Show that, for constant drift and volatility , and zero discount rate, that the price of

Show that, for constant drift μ and volatility σ, and zero discount rate, that the price of an up and in one-touch option, barrier height B ≥ S, maturity T is given by:

where

(Use (3.17) for the joint density of the terminal spot for a constant drift Brownian motion and its maximum over the interval of the trade.)
Show that the vega of this option is given by:

where

Transcribed Image Text:

(2/02-1) B 1 - N(-d2) + N(d') S