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615
Multilevel Monte Carlo path simulation
 OPERATIONS RESEARCH
, 2008
"... We show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using Monte Carlo path simulations. In the simplest case of a Lipschitz payoff and an Euler discretisation, the computational cost to achiev ..."
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Cited by 189 (23 self)
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We show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using Monte Carlo path simulations. In the simplest case of a Lipschitz payoff and an Euler discretisation, the computational cost to achieve an accuracy of O(É) is reduced from O(É â3) to O(É â2 (log É) 2). The analysis is supported by numerical results showing significant computational savings.
A stochastic mesh method for pricing highdimensional American options
 Journal of Computational Finance
, 1997
"... Highdimensional problems frequently arise in the pricing of derivative securities – for example, in pricing options on multiple underlying assets and in pricing term structure derivatives. American versions of these options, ie, where the owner has the right to exercise early, are particularly chal ..."
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Cited by 137 (8 self)
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Highdimensional problems frequently arise in the pricing of derivative securities – for example, in pricing options on multiple underlying assets and in pricing term structure derivatives. American versions of these options, ie, where the owner has the right to exercise early, are particularly challenging to price. We introduce a stochastic mesh method for pricing highdimensional American options when there is a finite, but possibly large, number of exercise dates. The algorithm provides point estimates and confidence intervals; we provide conditions under which these estimates converge to the correct values as the computational effort increases. Numerical results illustrate the performance of the method. 1
Exact Simulation of Stochastic Volatility and other
 Affine Jump Diffusion Processes, Working Paper
, 2004
"... The stochastic differential equations for affine jump diffusion models do not yield exact solutions that can be directly simulated. Discretization methods can be used for simulating security prices under these models. However, discretization introduces bias into the simulation results and a large nu ..."
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Cited by 121 (1 self)
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The stochastic differential equations for affine jump diffusion models do not yield exact solutions that can be directly simulated. Discretization methods can be used for simulating security prices under these models. However, discretization introduces bias into the simulation results and a large number of time steps may be needed to reduce the discretization bias to an acceptable level. This paper suggests a method for the exact simulation of the stock price and variance under Heston’s stochastic volatility model and other affine jump diffusion processes. The sample stock price and variance from the exact distribution can then be used to generate an unbiased estimator of the price of a derivative security. We compare our method with the more conventional Euler discretization method and demonstrate the faster convergence rate of the error in our method. Specifically, our method achieves an O(s− 1 2) convergence rate, where s is the total computational budget. The convergence rate for the Euler discretization method is O(s− 1 3) or slower, depending on the model coefficients and option payoff function. Subject Classifications: Simulation, efficiency: exact methods. Finance, asset pricing: computational methods. Acknowledgement: This paper was presented at seminars at Columbia University, the sixth Monte
Efficient simulation of the Heston stochastic volatility model. Working paper, Banc of America Securities
, 2007
"... Stochastic volatility models are increasingly important in practical derivatives pricing applications, yet relatively little work has been undertaken in the development of practical Monte Carlo simulation methods for this class of models. This paper considers several new algorithms for timediscreti ..."
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Cited by 69 (1 self)
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Stochastic volatility models are increasingly important in practical derivatives pricing applications, yet relatively little work has been undertaken in the development of practical Monte Carlo simulation methods for this class of models. This paper considers several new algorithms for timediscretization and Monte Carlo simulation of Hestontype stochastic volatility models. The algorithms are based on a careful analysis of the properties of affine stochastic volatility diffusions, and are straightforward and quick to implement and execute. Tests on realistic model parameterizations reveal that the computational efficiency and robustness of the simulation schemes proposed in the paper compare very favorably to existing methods. 1
Improved multilevel Monte Carlo convergence using the Milstein scheme. In: Monte Carlo and QuasiMonte Carlo methods 2006
, 2007
"... Summary. In this paper we show that the Milstein scheme can be used to improve the convergence of the multilevel Monte Carlo method for scalar stochastic dierential equations. Numerical results for Asian, lookback, barrier and digital options demonstrate that the computational cost to achieve a roo ..."
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Cited by 64 (14 self)
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Summary. In this paper we show that the Milstein scheme can be used to improve the convergence of the multilevel Monte Carlo method for scalar stochastic dierential equations. Numerical results for Asian, lookback, barrier and digital options demonstrate that the computational cost to achieve a rootmeansquare error of is reduced to O(2). This is achieved through a careful construction of the multilevel estimator which computes the dierence in expected payo when using dierent numbers of timesteps. 1
Pricing and Hedging Spread Options
 SIAM Review
, 2003
"... Abstract. We survey theoretical and computational problems associated with the pricing and hedging of spread options. These options are ubiquitous in the financial markets, whether they be equity, fixed income, foreign exchange, commodities, or energy markets. As a matter of introduction, we present ..."
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Cited by 59 (8 self)
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Abstract. We survey theoretical and computational problems associated with the pricing and hedging of spread options. These options are ubiquitous in the financial markets, whether they be equity, fixed income, foreign exchange, commodities, or energy markets. As a matter of introduction, we present a general overview of the common features of all spread options by discussing in detail their roles as speculation devices and risk management tools. We describe the mathematical framework used to model them, and we review the numerical algorithms actually used to price and hedge them. There is already extensive literature on the pricing of spread options in the equity and fixed income markets, and our contribution is mostly to put together material scattered across a wide spectrum of recent textbooks and journal articles. On the other hand, information about the various numerical procedures that can be used to price and hedge spread options on physical commodities is more difficult to find. For this reason, we make a systematic effort to choose examples from the energy markets in order to illustrate the numerical challenges associated with these instruments. This gives us a chance to discuss an interesting application of spread options to an asset valuation problem after it is recast in the framework of real options. This approach is currently the object of intense mathematical research. In this spirit, we review the two major avenues to modeling energy price dynamics. We explain how the pricing and hedging algorithms can be implemented in the framework of models for both the spot price dynamics and the forward curve dynamics.
Multilevel dual approach for pricing American style derivatives
, 2011
"... In this article we propose a novel approach to reduce the computational complexity of the dual method for pricing American options. We consider a sequence of martingales that converges to a given target martingale and decompose the original dual representation into a sum of representations that corr ..."
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Cited by 50 (4 self)
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In this article we propose a novel approach to reduce the computational complexity of the dual method for pricing American options. We consider a sequence of martingales that converges to a given target martingale and decompose the original dual representation into a sum of representations that correspond to different levels of approximation to the target martingale. By next replacing in each representation true conditional expectations with their Monte Carlo estimates, we arrive at what one may call a multilevel dual Monte Carlo algorithm. The analysis of this algorithm reveals that the computational complexity of getting the corresponding target upper bound, due to the target martingale, can be significantly reduced. In particular, it turns out that using our new approach, we may construct a multilevel version of the wellknown nested Monte Carlo algorithm of Andersen and Broadie (2004) that is, regarding complexity, virtually equivalent to a nonnested algorithm. The performance of this multilevel algorithm is illustrated by a numerical example.