Showing papers in "Monte Carlo Methods and Applications in 2011"
TL;DR: In this paper, the authors propose a scheme for unbiased estimation of the limiting value x ∞, together with estimates of standard error and apply this to examples including numerical integrals, root-finding and option pricing in a Heston Stochastic Volatility model.
Abstract: Consider a stochastic process Xn, n = 0, 1, 2, ...such that EXn → x∞ as n → ∞. The sequence {Xn} may be a deterministic one, obtained by using a numerical integration scheme, or obtained from Monte-Carlo methods involving an approximation to an integral, or a Newton-Raphson iteration to approximate the root of an equation but we will assume that we can sample from the distribution of X1, X2, ...Xm for finite m. We propose a scheme for unbiased estimation of the limiting value x∞, together with estimates of standard error and apply this to examples including numerical integrals, root-finding and option pricing in a Heston Stochastic Volatility model.
96 citations
TL;DR: De Gruyter et al. as mentioned in this paper developed algorithms for the fast generation of correlated Gaussian random fields on rectangular regions of ℝ d. The complexities of the algorithms are derived, simulation results and error analysis are presented.
Abstract: Fast Fourier transforms are used to develop algorithms for the fast generation of correlated Gaussian random fields on rectangular regions of ℝ d. The complexities of the algorithms are derived, simulation results and error analysis are presented. © de Gruyter 2011.
52 citations
TL;DR: In this article, the convergence and asymptotic normality of the adaptive Monte Carlo estimator were established under local assumptions which are easily verifiable in practice, and the optimal importance sampling parameter was approximated using a randomly truncated stochastic algorithm.
Abstract: Adaptive Monte Carlo methods are recent variance reduction techniques. In this work, we propose a mathematical setting which greatly relaxes the assumptions needed by for the adaptive importance sampling techniques presented by Vazquez-Abad and Dufresne, Fu and Su, and Arouna. We establish the convergence and asymptotic normality of the adaptive Monte Carlo estimator under local assumptions which are easily verifiable in practice. We present one way of approximating the optimal importance sampling parameter using a randomly truncated stochastic algorithm. Finally, we apply this technique to some examples of valuation of financial derivatives.
35 citations
TL;DR: Various alternative simulation methods within these algorithms on the basis of acceptance rate in acceptance-rejection sampling for both high- and low-frequency sampling are discussed, illustrating their advantage relative to the existing approximative simulation method based on infinite shot noise series representation.
Abstract: Exact yet simple simulation algorithms are developed for a wide class of Orn- stein-Uhlenbeck processes with tempered stable stationary distribution of finite varia- tion with the help of their exact transition probability between consecutive time points. Random elements involved can be divided into independent tempered stable and com- pound Poisson distributions, each of which can be simulated in the exact sense through acceptance-rejection sampling, respectively, with stable and gamma proposal distribu- tions. We discuss various alternative simulation methods within our algorithms on the ba- sis of acceptance rate in acceptance-rejection sampling for both high- and low-frequency sampling. Numerical results illustrate their advantage relative to the existing approxima- tive simulation method based on infinite shot noise series representation.
29 citations
TL;DR: The main purpose of the article consists in introducing and implementing a stochastic particle algorithm to approach the solution to PDE which also fits in the case when $\beta$ is possibly irregular, to predict some long-time behavior of the solution and in comparing with some recent numerical deterministic techniques.
Abstract: The object of this paper is a one-dimensional generalized porous media equation (PDE) with possibly discontinuous coefficient $\beta$, which is well-posed as an evolution problem in $L^1(\mathbb{R})$. In some recent papers of Blanchard et alia and Barbu et alia, the solution was represented by the solution of a non-linear stochastic differential equation in law if the initial condition is a bounded integrable function. We first extend this result, at least when $\beta$ is continuous and the initial condition is only integrable with some supplementary technical assumption. The main purpose of the article consists in introducing and implementing a stochastic particle algorithm to approach the solution to (PDE) which also fits in the case when $\beta$ is possibly irregular, to predict some long-time behavior of the solution and in comparing with some recent numerical deterministic techniques.
19 citations
TL;DR: In this article, the authors developed error expansions with computable lead- ing order terms for the global weak error in the tau-leap discretization of pure jump processes arising in kinetic Monte Carlo models.
