José Carlos Dias

The problem of integrating the Black, Scholes, and Merton (BSM) formula with respect to the time variable is paramount for an economist. Inspired by the real options literature, Shackleton and Wojakowski offer analytic formulae for valuing finite 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. 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 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.

We offer analytic formulae for valuing finite maturity profit caps and floors that are contingent on continuous flows without the need for subtracting the risk‐neutral expectation of the forward starting perpetual solution from the corresponding perpetual solution. The related price caps, floors, and collars are easily obtained from any analytic representation of profit caps and floors using some arbitrage‐free relations. Finally, we offer two novel methods for calculating the optimal triggers of investment projects in the presence of price floors and collars regimes in a way that is much simpler than the ones currently used.

This article derives simple closed-form solutions for computing Greeks of zero-coupon and coupon-bearing bond options under the CIR interest rate model, which are shown to be accurate, easy to implement, and computationally highly efficient. These novel analytical solutions allow us to extend the literature in two other directions. First, the static hedging portfolio approach is used for pricing and hedging American-style plain-vanilla zero-coupon bond options under the CIR model. Second, we derive analytically the comparative static properties of sinking-fund bonds under the same interest rate modeling setup.