Asymptotic behaviour of random maturity barrier options

In this paper, we study and analyse European barrier options in the Black‐Scholes framework with deterministic or random maturity. When the maturity is deterministic, we provide closed form formulas for (up and out) European barrier call and put options. When the maturity is random, we suppose that it follows a Poisson distribution. In this case, we provide an explicit expression for the European barrier call option, presented in the form of a numerical series. In order to make our results more explicit, we give a Taylor expansion's formula for the random maturity European barrier call option, as the volatility decreases to zero. Since there are no, or at least, it is hard to obtain closed‐form formulas for random maturity barrier contingent claims, the last formula should be very useful both in practice and theory of option pricing.