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

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.     

Long term spread option valuation and hedging

This paper investigates the valuation and hedging of spread options on two commodity prices which in the long run are in dynamic equilibrium (i.e., cointegrated). The spread exhibits properties different from its two underlying commodity prices and should therefore be modelled directly. This approach offers significant advantages relative to the traditional two price methods since the correlation between two asset returns is notoriously hard to model. In this paper, we propose a two factor model for the spot spread and develop pricing and hedging formulae for options on spot and futures spreads. Two examples of spreads in energy markets – the crack spread between heating oil and WTI crude oil and the location spread between Brent blend and WTI crude oil – are analyzed to illustrate the results.     

A class of Lévy process models with almost exact calibration to both barrier and vanilla FX options

Vanilla (standard European) options are actively traded on many underlying asset classes, such as equities, commodities and foreign exchange (FX). The market quotes for these options are typically used by exotic options traders to calibrate the parameters of the (risk-neutral) stochastic process for the underlying asset. Barrier options, of many different types, are also widely traded in all these markets but one important feature of the FX options markets is that barrier options, especially double-no-touch (DNT) options, are now so actively traded that they are no longer considered, in any way, exotic options. Instead, traders would, in principle, like to use them as instruments to which they can calibrate their model. The desirability of doing this has been highlighted by talks at practitioner conferences but, to our best knowledge (at least within the realm of the published literature), there have been no models which are specifically designed to cater for this. In this paper, we introduce such a model. It allows for calibration in a two-stage process. The first stage fits to DNT options (or other types of double barrier options). The second stage fits to vanilla options. The key to this is to assume that the dynamics of the spot FX rate are of one type before the first exit time from a ‘corridor’ region but are allowed to be of a different type after the first exit time. The model allows for jumps (either finite activity or infinite activity) and also for stochastic volatility. Hence, not only can it give a good fit to the market prices of options, it can also allow for realistic dynamics of the underlying FX rate and realistic future volatility smiles and skews. En route, we significantly extend existing results in the literature by providing closed-form (up to Laplace inversion) expressions for the prices of several types of barrier options as well as results related to the distribution of first passage times and of the ‘overshoot’.     

No-arbitrage conditions for storable commodities and the modeling of futures term structures

One distinguishable feature of storable commodities is that they relate to two markets: cash market and storage market. This paper proves that, if no arbitrage exists in the storage-cash dual markets, the commodity convenience yield has to be non-negative. However, classical reduced-form models for futures term structures could allow serious arbitrages due to the high volatility of the convenience yield. To avoid negative convenience yield, this paper proposes a semi-affine arbitrage-free model, which prices futures analytically and fits futures term structures reasonably well. Importantly, our model prices commodity-related contingent claims (such as calendar spread options) quite differently with classical models.     

Strategic commodity allocation

This article extends the study of the financialization of commodities (Rouwenhorst and Tang [Annu. Rev. Financ. Econ., 2012, 4, 447–467]) by considering an investment in the term structure of commodity futures prices. Specifically, we analyse the benefits of adding a distant commodity futures contract and/or a spot commodity (near month futures contract) to a portfolio of bonds and stocks in a setting similar to Brennan and Schwartz [The use of treasury bill futures in strategic asset allocation programs. In Worldwide Asset and Liability Modeling, edited by W.T. Ziemba and J.M. Mulvey, pp. 205–230, 1998 (Cambridge University Press: Cambridge)]. Our analysis employs an empirical study that covers the post financial crisis period. We show that the spot commodity considerably improves the value of the portfolio. However, an investment in the whole term structure of futures contracts is optimally achieved through high opposite positions in the spot commodity and distant futures contracts. We find that these extreme calendar spreads can result in an inappropriate investment.     

A factor model for co-movements of commodity prices

We fit a factor model to two monthly panels of deflated prices of energy, metals and agricultural commodities. Prices consistently display a tendency to revert towards the factor, though the speed of reversion to the factor is slow. Using both in- and out-of-sample metrics, we compare the factor model to that of a “no change” model and to two simple models that tie changes in commodity prices to percentage change in either global industrial production or the U.S. dollar. The factor model does relatively well at long (12 month) horizons. In terms of commodities, the factor model's performance is best for energy prices, worst for metals, with agricultural prices falling in between.     

A two-factor cointegrated commodity price model with an application to spread option pricing

In this paper, we propose an easy-to-use yet comprehensive model for a system of cointegrated commodity prices. While retaining the exponential affine structure of previous approaches, our model allows for an arbitrary number of cointegration relationships. We show that the cointegration component allows capturing well-known features of commodity prices, i.e., upward sloping (contango) and downward sloping (backwardation) term-structures, smaller volatilities for longer maturities and an upward sloping correlation term structure. The model is calibrated to futures price data of ten commodities. The results provide compelling evidence of cointegration in the data. Implications for the prices of futures and options written on common commodity spreads (e.g., spark spread and crack spread) are thoroughly investigated.     

