Closed Form Spread Option Valuation

Abstract

This paper considers the valuation of a spread call when asset prices are lognormal. The implicit strategy of the Kirk formula is to exercise if the price of the long asset exceeds a given power function of the price of the short asset. We derive a formula for the spread call value, conditional on following this feasible but non-optimal exercise strategy. Numerical investigations indicate that the lower bound produced by our formula is extremely accurate. The precision is much higher than the Kirk formula. Moreover, optimizing with respect to the strategy parameters (which corresponds to the Carmona-Durrleman procedure) yields only a marginal improvement of accuracy (if any).Keywords: Spread option, closed form, valuation formula, lognormal asset prices.JEL code: G12, G13, D81, C63. Closed form spread option valuation AbstractThis paper considers the valuation of a spread when asset prices are lognormal. The implicit strategy of the Kirk formula is to exercise if the price of the long asset exceeds a given power function of the price of the short asset. We derive a formula for the spread call value, conditional on following this feasible but non-optimal exercise strategy. Numerical investigations indicate that the lower bound produced by our formula is extremely accurate. The precision is much higher than the Kirk formula. Moreover, optimizing with respect to the strategy parameters (which corresponds to the Carmona-Durrleman procedure) yields only a marginal improvement of accuracy (if any).

A general closed-form spread option pricing formula

“…Again, in practice, this is not necessary because the educated guesses proposed by Bjerksund and Stensland (2011) for the log-normal setting and generalized here to the non-Gaussian case, expressions (7) and (8), turn out to be very effective.…”

“…We now present our lower bound, extending to a non Gaussian framework the idea of Bjerksund and Stensland (2011). Let us define the event A…”

“…The main contribution of the present work is the derivation of a lower bound, as in Bjerksund and Stensland (2011), but for general processes. The only quantity we need to know explicitly is the joint characteristic function of the log-returns of the two assets.…”

“…Section 2 generalizes the lower bound of Bjerksund and Stensland (2011) to non-Gaussian models. Section 3 compares the lower bound with the method of Hurd and Zhou (2010).…”

A Two-Factor Cointegrated Commodity Price Model with an Application to Spread Option Pricing

“…Alternative approximation formulae can be found in Carmona and Durrleman (2003) and Bjerksund and Stensland (2011). These formulae typically improve upon Kirk's formula in terms of numerical accuracy and efficiency, but they come at the cost of higher computational complexity.…”