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59
High order discretization schemes for stochastic volatility models
, 2009
"... In usual stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous onedimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using Itô’s formula, we get rid, in the asset price dynamics, ..."
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In usual stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous onedimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using Itô’s formula, we get rid, in the asset price dynamics, of the stochastic integral with respect to the Brownian motion driving this SDE. Taking advantage of this structure, we propose a scheme, based on the Milstein discretization of this SDE, with order one of weak trajectorial convergence for the asset price, a scheme, based on the NinomiyaVictoir discretization of this SDE, with order two of weak convergence for the asset price. We also propose a specific scheme with improved convergence properties when the volatility of the asset price is driven by an OrsteinUhlenbeck process. We confirm the theoretical rates of convergence by numerical experiments and show that our schemes are well adapted to the multilevel Monte Carlo method introduced by Giles [2008a,b].
Complexity of Multilevel Monte Carlo TauLeaping, in progress
, 2013
"... Tauleaping is a popular discretization method for generating approximate paths of continuous time, discrete space, Markov chains, notably for biochemical reaction systems. To compute expected values in this context, an appropriate multilevel Monte Carlo form of tauleaping has been shown to improve ..."
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Tauleaping is a popular discretization method for generating approximate paths of continuous time, discrete space, Markov chains, notably for biochemical reaction systems. To compute expected values in this context, an appropriate multilevel Monte Carlo form of tauleaping has been shown to improve efficiency dramatically. In this work we derive new analytic results concerning the computational complexity of multilevel Monte Carlo tauleaping that are significantly sharper than previous ones. We avoid taking asymptotic limits, and focus on a practical setting where the system size is large enough for many events to take place along a path, so that exact simulation of paths is expensive, making tauleaping an attractive option. We use a general scaling of the system components that allows for the reaction rate constants and the abundances of species to vary over several orders of magnitude, and we exploit the random time change representation developed by Kurtz. The key feature of the analysis that allows for the sharper bounds is that when comparing relevant pairs of processes we analyze the variance of their difference directly rather than bounding via the second moment. Use of the second moment is natural in the setting of a diffusion equation, where multilevel was first developed and where strong convergence results for numerical methods are readily available, but is not optimal for the Poissondriven jump systems that we consider here. We also present computational results that illustrate the new analysis. 1
Multilevel Monte Carlo methods
"... An outline history inspired by undergraduate numerical projects course at Cambridge, and summer projects at RollsRoyce this was one of my first textbooks after 25 years working on CFD, 10 years ago I switched to Monte Carlo methods for computational finance and other application areas ..."
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An outline history inspired by undergraduate numerical projects course at Cambridge, and summer projects at RollsRoyce this was one of my first textbooks after 25 years working on CFD, 10 years ago I switched to Monte Carlo methods for computational finance and other application areas
Central limit theorem for the multilevel monte carlo euler method and applications to asian options
, 2012
"... This paper focuses on studying the multilevel Monte Carlo method recently introduced by Giles [8] and significantly more efficient than the classical Monte Carlo one. Our aim is to prove a central limit theorem of Lindeberg Feller type for the multilevel Monte Carlo method associated to the Euler di ..."
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This paper focuses on studying the multilevel Monte Carlo method recently introduced by Giles [8] and significantly more efficient than the classical Monte Carlo one. Our aim is to prove a central limit theorem of Lindeberg Feller type for the multilevel Monte Carlo method associated to the Euler discretization scheme. To do so, we prove first a stable law convergence theorem, in the spirit of Jacod and Protter [15], for the Euler scheme error on two consecutive levels of the algorithm. This leads to an accurate description of the optimal choice of parameters and to an explicit characterization of the limiting variance in the central limit theorem of the algorithm. We investigate the application of the Multilevel Monte Carlo method to the pricing of Asian options, by discretizing the integral of the payoff process using Riemann and trapezoidal schemes. In particular, we prove stable law convergence for the error of these second order schemes. This allows us to prove two additional central limit theorems providing us the optimal choice of the parameters with an explicit representation of the limiting variance. For this setting of second order schemes, we give new optimal parameters leading to the convergence of the central limit theorem. Complexity analysis of the Multilevel Monte Carlo algorithm were processed.
Multilevel monte carlo for exponential L\’{e} vy models. arXiv preprint arXiv:1403.5309
, 2014
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MULTILEVEL STOCHASTIC APPROXIMATION ALGORITHMS
, 2013
"... Abstract. This paper studies multilevel stochastic approximation algorithms. Our aim is to extend the scope of the multilevel Monte Carlo method recently introduced by Giles [Gil08b] to the framework of stochastic optimization by means of stochastic approximation algorithm. We first introduce and s ..."
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Abstract. This paper studies multilevel stochastic approximation algorithms. Our aim is to extend the scope of the multilevel Monte Carlo method recently introduced by Giles [Gil08b] to the framework of stochastic optimization by means of stochastic approximation algorithm. We first introduce and study a twolevel method, also referred as statistical Romberg stochastic approximation algorithm. Then, its extension to multilevel is proposed. We prove a central limit theorem for both methods and describe the possible optimal choices of step size sequence. Numerical results confirm the theoretical analysis and show a significant reduction in the initial computational cost.
CONSISTENCY AND STABILITY OF A MILSTEINGALERKIN FINITE ELEMENT SCHEME FOR SEMILINEAR SPDE
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A micro/macro algorithm to accelerate Monte CARLO SIMULATION OF STOCHASTIC DIFFERENTIAL EQUATIONS
, 2010
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Radon Series Comp. Appl. Math 8, 1–18 c © de Gruyter 2009 Multilevel quasiMonte Carlo path simulation
"... Abstract. This paper reviews the multilevel Monte Carlo path simulation method for estimating option prices in computational finance, and extends it by combining it with quasiMonte Carlo integration using a randomised rank1 lattice rule. Using the Milstein discretisation of the stochastic differe ..."
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Abstract. This paper reviews the multilevel Monte Carlo path simulation method for estimating option prices in computational finance, and extends it by combining it with quasiMonte Carlo integration using a randomised rank1 lattice rule. Using the Milstein discretisation of the stochastic differential equation, it is demonstrated that the combination has much lower computational cost than either one on its own for evaluating European, Asian, lookback, barrier and digital options.