An Asian option softens the payoff by averaging the path. A lookback option does the opposite: it rewards the holder with the single best price the path ever reached. A lookback call pays the terminal price minus the lowest price seen along the way — as if you had somehow known, in advance, exactly which day to buy. A lookback put pays the highest price the path ever reached, minus the terminal price — as if you had known exactly which day to sell.

Nobody actually gets to time the market that well. A lookback option is the closest thing finance offers to selling that impossible skill as a product — and, unsurprisingly, it is priced accordingly.

Floating vs fixed lookbacks

As with Asian options, there are two families. The more famous one is the floating-strike lookback, which has no strike at all — the "strike" is whatever extreme the path itself produced:

A floating lookback call effectively lets the holder buy at the path's minimum and sell at the terminal price; a floating lookback put lets them sell at the path's maximum. Both quantities are non-negative by construction — the minimum can never exceed S_T, and the maximum can never fall below it — so a floating lookback never expires worthless, unlike almost everything else in this module.

A fixed-strike lookback keeps an ordinary strike K, but replaces the terminal price with the path's best extreme:

This version can expire worthless — if the price never even reaches the strike — but it is still strictly better than the matching vanilla option, since it locks in the best price reached, not the (generally worse) terminal price.

Worked example: the best entry price in hindsight

Suppose a stock starts at 100, dips to 82 midway through the year on a scare, recovers, and finishes at 118. A vanilla call struck at 100 pays 118 - 100 = 18. A floating-strike lookback call pays 118 - 82 = 36 — exactly double, because it silently swapped the strike for the lowest price the stock ever touched, with no skill or luck required from the holder. The chart below traces one such path together with its running minimum — the "strike" a floating lookback call effectively locks in, updated continuously as the path makes new lows.

Notice the running minimum only ever falls or stays flat — it can never rise, since a running minimum is by definition never beaten by a later, higher price. That one-way ratchet is exactly what makes the floating lookback call so valuable: every fresh low along the way permanently improves the payoff, and nothing the path does afterwards can undo it.

Why lookbacks are the priciest exotic in this module

Compare the three path-dependent contracts side by side. A barrier option is cheaper than the vanilla it modifies, because it removes value in some scenarios. An Asian option is cheaper still relative to a comparable notional exposure, because averaging shrinks the variance of the payoff variable. A lookback runs in the opposite direction on both counts: it never removes value, and it actively picks out the single most favourable moment on every path, every time. Selling that guarantee — "you will always get the best possible price, no forecasting required" — is expensive precisely because the seller is on the hook for the market's single best (for the buyer) moment on every path, with certainty, rather than merely offering a chance at a good outcome.