Calibrating a market model with stochastic volatility to commodity and interest rate risk

Abstract

Based on the multi-currency LIBOR Market Model, this paper constructs a hybrid commodity interest rate market model with a 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

Acknowledgements

The first author work was carried out while Patrik was a PhD student at Lund University, Sweden and a visiting scholar position at the Quantitative Finance Research Centre (QFRC) at University of Technology Sydney, Australia, Oct 2011–Jan 2012. He wishes to thank Caroline Dobson and the QFRC for their hospitality. Moreover, thanks to Hans Byström for connecting him with the QFRC. The second author work of this paper was carried out while being a visitor at the Quantitative Finance Research Centre (QFRC) at University of Technology Sydney, Australia. Kay Pilz wishes to thank the QFRC for their hospitality.

Notes

No potential conflict of interest was reported by the authors.

1 See the seminal papers by Miltersen et al. (1997), Brace et al. (1997) and Jamshidian (1997).

2 See Schlögl (2002b).

3 Grzelak and Oosterlee (2012) presented an extensions of Schlögl (2002b) with stochastic volatility.

4 See, for example, Gibson and Schwartz (1990), Cortazar and Schwartz (1994), Schwartz (1997), Miltersen and Schwartz (1998), and Miltersen (2003).

5 See Golub and Van Loan (1996).

6 This algorithm is given as algorithm 8.1 in Gower and Dijksterhuis (2004).

7 This is reported, for example, by Derman (2003).

8 The stochastic interest rate dynamics used in these papers are simpler than the SLV–LMM dynamics considered here, but nevertheless demonstrate the relevance of incorporating interest rate risk into a commodity derivatives model.

9 See section 3.10 of Rebonato et al. (2009).

10 Although swaption prices depend in theory on correlations between forward rates, in practice this dependence is too weak for these correlations to be extracted in a meaningful way; see, e.g. Choy et al. (2004).

11 This forward measure is the equivalent martingale measure associated with taking the zero coupon bond as the numeraire, and under this measure (the existence of which is assured under the model assumptions below) forward LIBOR is necessarily a martingale, i.e. driftless—see, e.g. Musiela and Rutkowski (1997b).

12 Thus is a d-dimensional vector, each component , , is a Brownian motion under the -forward measure, and the quadratic covariation between the components is zero: .

13 See for instance section 4.2.3 of Andersen and Piterbarg (2010).

14 Thus, the C(t, T) represent the effect of the convenience yield net of storage cost.

15 The futures vs. forward relation will be discussed in section 4.

16 See equation (11) in Schlögl (2002b).

17 For notational simplicity, we assume that the option expires at the same time the futures does. In most cases the option expires a few days before the futures expiry. In some cases, like for EUA carbon emission futures, the option can even expire several months before the underlying futures.

18 For details, see e.g. Andersen and Piterbarg (2010) (Chapter 9).

19 Note that the roles of and can be interchanged, and the sufficient assumption actually is . Furnishing the model with more stochastic factors than contributes only spurious complexity to the model. From a practical point of view, the aim is to keep the number of stochastic factors small.

20 Note that and .

21 Note that we must iterate over repeated calibration to the commodity market and to the cross-correlations, as the conversion of commodity futures into forwards depends on cross-correlations.

22 The origins of this parametric form can be traced back to Rebonato (1999); see Chapter 22 of Rebonato (2004) for a detailed discussion of the rationale behind a parametric form of this type. Our notation follows Andersen and Piterbarg (2010).

23 These are a selection of current expiries, i.e. the time between the calibration date 13 January 2015 and the various expiry dates, for the standardized commodity options.

24 Specifically, mean reversion would manifest itself in the market as a downward sloping term structure of commodity option implied volatilities.