Joint distribution of Brownian motion and its maximum, with a generalization to correlated BM and applications to barrier options
Let \(W_t\) be a standard one-dimensional Brownian motion, \(B_t = \sigma W_t + \mu t\) a Brownian motion with drift, \(M_s = \max_{0 \leq u \leq s} B_u\). The author derives the joint distribution of \((M_s, B_t)\) for all \(s\). The explicit formula for \(P (M_s \geq a\), \(B_t \leq z)\), \(s \leq t\), is used to study the joint distribution of \((M_s^{(1)}, B_t^{(2)})\) for the so-called ``correlated Brownian motion \((B_t^{(1)}, B_t^{(2)})\) in the plane'' defined as \[ B_t^{(1)} = \sigma_1 W_t^{(1)} + \mu_1 t, \quad B_t^{(2)} = \sigma_2 (\rho W_t^{(1)} + \sqrt {1 - \rho^2} W_t^{(2)}) + \mu_2 t, \] where \(W_t^{(1)}, W_t^{(2)}\) are independent standard Brownian motions, \(M_s^{(1)} = \sup_{0\leq u\leq s}B_u^{(1)}\). These distributions are applied to study the price barrier options. Some numerical examples are given.
scientific article; zbMATH DE number 896113