Digital barrier option contract with exponential random time

Barrier options are a widely used class of path-dependent derivative securities. These options either cease to exist or come into existence when some pre-specified asset price barrier is hit during the option’s life. Merton (1973) has derived a down-and-out call price by solving the corresponding partial differential equation with some boundary conditions. Reiner & Rubinstein (1991) published closed-form pricing formulas for various types of single barrier options. Rich (1997) also provided a mathematical framework to value barrier options. Moreover, Kunitomo & Ikeda (1992) derived a pricing formula for double barrier options with curved boundaries as the sum of an infinite series. Geman & Yor (1996) followed a probabilistic approach to derive the Laplace transform of the double barrier option price. Pelsser (2000) used contour integration for Laplace transform inversion to price new types of double barrier options. In these papers, the underlying asset price is monitored for barrier hits or crossings during the entire life of the option. Heynen & Kat (1994) studied partial barrier options where the underlying price is monitored during only part of the option’s lifetime. Partial barrier options have two classes. One is forward starting barrier options where the barrier appears at a fixed date strictly after the option’s initial starting date. The other is early ending barrier options where the barrier disappears at a specified date strictly before the expiry date. They can be applied for various types of options according to the clients’ needs as controlling the starting or ending time of the monitoring period. Also, they can be used as components to synthetically create other types of exotic options. Digital barrier options (cash-or-nothing barrier options) of knock-in type, a part of standard barrier options, pay out a fixed amount of money if the underlying asset price crosses a given barrier and arrives above a certain value at expiry date. For digital barrier options of knock-out type, the payoff is a fixed amount if the underlying asset price never hits a given barrier and is above a certain level at expiry.