Options Under a Defended Currency Floor

When a central bank defends a currency floor, the exchange rate does not bounce off it: it sits on it. We model the log distance to the floor as a diffusion that is sticky at zero, pushed up at rate $\theta$ while it sits there, and let the bank abandon the floor at hazard $\kappa$ per unit of floor time, the rate jumping by $e^{J}$. The discounted rate is a martingale exactly when the drift above the floor is the carry less half the variance and $\kappa(1 – e^{J}) = \theta – q$, with $q$ the interest differential. Because $\theta$ is a path functional, local time over twice the floor time, it is the same under every equivalent measure, and the integrated break hazard equals the floor’s push net of carry divided by the relative jump. Calls, puts and digitals are explicit Laplace transforms; they agree with finite differences to $1.2 \times 10^{-5}$ and with Monte Carlo within $2.83$ standard errors, and the floor puts a steep downside skew on options. On $858$ ECB fixings of the EUR/CHF floor of 2011 to 2015 the stickiness is weakly identified, $0.20$ to $1.71$ per year across bands, yet the risk-neutral twelve-month probability of a rate below $1.20$ from the floor stays between $16.9$ and $18.7$ per cent with the realized jump to $1.028$. With the realignment to $1.15$ that Hertrich and Zimmermann describe, the same path gives $35.6$ per cent, inside their option-implied range: the path prices the product of break probability and jump size, not either factor alone.