Using the result that the expected maximum of a driftless Brownian path over [0,T] equals sigma*sqrt(2T/pi), compare a one-year lookback call struck at the start level against the ordinary at-the-money call, for a $100 stock at 20% volatility.

Using the result that the expected maximum of a driftless Brownian path over [0,T] equals sigma*sqrt(2T/pi), compare a one-year lookback call struck at the start level against the ordinary at-the-money call, for a $100 stock at 20% volatility.

Approach: Value the lookback through the expected running maximum in log terms and the vanilla through the at-the-money approximation, then take the ratio of the two.

About twice. The expected running maximum of a driftless path is sigma*sqrt(2T/pi) = 0.20*sqrt(2/3.1416) = 0.20*0.7979 = 0.1596, so the lookback call that pays the maximum minus the start level is worth about 15.96 on a $100 stock. The vanilla at-the-money call is 0.4*sigma*sqrt(T)*S = 7.98, so the lookback is 2.0 times the vanilla, which follows directly from sqrt(2/pi) being twice 1/sqrt(2*pi). The gap is the value of path dependence: the lookback pays on the best level the stock ever reached rather than where it happened to close, and it therefore has no gamma-free region and cannot be hedged with a static option position. A dealer short one is short every intermediate high, so hedging error accumulates with the number of new maxima rather than with the terminal price.

Follow-up: How does discrete monitoring, say daily rather than continuous, change the price and in which direction?

Key concepts: lookback option, running maximum, at-the-money approximation, path dependence.