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Abstract
Another encounter with a three-legged spread option may occur when the strike of the standard two-legged option is denominated in a currency different from the currency of the spread leg prices. One can easily come up with an example of a crack option in which both crude and refined products are USD denominated (the most common situation) and the strike - that is, variable costs - are denominated in some local currency. In this case, the first two prices in the resulting three-legged spread option are the commodity prices, while the third is the foreign exchange rate. Alternatively, the [Pearson] algorithm computes the two-dimensional integral representing the expected value of the spread option payoff by integrating that payoff analytically along the first dimension and numerically along the second dimension. Among the approximate formulas, the most widely used is Kirk's approximation, which is described in the next section.
As discussed in Carmona 6c Durleman's in-depth survey and in the note by Bjerksund oc Stensland (2006), this formula is remarkably accurate for a formula of its size. In this paper, we will present two formulas that are equally efficient in the case of three assets. The first, based on a technique introduced by one of the authors of this paper (Alòs, 2011), called the decomposition method, is a direct and natural generalisation of [Kirk]'s formula. It has the distinct advantage of providing a well-defined systematic technique for deriving such approximations and sharper versions thereof. Kirk's original formula in the two-asset case is recovered as a special case. The second formula is based on the Laplace approximation technique and is related to the steepest ascent technique introduced for index options by [Avellaneda M] et al '(2002). The index options considered there are basket options with positive weights, whereas in the case considered here, two of the weights are negative. In the lognormal setting, as we will see, certain additional simplifications (near-the-money, doubleslope relation between implied and local volatility) used in Avellaneda et al (2002) can be circumvented and a highly accurate formula can be found, which is in closed form. The second formula is longer than the first but is explicit, so both computational times are instantaneous. As is the case for the first formula, the formula presented here is one in a hierarchy of successively more accurate approximations.
The tolling contracts do not belong exclusively to the world of power. In fact, the word 'tolling' comes from the oil industry, where similar arrangements allow the toller to pay a certain fee to have the right to use refinery (supply crude and receive refined products). Viewing the tolling agreement as a call option with a fuel and emission-linked strike price allows us to write the payoff of this option at expiration as: Π^sub tolling^ = max(P^sub power^ - P^sub fuel^ - P^sub emission^ - VC, 0),
Now the strategy of the rest of the proof is to try to asymptotically match the price of spread option to that of a Bachelier model with the same strike and an appropriate normal volatility, which will correspond to the implied normal volatility for the spread option. The reason, as mentioned earlier, it is more appropriate to take a normal implied volatility as opposed to a standard Black-Scholes (lognormal) volatility, is that the spread itself can of course be negative and lognormal variables on the other hand are always positive. We have already defined the Bachelier price above. Using again the asymptotics for the error function for large arguments, we find that the asymptotic value in the Bachelier model with the spot price equal to the spread spot value F1-F2- F3 is hence given by an expression