Commodity price modelling that matches current observables: a new approach

We develop a stochastic model of the spot commodity price and the spot convenience yield such that the model matches the current term structure of forward and futures prices, the current term structure of forward and futures volatilities, and the inter-temporal pattern of the volatility of the forward and futures prices. We let the underlying commodity price be a geometric Brownian motion and we let the spot convenience yield have a mean-reverting structure. The flexibility of the model, which makes it possible to simultaneously achieve all these goals, comes from allowing the volatility of the spot commodity price, the speed of mean-reversion parameter, the mean-reversion parameter, and the diffusion parameter of the spot convenience yield all to be time-varying deterministic functions.     

Pricing Commodity Spread Options with Stochastic Term Structure of Convenience Yields and Interest Rates

The purpose of this research is to provide a valuation formula for commodity spread options. Commodity spread options are options written on the difference of the prices (spread) of two commodities. From the aspect of commodity contingent claims, it is considered that commodity spread options are difficult to evaluate with accuracy because of the existence of the convenience yield. Hence, the model of the convenience yield is the key factor to price commodity spread options. We use the concept of future convenience yields to develop the model that enriches the stochastic behavior of convenience yield. We also introduce Heath-Jarrow-Morton interest rate model to the valuation framework. This general model not only captures the mean reverting feature of the convenience yield, but also allows us to handle a very wide range of shape that the term structure of convenience yield can take. Therefore our model provides various types of models. The numerical analysis presented in this paper provides some unique features of commodity spread options in contrast to normal options. These characteristics have never been addressed in previous studies. Moreover, it suggests that the existing model overprice commodity spread options through neglecting the effect of interest rates.     

Equilibrium Forward Curves for Commodities

We develop an equilibrium model of the term structure of forward prices for storable commodities. As a consequence of a nonnegativity constraint on inventory, the spot commodity has an embedded timing option that is absent in forward contracts. This option's value changes over time due to both endogenous inventory and exogenous transitory shocks to supply and demand. Our model makes predictions about volatilities of forward prices at different horizons and shows how conditional violations of the 'Samuelson effect' occur. We extend the model to incorporate a permanent second factor and calibrate the model to crude oil futures data.     

The role of speculation in international futures markets on commodity prices

The role of futures contracts on spot prices has been one of the key focus areas of research since the recent surge in commodity prices and increase in the volatility of commodity returns. However, no consensus arises from this literature, and hence it is difficult to link the use of futures contracts in agricultural commodities by non-hedgers and the growing food insecurity within developing countries. The purpose of this paper is to highlight causal relationships from futures contracts to spot prices of underlying assets, namely agricultural commodities. As research that focus on exchange-traded funds do not provide any clear conclusions, we focus on the imbalance between short- and long-open positions, this imbalance being caused by the exchange traded funds’ participation in futures markets. In this paper, we estimate relationships between financial variables including indicators for speculation in futures markets and the returns of cocoa, corn, soybean, wheat, coffee, rice, and sugar on a weekly basis from 1998 to 2013. Significant results lead to Granger-causality tests that in turn validate the hypothesis of a positive impact of speculation in futures markets to returns on the underlying commodities.     

A continuous time equilibrium model of forward prices and futures prices in a multigood economy

This paper is a theoretical investigation of equilibrium forward and futures prices. We construct a rational expectations model in continuous time of a multigood, identical consumer economy with constant stochastic returns to scale production. Using this model we find three main results. First, we find formulas for equilibrium forward, futures, discount bond, commodity bond and commodity option prices. Second, we show that a futures price is actually a forward price for the delivery of a random number of units of a good; the random number is the return earned from continuous reinvestment in instantaneously riskless bonds until maturity of the futures contract. Third, we find and interpret conditions under which normal backwardation or contango is found in forward or futures prices; these conditions reflect the usefulness of forward and futures contracts as consumption hedges.     

A Model of Financialization of Commodities

We analyze how institutional investors entering commodity futures markets, referred to as the financialization of commodities, affect commodity prices. Institutional investors care about their performance relative to a commodity index. We find that all commodity futures prices, volatilities, and correlations go up with financialization, but more so for index futures than for nonindex futures. The equity‐commodity correlations also increase. We demonstrate how financial markets transmit shocks not only to futures prices but also to commodity spot prices and inventories. Spot prices go up with financialization, and shocks to any index commodity spill over to all storable commodity prices.     

