Brownian bridge construction and a Richardson extrapolation on the weak error. By uncovering the available regularity, our hierarchical methods demonstrate substantial computational gains with respect to the standard MC method. They reach a sufficiently small relative error tolerance in the price estimates across different parameter constellations, even for very small values of the Hurst parameter. Our work opens a new research direction in this field, i.e. to investigate the performance of methods other than Monte Carlo for pricing and calibrating under the rBergomi model.

High-dimensional integration on [formula omitted], weighted Hermite spaces, and orthogonal transforms

It has been found empirically that quasi-Monte Carlo methods are often efficient for very high-dimensional problems, that is, with dimension in the hundreds or even thousands. The common explanation for this surprising fact is that those functions for which this holds true behave rather like low-dimensional functions in that only few of the coordinates have a sizeable influence on its value. However, this statement may be true only after applying a suitable orthogonal transform to the input data, like utilizing the Brownian bridge construction or principal component analysis construction.We study the effect of general orthogonal transforms on functions on Rd which are elements of certain weighted reproducing kernel Hilbert spaces. The notion of Hermite spaces is defined and it is shown that there are examples which admit tractability of integration. We translate the action of the orthogonal transform of Rd into an action on the Hermite coefficients and we give examples where orthogonal transforms have a dramatic effect on the weighted norm, thus providing an explanation for the efficiency of using suitable orthogonal transforms.

Efficient Monte Carlo simulation for integral functionals of Brownian motion

In the paper, we develop a variance reduction technique for Monte Carlo simulations of integral functionals of a Brownian motion. The procedure is based on a new method of sampling the process, which combines the Brownian bridge construction with conditioning on integrals along paths of the process. The key element in our method is the identification of a low-dimensional vector of variables that reduces the dimension of the integration problem more effectively than the Brownian bridge. We illustrate the method by applying it in conjunction with low-discrepancy sequences to the problem of pricing Asian options.

An Analysis of Sobol Sequence and the Brownian Bridge

We investigate the loss of efficiency of Sobol low-discrepancy sequence at high dimensions and the apparent improvement provided by the use of the Brownian bridge construction of Brownian motion paths. We show numerically that some often cited potential causes for these phenomena such as low quality of Sobol coordinates at high dimensions are not to blame and instead isolate a bias in Sobol sequence which we conjecture to be the main cause of the problem. We motivate this conjecture by an analysis of the equations defining both the Incremental and Brownian bridge constructions in a simplified setting, showing how the identified bias is removed by the use of the Brownian bridge. We further give numerical evidence that randomizing Sobol sequence can remove most of this bias and achieve a good convergence at high dimensions. We explain why this is particularly relevant for efficient inline implementations in massively parallel environments such as GPUs under the programming language CUDA. Tested products and models include vanilla and barrier options as well as TARN PRDCs in Local Volatility. We also provide proofs of the homogeneity properties of the Gaussian deviates derived from Sobol sequence for particular numbers of iterations.

New Brownian bridge construction in quasi-Monte Carlo methods for computational finance

Quasi-Monte Carlo (QMC) methods have been playing an important role for high-dimensional problems in computational finance. Several techniques, such as the Brownian bridge (BB) and the principal component analysis, are often used in QMC as possible ways to improve the performance of QMC. This paper proposes a new BB construction, which enjoys some interesting properties that appear useful in QMC methods. The basic idea is to choose the new step of a Brownian path in a certain criterion such that it maximizes the variance explained by the new variable while holding all previously chosen steps fixed. It turns out that using this new construction, the first few variables are more “important” (in the sense of explained variance) than those in the ordinary BB construction, while the cost of the generation is still linear in dimension. We present empirical studies of the proposed algorithm for pricing high-dimensional Asian options and American options, and demonstrate the usefulness of the new BB.

On the tractability of the Brownian Bridge algorithm

Recent results in the theory of quasi-Monte Carlo methods have shown that the weighted Koksma–Hlawka inequality gives better estimates for the error of quasi-Monte Carlo algorithms. We present a method for finding good weights for several classes of functions and apply it to certain algorithms using the Brownian Bridge construction, which are important for financial applications.

Path Generation for Quasi-Monte Carlo Simulation of Mortgage-Backed Securities

Monte Carlo simulation is playing an increasingly important role in the pricing and hedging of complex, path dependent financial instruments. Low discrepancy simulation methods offer the potential to provide faster rates of convergence than those of standard Monte Carlo methods; however, in high dimensional problems special methods are required to ensure that the faster convergence rates hold. Indeed, Ninomiya and Tezuka (1996) have shown highdimensional examples, in which low discrepancy methods perform worse than Monte Carlo methods. The principal component construction introduced by Acworth et al. (1998) provides one solution to this problem. However, the computational effort required to generate each path grows quadratically with the dimension of the problem. This article presents two new methods that offer accuracy equivalent, in terms of explained variability, to the principal components construction with computational requirements that are linearly related to the problem dimension. One method is to take into account knowledge about the payoff function, which makes it more flexible than the Brownian Bridge construction. Numerical results are presented that show the benefits of such adjustments. The different methods are compared for the case of pricing mortgage backed securities using the Hull-White term structure model.

Numerical integration using sparse grids

We present new and review existing algorithms for the numerical integration of multivariate functions defined over d-dimensional cubes using several variants of the sparse grid method first introduced by Smolyak [49]. In this approach, multivariate quadrature formulas are constructed using combinations of tensor products of suitable one-dimensional formulas. The computing cost is almost independent of the dimension of the problem if the function under consideration has bounded mixed derivatives. We suggest the usage of extended Gauss (Patterson) quadrature formulas as the one‐dimensional basis of the construction and show their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, Clenshaw–Curtis and Gauss rules in several numerical experiments and applications. For the computation of path integrals further improvements can be obtained by combining generalized Smolyak quadrature with the Brownian bridge construction.