Robust Product Markovian Quantization

19 Pages Posted: 21 Jul 2020

See all articles by Ralph Rudd

Ralph Rudd

The African Institute of Financial Markets and Risk Management

Thomas McWalter

University of Cape Town (UCT); University of Johannesburg

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: June 24, 2020

Abstract

Recursive marginal quantization (RMQ) allows the construction of optimal discrete grids for approximating solutions to stochastic differential equations in d-dimensions. Product Markovian quantization (PMQ) reduces this problem to d one-dimensional quantization problems by recursively constructing product quantizers, as opposed to a truly optimal quantizer. However, the standard Newton-Raphson method used in the PMQ algorithm suffers from numerical instabilities, inhibiting widespread adoption, especially for use in calibration. By directly specifying the random variable to be quantized at each time step, we show that PMQ, and RMQ in one dimension, can be expressed as standard vector quantization. This reformulation allows the application of the accelerated Lloyd's algorithm in an adaptive and robust procedure. Furthermore, in the case of stochastic volatility models, we extend the PMQ algorithm by using higher-order updates for the volatility or variance process. We illustrate the technique for European options, using the Heston model, and more exotic products, using the SABR model.

Keywords: vector quantization, option pricing, stochastic volatility, calibration

JEL Classification: C63, G12, G13

Suggested Citation

Rudd, Ralph and McWalter, Thomas and Kienitz, Joerg and Platen, Eckhard, Robust Product Markovian Quantization (June 24, 2020). Available at SSRN: https://ssrn.com/abstract=3637784 or http://dx.doi.org/10.2139/ssrn.3637784

Ralph Rudd

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

Leslie Commerce Building
Rondebosch
Cape Town, Western Cape 7700
South Africa
+27 21 650 2474 (Phone)

Thomas McWalter (Contact Author)

University of Cape Town (UCT) ( email )

Private Bag X3
Rondebosch, Western Cape 7701
South Africa

University of Johannesburg ( email )

PO Box 524
Auckland Park
Johannesburg, Gauteng 2006
South Africa

Joerg Kienitz

University of Wuppertal - Applied Mathematics ( email )

Gaußstraße 20
42097 Wuppertal
Germany

University of Cape Town (UCT) ( email )

Private Bag X3
Rondebosch, Western Cape 7701
South Africa

Quaternion Risk Management ( email )

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Dublin, D02X308
Ireland

Eckhard Platen

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

Broadway
GPO Box 123
Sydney, NSW 2007, 2007
Australia
+61 2 9514 7759 (Phone)

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 )

P.O. Box 123
Broadway
Sydney, New South Wales 2007
Australia
+61 (02) 9514 2271 (Phone)

Financial Research Network (FIRN)

C/- University of Queensland Business School
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Queensland
Australia

HOME PAGE: http://www.firn.org.au

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