Step options. (Q2757294)

Step options are a variant of standard barrier options; their payoff is of the form \(f(\tau_B^\pm) (S_T-K)^+\) or \(f(\tau_B^\pm) (K-S_T)^+\) where \(S\) denotes the underlying stock price process, \(T\) is maturity, \(K\) is the strike price and \(\tau_B^\pm\) is the time in \([0,T]\) spent by \(S\) above/below the barrier level \(B\). This paper derives in the Black-Scholes model of geometric Brownian motion closed-form expressions for prices and hedge ratios for a number of such options. The key result is an explicit expression for the transition density of Brownian motion started at \(x\) and killed at rate \(\varrho\) below zero.