ارزیابی گزینه گسترش بلند مدت و مصون سازی

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.

Commodity spreads are important for both investors and manufacturers. For example, the price spread between heating oil and crude oil (crack spread) represents the value of production (including profit) for a refinery firm. If an oil refinery in Singapore can deliver its oil both to the US and the UK, then it possesses a real option of diversion which directly relates to the spread of WTI and Brent crude oil prices. There are four commonly used spreads: spreads between prices of the same commodity at two different locations (location spreads) or times (calendar spreads), between the prices of inputs and outputs (production spreads) or between the prices of different grades of the same commodity (quality spreads). 1 A spread option is an option written on the difference (spread) of two underlying asset prices S1 and S2, respectively. We consider European options with payoff the greater or lesser of S2(T)–S1(T)–K and 0 at maturity T for strike price K and focus on spreads in the commodity (especially energy) markets (for both spot and futures). In pricing spread options it is natural to model the spread by modelling each asset price separately. Margrabe (1978) was the first to treat spread options and gave an analytical solution for strike price zero (the exchange option). Closed form valuation of a spread option is not available if the two underlying prices follow geometric Brownian motions (see Eydeland and Geman, 1998). Hence various numerical techniques have been proposed to price spread options, such as for example the Dempster and Hong (2000) fast Fourier transformation approach. Carmona and Durrleman (2003) offer a good review of spread option pricing. Many researchers have modelled the spread using two underlying commodity spot prices (the two price method) in the unique risk neutral measure as where S1 and S2 are the spot prices of the commodities and δ1 and δ2 are their convenience yields, and W1,1, W1,2, W2,1 and W2,2 are four correlated Wiener processes. This is the classical Gibson and Schwartz (1990) model for each commodity price in a complete market. 3 The return correlation ρ13 := E[dW1,1 dW2,1]/dt plays a substantial role in valuing a spread option; trading a spread option is equivalent to trading the correlation between the two asset returns. However, Kirk, 1995, Mbanefo, 1997 and Alexander, 1999 have suggested that return correlation is very volatile in energy markets. Thus assuming a constant correlation in (1) is inappropriate. But there is another longer term relationship between two asset prices, termed cointegration, which has been little studied by asset pricing researchers. If a cointegration relationship exists between two asset prices the spread should be modelled directly over the long term horizon. Soronow and Morgan (2002) proposed a one factor mean reverting process to model the location spread directly, but do not explain under what conditions this is valid nor derive any results. 4 See also Geman (2005a) where diffusion models for various types of spread option are discussed. In this paper, we use two factors to model the spot spread process and fit the futures spread term structure. Our main contributions are threefold. First, we give the first statement of the economic rationale for mean reversion of the spread process and support it statistically using standard cointegration tests on data. Second, the paper contains the first test of mean-reversion of latent spot spreads in both the risk neutral and market measures. Third, we give the first latent multi-factor model of the spread term structure which is calibrated using standard state-space techniques, i.e. Kalman filtering. The paper is organized as follows. Section 2 gives a brief review of price cointegration together with the principal statistical tests for cointegration and the mean reversion of spreads. Section 3 proposes the two factor model for the underlying spot spread process and shows how to calibrate it. Section 4 presents option pricing and hedging formulae for options on spot and futures spreads. Sections 5 and 6 provide two examples in energy markets which illustrate the theoretical work and Section 7 concludes.

In this paper we have developed spread option pricing models for the situation when the two underlying prices of the spread are cointegrated. Since the cointegration relationship is important for the long run dynamical relationship between the two prices, contingent claim evaluation based on spreads should take account of this relationship for long maturities. We model the spread process directly using a two factor model, i.e. we model directly the dynamic deviation from the long run equilibrium which cannot be specified correctly by modelling the two underlying assets separately. We also propose two methods (risk neutral and market) for testing data for mean-reversion of the spread process. The corresponding spread option can be priced and hedged analytically. In order to illustrate the theory we study two examples – of crack and location spreads, respectively. Both spread processes are found to be mean reverting. From likelihood ratio tests the second y factor is found to be important in explaining the crack and location spread data. The option values from our model are quite different from those of standard models, but they are consistent with the practical observations of Mbanefo (1997).