Calibrating a Market Model to Commodity and Interest Rate Risk

24 Pages Posted: 4 May 2016 Last revised: 2 Jul 2016

See all articles by Patrik Karlsson

Patrik Karlsson

drkarlsson.com

Kay F. Pilz

RIVACON

Erik Schlögl

University of Technology Sydney (UTS), Quantitative Finance Research Centre; University of Cape Town (UCT) - The African Institute of Financial Markets and Risk Management; Faculty of Science, Department of Statistics, University of Johannesburg; Financial Research Network (FIRN)

Date Written: May 3, 2016

Abstract

Based on the multi-currency LIBOR Market Model (LMM) this paper constructs a hybrid commodity interest rate market model with a time-dependent stochastic local volatility function allowing the model to simultaneously fit the implied volatility surfaces of commodity and interest rate options. Since liquid market prices are only available for options on commodity futures, rather than forwards, a convexity correction formula for the model is derived to account for the difference between forward and futures prices. A procedure for efficiently calibrating the model to interest rate and commodity volatility smiles is constructed. Finally, the model is fitted to an exogenously given correlation structure between forward interest rates and commodity prices (cross-correlation). When calibrating to options on forwards (rather than futures), the fitting of cross-correlation preserves the (separate) calibration in the two markets (interest rate and commodity options), while in the case of futures a (rapidly converging) iterative fitting procedure is presented. The fitting of cross-correlation is reduced to finding an optimal rotation of volatility vectors, which is shown to be an appropriately modified version of the "orthonormal Procrustes" problem in linear algebra. The calibration approach is demonstrated in an application to market data for oil futures.

Keywords: Calibration, Commodity markets, Derivative pricing, Interest rate modelling, Interest rate derivatives, Oil futures, Energy derivatives

JEL Classification: G13

Suggested Citation

Karlsson, Patrik and Pilz, Kay F. and Schloegl, Erik, Calibrating a Market Model to Commodity and Interest Rate Risk (May 3, 2016). Available at SSRN: https://ssrn.com/abstract=2773974 or http://dx.doi.org/10.2139/ssrn.2773974

Patrik Karlsson

drkarlsson.com ( email )

Sweden

Kay F. Pilz

RIVACON ( email )

Im Apfelgrund 4
Friedrichsdorf, 61381
Germany

HOME PAGE: http://www.rivacon.com

Erik Schloegl (Contact Author)

University of Technology Sydney (UTS), Quantitative Finance Research Centre ( email )

Ultimo
PO Box 123
Sydney, NSW 2007
Australia
+61 2 9514 2535 (Phone)

HOME PAGE: http://www.schlogl.com

University of Cape Town (UCT) - The African Institute of Financial Markets and Risk Management ( email )

Leslie Commerce Building
Rondebosch
Cape Town, Western Cape 7700
South Africa

Faculty of Science, Department of Statistics, University of Johannesburg ( email )

Auckland Park, 2006
South Africa

Financial Research Network (FIRN)

C/- University of Queensland Business School
St Lucia, 4071 Brisbane
Queensland
Australia

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

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