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Markov models for commodity futures: theory and practice

2010; Taylor & Francis; Volume: 10; Issue: 8 Linguagem: Inglês

10.1080/14697680903493599

1469-7696

Leif B. G. Andersen,

Financial Risk and Volatility Modeling

Abstract The objective of this paper is to develop a generic, yet practical, framework for the construction of Markov models for commodity derivatives. We aim for sufficient richness to permit applications to a broad variety of commodity markets, including those that are characterized by seasonality and by spikes in the spot process. In the first, largely theoretical, part of the paper we derive a series of useful results concerning the low-dimensional Markov representation of the dynamics of an entire term structure of futures prices. Extending previous results in the literature, we cover jump-diffusive models with stochastic volatility as well as several classes of regime-switching models. To demonstrate the process of building models for a specific commodity market, the second part of the paper applies a selection of our theoretical results to the exercise of constructing and calibrating derivatives trading models for USD natural gas. Special attention is paid to the incorporation of empirical seasonality effects in futures prices, in implied volatilities and their 'smile', and in correlations between futures contracts of different maturities. European option pricing in our proposed gas model is closed form and of the same complexity as the Black–Scholes formula. Keywords: Quantitative financeProbability theoryPricing modelsPricing of derivatives securities Acknowledgements I am indebted to Jesper Andreasen, Alex Levin, John Crosby and the two referees for their comments and suggestions. Notes †A similar split exists in interest derivative applications, where traditional short-rate models have largely been superseded by models that operate directly on forward curve dynamics (e.g. the famous HJM model (Heath et al. Citation1992)). †Of course, our usage of a standard (uncorrelated) vector Brownian motion W(t) involves no loss of generality as a suitable rotation (e.g. by a Cholesky matrix) can induce any correlation structure. †Loosely speaking, this happens when there are more mean-reversion speeds than Brownian motions. †Lemma 3.1 can, if convenient, be restated using deterministic transformation of the state variables x* and y*, as in sections 2.1 and 2.2. ‡We enable time dependence to allow for seasonality effects (e.g. that the volatility-of-volatility is higher in winters than in summers, and so forth). †In many commodity markets, we expect that both the arrival rate and the severity of jumps may exhibit seasonality, i.e. are functions of calendar time. ‡The use of a vector-valued jump process is primarily useful for applications where we model many commodities simultaneously, as some components of the jump vector can be considered idiosyncratic and some can be shared across commodities; the latter type can be used to create 'jump correlation' (Crosby Citation2008a). In most single-commodity applications, one will often use only a small number of jump processes. †In the expression for φ J in lemma 4.3, notice that integrals cannot be reused across option maturities due to the presence of β(T) in the integrand. An exception occurs when β(T) is constant, including the case where jumps do not mean-revert (a case that is of no interest for markets with spiky price behavior). ‡In fact, we can also allow for jump-models of the type in section 4. However, it is unlikely that we, in practice, would ever use both regime-switching and ordinary jump-diffusions in the same model. †We list extensions to multiple jump processes shortly. ‡We thank a referee for emphasizing this point. †If we are interested in having multiple average spike durations in the same model, we simply add more Markov chains, as described in section 5.3. ‡In particular, we break the independence of the elements in the matrix H by effectively forcing the jump in J associated with a move from state e j to state e 1 to be precisely opposite of the most recent jump in J (that took place when c transitioned into state e j ). §If we elect to use M independent jump processes in our model, the number of Markov state variables associated with spike generation will therefore be M. †Henry Hub natural gas futures, as marked by the trading desk at Banc of America Securities. ‡Traded gas futures contracts mature at fixed time of maturity dates T spaced one month apart. We have used interpolation between neighboring futures contracts to construct the graph of futures with constant time to maturity T − t in figure 3. †Notice that our model allows for closed-form computation of the deterministic variables y(t) in proposition 2.2. ‡Miltersen (Citation2003) is one of the few references in the literature that proposes a specific procedure for actual model calibration; his algorithm is less market-oriented (and less convenient) than ours, but shares some of the principles on display in our section 7.5. §Overall, implied volatilities for gas futures prices with locked time-of-maturity T here displayed no clear-cut seasonality effects. †Notice that, in lemma 2.5, . †For the case where d(T) is found by fitting to l(t) in (Equation59), the stationarity properties of the model are, if anything, better than those in figure 10, so we omit the graph for this case. †In addition, we have the ability to induce further skew seasonality by letting intensities n 1 and n 2 vary periodically with time. In practice, however, the amount of volatility seasonality that can be induced this way is quite limited, except from the short end of the volatility term structure. †Write down an expression for dA(t) and set its expectation equal to zero.