Lecture 8: the Cameron–martin Formula and Barrier Options

Thus far in our study of continuous-time markets, we have considered only very simple derivative securities, the European contingent claims (contingent claims whose payoffs are functions only of the terminal share price of the underlying asset). In this lecture, we shall study several exotic options – the knockins/knockouts and barrier options – whose payoffs depend on the entire history of the share price up to termination. These options are “activated” (or, in some cases, “deactivated”) when the share price of the underlying asset reaches a certain threshold value. If, as in the simple Black-Scholes model, the share price process behaves as a geometric Brownian motion under the risk-neutral measure, then the time at which the option is activated is the first-passage time of the driving Brownian motion to a linear boundary. Thus, it should be no surprise that the exponential martingales of Lecture 5 play a central role in the pricing and hedging of barrier and knockin/knockout options. The use of these martingales is greatly facilitated by the Cameron– Martin theorem, a precursor to the Girsanov theorem, which will be discussed in a subsequent lecture.