Abstract: This work develops novel error expansions with computable lead- ing order terms for the global weak error in the tau-leap discretization of pure jump processes arising in kinetic Monte Carlo models. Accurate computable a posteriori error approximations are the basis for adaptive algorithms; a fun- damental tool for numerical simulation of both deterministic and stochastic dynamical systems. These pure jump processes are simulated either by the tau-leap method, or by exact simulation, also referred to as dynamic Monte Carlo, the Gillespie algorithm or the Stochastic simulation algorithm. Two types of estimates are presented: an a priori estimate for the relative error that gives a comparison between the work for the two methods depending on the propensity regime, and an a posteriori estimate with computable leading order term.
14 citations
TL;DR: The main result is a derivation of an error expansion for the resulting model error, , with computable leading order term, which depends both on the choice of truncation level ∈ and the contract payoff g, and it is valid even when g is not continuous.
Abstract: This thesis is a study of approximation and calibration of stochastic processes with applications in finance. It consists of an introduction and four research papers. The introduction is as an overview of the role of mathematics incertain areas of finance. It contains a brief introduction to the mathematicaltheory of option pricing, as well as a description of a mathematicalmodel of a financial exchange. The introduction also includessummaries of the four research papers. In Paper I, Markov decision theory is applied to design algorithmic trading strategies in an order driven market. A high dimensional Markov chain is used to model the state and evolution of the limit order book. Trading strategies are formulated as optimal decision problems. Conditions that guarantee existence of optimal strategies are provided, as well as a value-iterative algorithm that enables numerical construction of optimal strategies. The results are illustrated with numerical experiments on high frequency data from a foreign exchange market. Paper II focuses on asset pricing with Levy processes. The expected value E[g(XT )] is estimated using a Monte Carlo method, when Xt is a d-dimensional Levy process having infinite jump activity and a smooth density. Approximating jumps smaller then a parameter e > 0 by diffusion results in a weak approximation, Xt, of Xt. The main result of the paper is an estimate of the resulting model error E[g(XT )] − E[g(T )], with a computable leading order term. Option prices in exponential Levy models solve certain partia lintegro-differential equations (PIDEs). A finite difference scheme suitable for solving such PIDEs is studied in Paper III. The main resultsare estimates of the time and space discretization errors, with leading order terms in computable form. If the underlying Levy process has infinite jump activity, the jumps smaller than some e > 0 are replacedby diffusion. The size of this diffusion approximation is estimated, as well as its effect on the space and time discretization errors. Combined, the results of the paper are enough to determine how to jointly choose the grid size and the parameter e. In Paper IV it is demonstrated how optimal control can be used to calibrate a jump-diffusion process to quoted option prices. The calibration problem is formulated as an optimal control problem with the model parameter as a control variable. The corresponding regularized Hamiltonian system is solved with a symplectic Euler method.
13 citations
TL;DR: In this article, the authors proposed an optimal quadratic quantization method for the pricing of barrier options by using a known useful representation of the premium of the barrier options and then deduced an algorithm similar to one used to estimate nonlinear filter using quad-ratic optimal functional quantization.
Abstract: This paper is devoted to the pricing of Barrier options by optimal quadratic quantization method. From a known useful representation of the premium of barrier options one deduces an algorithm similar to one used to estimate nonlinear filter using quadratic optimal functional quantization. Some numerical tests are fulfilled in the Black-Scholes model and in a local volatility model and a comparison to the so called Brownian Bridge method is also done.
11 citations
TL;DR: A wavelet-based method for simulation of strictly sub-Gaussian processes with given accuracy and reliability in C([0, T]) is proposed.
Abstract: A wavelet-based method for simulation of strictly sub-Gaussian processes with given accuracy and reliability in C([0, T]) is proposed.
10 citations
TL;DR: In the area of financial mathematics, Monte Carlo simulation is often successfully used to estimate the prices of certain products as mentioned in this paper, however in many cases calibrating Monte Carlo based models to market prices turns out to be difficult due to stochastic noise arising in the objective functionals.
Abstract: In the area of financial mathematics Monte Carlo simulation is often successfully used to estimate the prices of certain products. However in many cases calibrating Monte Carlo based models to market prices turns out to be difficult due to stochastic noise arising in the objective functionals.
5 citations
TL;DR: In this article, the authors run simulations regarding an open problem aboutd -dimensional critical branching random walks in a random IID environment, given by the rule that at every site independently, with probability p 2 Œ0;1�, there is a cookie, completely suppressing the branching of any particle located there.
