Recursive Marginal Quantization of Higher-Order Schemes

28 Pages Posted: 9 Jan 2017

See all articles by Thomas McWalter

Thomas McWalter

University of Cape Town (UCT); University of Johannesburg

Ralph Rudd

The African Institute of Financial Markets and Risk Management

Joerg Kienitz

University of Wuppertal - Applied Mathematics; University of Cape Town (UCT); Quaternion Risk Management

Eckhard Platen

University of Technology, Sydney (UTS) - Finance Discipline Group; University of Technology Sydney, School of Mathematical and Physical Sciences; Financial Research Network (FIRN)

Date Written: January 5, 2017

Abstract

Quantization techniques have been applied in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and efficient calibration of large derivative books. Recursive Marginal Quantization of the Euler scheme has recently been proposed as an efficient numerical method for evaluating functionals of solutions of stochastic differential equations. This method involves recursively quantizing the conditional marginals of the discrete-time Euler approximation of the underlying process. By generalizing this approach, we show that it is possible to perform recursive marginal quantization for two higher-order schemes: the Milstein scheme and a simplified weak order 2.0 scheme. As part of this generalization a simple matrix formulation is presented, allowing efficient implementation. We further extend the applicability of recursive marginal quantization by showing how absorption and reflection at the zero boundary may be incorporated, when this is necessary. To illustrate the improved accuracy of the higher order schemes, various computations are performed using geometric Brownian motion and its generalization, the constant elasticity of variance model. For both processes, we show numerical evidence of improved weak order convergence and we compare the marginal distributions implied by the three schemes to the known analytical distributions. By pricing European, Bermudan and Barrier options, further evidence of improved accuracy of the higher order schemes is demonstrated.

Keywords: quantization, option pricing

Suggested Citation

McWalter, Thomas and Rudd, Ralph and Kienitz, Joerg and Platen, Eckhard, Recursive Marginal Quantization of Higher-Order Schemes (January 5, 2017). Available at SSRN: https://ssrn.com/abstract=2894753 or http://dx.doi.org/10.2139/ssrn.2894753

Thomas McWalter (Contact Author)

University of Cape Town (UCT) ( email )

Private Bag X3
Rondebosch, Western Cape 7701
South Africa

University of Johannesburg ( email )

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Ralph Rudd

The African Institute of Financial Markets and Risk Management ( email )

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South Africa
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Joerg Kienitz

University of Wuppertal - Applied Mathematics ( email )

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Germany

University of Cape Town (UCT) ( email )

Private Bag X3
Rondebosch, Western Cape 7701
South Africa

Quaternion Risk Management ( email )

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Eckhard Platen

University of Technology, Sydney (UTS) - Finance Discipline Group ( email )

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Sydney, NSW 2007, 2007
Australia
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HOME PAGE: http://datasearch.uts.edu.au/business/finance/staff/StaffDetails.cfm?UnitStaffId=90

University of Technology Sydney, School of Mathematical and Physical Sciences ( email )

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Australia
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Financial Research Network (FIRN)

C/- University of Queensland Business School
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Queensland
Australia

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