Lookback Options

Imagine being allowed to buy a stock at the lowest price it touched at any point over the past year, decided with perfect hindsight after the fact. Nobody can actually time a market that well, but a lookback option manufactures exactly that outcome contractually. Where a plain option's payoff depends only on where the underlying ends up at expiry, a lookback option's payoff depends on the entire path it traveled — specifically, on the running minimum or running maximum recorded along the way.

The payoff

A floating-strike lookback call pays the difference between the final price and the lowest price seen during the option's life:

$$\text{Payoff} = S_T - \min_{0 \le t \le T} S_t.$$

In words: $S_T$ is the underlying's price at expiry, and $\min_{0 \le t \le T} S_t$ is the lowest price it touched at any moment from inception to expiry — its running minimum. The holder effectively buys at the best (lowest) price the underlying ever offered and sells at the final price. Because the running minimum can only ever be less than or equal to every price the underlying reaches — including $S_T$ itself — this payoff is always zero or positive; it can never expire worthless in the sense of paying nothing when the stock rose, since the minimum is locked in the past.

Worked example 1 — a clean path

A stock starts at $100. Over three months it moves: $100 → $85 → $110 → $95 (final). The running minimum recorded along that path is $85. A floating-strike lookback call pays $S_T - \min S_t = 95 - 85 =$ $10. Compare that to an ordinary at-the-money call struck at $100: it pays $\max(95 - 100, 0) =$ $0, because the stock ended below where it started. The lookback holder profits precisely because the path dipped to $85 before recovering to $95 — the ordinary call holder gets nothing from that same dip-and-recover shape.

Worked example 2 — a fixed-strike variant

A fixed-strike lookback call, struck at $90, pays $\max(\max_{0\le t\le T} S_t - K,\ 0)$ — the best price the underlying reached, minus a strike fixed today, floored at zero. Using a path $90 → $120 → $105 → $115 (final), the running maximum is $120. Payoff $= \max(120 - 90, 0) =$ $30, even though the underlying closed at $115, ten points below its peak. An ordinary call struck at $90 pays $\max(115 - 90, 0) =$ $25 — five points less, because it can only ever use the closing price, missing the extra value the fixed-strike lookback captured by locking in the path's high point.

Watch a simulated price path above and imagine tracking its running high and low as it unfolds — that running extremum, not the final dot on the chart, is what a lookback option's payoff is built from. Regenerate a few paths and notice how a choppier path with a deep early dip or a sharp early spike can make the lookback payoff much larger than the ordinary option payoff on the same path.

What this means in practice

Lookbacks are priced almost exclusively by Monte Carlo simulation or specialized closed-form formulas (a Black-Scholes-style formula exists for the floating-strike case under lognormal assumptions), because the payoff depends on the entire simulated path's extremum, not just its endpoint. They trade rarely as standalone contracts — the premium for perfect hindsight is expensive — but the running-extremum logic reappears inside barrier options, cliquets, and path-dependent structured notes, so understanding it here pays off elsewhere.