Abstract: Using a high performance computer cluster, we run simulations regarding an open problem aboutd -dimensional critical branching random walks in a random IID en- vironment The environment is given by the rule that at every site independently, with probability p 2 Œ0;1� , there is a cookie, completely suppressing the branching of any particle located there. The simulations suggest self averaging: the asymptotic survival probability inn steps is the same in the annealed and the quenched case; it is 2 qn ,w hereq WD1� p. This particular asymptotics indicates a non-trivial phenomenon: the tail of the survival probability (both in the annealed and the quenched case) is the same as in the case of non-spatial unit time critical branching, where the branching rule is modified: branching only takes place with probabilityq for every particle at every iteration.
TL;DR: This work approximates the distribution functions of cumulative sum (cusum) and cumulative sum of square (cusumsq) test statistics using the Hansen's (Journal of Business & Economic Statistics 15: 60–67, 1997) method.
Abstract: In this note, we are concerned with approximating the distribution functions of cumulative sum (cusum) and cumulative sum of square (cusumsq) test statistics using the Hansen's (Journal of Business & Economic Statistics 15: 60–67, 1997) method. The coefficients of Hansen's polynomial are estimated by least square method. Error analysis shows that approximations are accurate. We first assume that the data are independent with common normal distribution. Then extensions to non-normal and dependent observations are considered. A conclusion section is also given.
TL;DR: Estimates for the probability of deviation in the uniform metric of sums of independent identically distributed fields, which belong to Orlicz spaces, were found.
Abstract: This paper is devoted to the accuracy and reliability estimation (in uniform metrics) of calculation of improper integrals depending on a parameter t , using the Monte Carlo method. For this, estimates for the probability of deviation in the uniform metric of sums of independent identically distributed fields, which belong to Orlicz spaces, were found.
TL;DR: In this paper, the problem of finding a non-negative pseudo-solution of systems of linear algebraic equations is treated and a probabilistic method for regularization is proposed.
Abstract: In this paper we treat the problem of finding a non-negative pseudo-solution of systems of linear algebraic equations. A probabilistic method for regularization is proposed. A theorem of existence of the best possible solution is proved. Some extensions of the method are proposed.
TL;DR: A hybrid model (stochastic/deterministic) that describes the time evolution of chemical species in a homogenous gas-phase combustion reaction process at constant volume and the effect of multiple runs on auto-ignition time is investigated.
Abstract: This paper presents a hybrid model (stochastic/deterministic) that describes the time evolution of chemical species in a homogenous gas-phase combustion reaction process at constant volume. First, the paper briefly introduces currently employed stochastic algorithms. Next, the development of the hybrid algorithm is detailed. The model is then validated and tested using a reduced reaction mechanism for methane combustion. The effect of user-input performance parameters on stochastic behavior and computational time is studied. The computational time of the algorithm compared to the stochastic simulation algorithm is then compared, and finally, the effect of multiple runs on auto-ignition time is investigated.
TL;DR: The WD method has the same advantageous property of the well-known score function method that the form of the Greek estimator does not depend on the details of the payoff function but only on the probability density of the underlying model.
Abstract: The stochastic gradient estimation method of weak derivatives (WD) is presented with the aim of constructing efficient algorithms for the estimation of the “Greeks” of financial derivatives. The key idea is to replace the derivative of the probability measure of the underlying model by its WD. The WD method has the same advantageous property of the well-known score function method that the form of the Greek estimator does not depend on the details of the payoff function but only on the probability density of the underlying model. Simulation studies indicate that the WD estimator has significantly lower variance than the score function and finite difference estimator, however, the associated computational burden in certain cases may not be negligible.
TL;DR: This article considers in detail the most simple estimator for functionals of the solutions to second type integral equations, the so-called “estimator on absorbtion”, and establishes some new facts about reduction of its variance in the sign-changing case.
Abstract: The variance minimization problem is very important in the theory of Monte Carlo methods. In the case of integral equations with sign-changing kernel this problem is comparatively little studied. In this article the author points out some methods which allow to avoid difficulties emerging in the sign-changing case. The simplest (in the computational sense) estimators for functionals of the solutions to second type integral equations play an important role in theory and applications of the Monte Carlo method. We consider in detail the most simple, the so-called “estimator on absorbtion”, and establish some new facts about reduction of its variance in the sign-changing case. In conclusion some observations considering other types of estimators are made which allow to improve some known results in the sigh-changing case.