A simple derivation of Kirk’s approximation for spread options

Ever since Kirk proposed an approximate price formula for a European call spread option in 1995, Kirk’s approximation has become the most widely used among the practitioners, especially in the energy markets. It is well known that Kirk’s approximation extends from Margrabe’s exchange option formula but no explicit derivation is available or has ever been published. In this paper we apply the idea of WKB method to provide a simple derivation of Kirk’s approximation and discuss its validity.