Time integrals under the Black-Scholes-Merton and Margrabe economies

Dias, Jose Carlos and Shackleton, Mark and da Silva, Fernando and Wojakowski, Rafal (2026) Time integrals under the Black-Scholes-Merton and Margrabe economies. Journal of Futures Markets. ISSN 0270-7314 (In Press)

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Abstract

The problem of integrating the Black and Scholes (1973) and Merton (1973) (BSM) formula with respect to the time variable is paramount for an economist. Inspired by the real options literature, Shackleton and Wojakowski (2007) offer analytic formulae for valuing f inite maturity (profit) caps and floors that are contingent on continuous flows following a lognormal distribution. Alternative, but equivalent, closed-form solutions have been recently proposed in Dias et al. (2024b) by solving the time integral of options using a direct approach that does not rely on the real options intuition. This paper further extends and simplifies the computation of time integrals under the BSM world, considering not only plain-vanilla but also several exotic, including path-dependent options. We also provide a new closed-form solution of the time integral under the Margrabe (1978) economy. The method proposed in this paper makes the evaluation easier, cements the “non-real options” route and opens the way for more analytical work in BSM, Margrabe and other areas.