The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging

In this article we compare three models of the stochastic behavior of commodity prices that take into account mean reversion, in terms of their ability to price existing futures contracts, and their implication with respect to the valuation of other financial and real assets. The first model is a simple one-factor model in which the logarithm of the spot price of the commodity is assumed to follow a mean reverting process. The second model takes into account a second stochastic factor, the convenience yield of the commodity, which is assumed to follow a mean reverting process. Finally, the third model also includes stochastic interest rates. The Kalman filter methodology is used to estimate the parameters of the three models for two commercial commodities, copper and oil, and one precious metal, gold. The analysis reveals strong mean reversion in the commercial commodity prices. Using the estimated parameters, we analyze the implications of the models for the term structure of futures prices and volatilities beyond the observed contracts, and for hedging contracts for future delivery. Finally, we analyze the implications of the models for capital budgeting decisions.     

Futures-Style Options on Euro-deposit Futures: Nihil sub Sole Novi?

Euro-deposit futures play a relevant role among the derivative products traded in official markets. As opposed to most futures contracts, the underlying instrument is not represented by a traded asset but by a linear transformation of an interest rate, the Libor. The options written on Euro-deposit futures that are traded at the London International Financial Futures & Options Exchange (LIFFE) are subject to daily marking to market, as the underlying futures; thus, they are called futures-style options or pure futures options. These options are often priced with the Black (1976) formula, whose use entails several shortcomings. A more realistic alternative is represented by the univariate Cox, Ingersoll and Ross (1985) model. The closed-form solutions for the prices of Euro-deposit futures and futures-style options on Euro-deposit futures obtained in the CIR model are two major original contributions presented in this paper. Other original contributions involve the determination of the relation between futures rates and forward rates and the derivation of the equivalent portfolio for the hedging of futures-style options on Euro-deposit futures.     

Effects of speculation and interest rates in a “carry trade” model of commodity prices

The paper presents and estimates a model of the prices of oil and other storable commodities, a model that can be characterized as reflecting the carry trade. It focuses on speculative factors, here defined as the trade-off between interest rates on the one hand and market participants' expectations of future price changes on the other hand. It goes beyond past research by bringing to bear new data sources: survey data to measure expectations of future changes in commodity prices and options data to measure perceptions of risk. Some evidence is found of a negative effect of interest rates on the demand for inventories and thereby on commodity prices and positive effects of expected future price gains on inventory demand and thereby on today's commodity prices.     

Macroeconomic impacts on commodity prices: China vs. the United States

This study compares the macroeconomic impacts of China and the United States on international commodity markets using a factor-augmented vector auto-regression (FAVAR) model with latent factors extracted from a rich data set that includes various macroeconomic and financial indicators at monthly frequency. The main results suggest that whether or not the Chinese demand cause commodity prices to soar depends. Macroeconomic factors of China do have significant impact on commodity markets, but the impacts of the United States outperform those of China in terms of the size of coefficients and their level of significance, as well as the direction and magnitude of directional return spillovers. Moreover, the effects of these factors on individual commodity futures are not a universal phenomenon. Therefore, there is no systematic evidence of a relationship between strong growth in the emerging economy and the boom in commodity futures prices, either statistically or economically.     

Price dynamics in refined petroleum spot and futures markets

This article characterizes the spot and futures price dynamics of two important physical commodities, gasoline and heating oil. Using a non-linear error correction model with time-varying volatility, we demonstrate many new results. Specifically, the convergence of spot and futures prices is asymmetric, non-linear, and volatility inducing. Moreover, spreads between spot and futures prices explain virtually all spot return volatility innovations for these two commodities, and spot returns are more volatile when spot prices exceed futures prices than when the reverse is true. Furthermore, there are volatility spillovers from futures to spot markets (but not the reverse), futures volatility shocks are more persistent than spot volatility shocks, and the convergence of spot and futures prices is asymmetric and non-linear. These results have important implications. In particular, since the theory of storage implies that spreasd vary with fundamental supply and demand factors, the strong relation between spreads and volatility suggests that these fundamentals — rather than trading induced noise — are the primary determinants of spot price volatility. The volatility spillovers, differences in volatility persistence, and lead-lag relations are consistent with the view that the futures market is the primary locus of informed trading in refined petroleum product markets. Finally, our finding that error correction processes may be non-linear, asymmetric, and volatility inducing suggests that traditional approaches to the study of time series dynamics of variables that follow a common stochastic trend that ignore these complexities may be mis-specified.     

Valuation of commodity derivatives in a new multi-factor model

This paper extends existing commodity valuation models to allow for stochastic volatility and simultaneous jumps in the spot price and spot volatility. Closed-form valuation formulas for forwards, futures, futures options, geometric Asian options and commodity-linked bonds are obtained using the Heston (1993) and Bakshi and Madan (2000) methodology. Stochastic volatility and jumps do not affect the futures price at a given point in time. However, numerical examples indicate that they play important roles in pricing options on futures. This revised version was published online in June 2006 with corrections to the Cover Date.     

Efficient Analytic Approximation of American Option Values

This paper provides simple, analytic approximations for pricing exchange-traded American call and put options written on commodities and commodity futures contracts. These approximations are accurate and considerably more computationally efficient than finite-difference, binomial, or compound-option pricing methods.