Finite maturity caps and floors on continuous flows

Models of interest rate caps and floors are typically based on discrete rates over finite horizons while existing real option models describe perpetual claims on the maximum of two continuous flows. In this paper, we produce formulae for finite maturity caps and floors that are contingent on continuous flows. We present hedge ratios and discuss applications where a lognormally distributed flow variable is suitable. For other situations where practitioners use proprietary models, the formula presented is useful as a quick, tractable and universal means for mapping quoted implied to prices and vice versa.