A stock is at 100 and its log price is modelled as driftless Brownian motion with 20% annual volatility. Price a one-year one-touch that pays $1 if the stock ever trades at 110, with zero rates, and say why the answer is exactly twice something.

A stock is at 100 and its log price is modelled as driftless Brownian motion with 20% annual volatility. Price a one-year one-touch that pays $1 if the stock ever trades at 110, with zero rates, and say why the answer is exactly twice something.

Approach: Use the reflection principle for driftless Brownian motion to relate the distribution of the running maximum to the distribution of the terminal value.

0.634. In log space the barrier sits at ln(1.10) = 0.09531 and the one-year standard deviation is 0.20, so the terminal value exceeds the barrier with probability N(-0.09531/0.20) = N(-0.4766) = 0.3168. The reflection principle says every path that finishes below the barrier after touching it is matched one for one with a path that finishes above it, so P(running maximum >= b) = 2*P(X_T >= b), giving 0.6336. The one-touch is therefore worth exactly twice the corresponding digital, which is the cleanest statement of why a barrier option is so much more valuable than its terminal-payoff cousin. The doubling is exact only with zero drift: the real lognormal drift of -sigma^2/2 in log space makes the up-barrier strictly less likely and the factor falls below 2.

Follow-up: How does the price change once you keep the -sigma^2/2 drift in log space, and in which direction?

Key concepts: reflection principle, running maximum, one-touch, barrier option.