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Floating-strike lookback option

A floating-strike lookback call uses the monitored minimum as its strike; a put uses the maximum. Trace the cash payoff and see why monitoring dates matter.

A floating-strike lookback option is a lookback option whose strike is set retrospectively by an eligible price extremum: the monitored minimum for a call, or the monitored maximum for a put. The other side of the comparison is the terminal underlying price. A call therefore captures recovery from the recorded low; a put captures retreat from the recorded high, even when the holder did not trade at that turning point.

The terminal spread ledger

Consider two separate European, cash-settled contracts, each covering exactly one underlying unit and paying only at expiry. The following synthetic prices are in USD per unit. All four dates, including start and expiry, count as contractual observations.
Observation dateRoleUnderlying price
26 October 2026Start100
26 February 2027Intermediate observation80
26 June 2027Intermediate observation120
26 October 2027Expiry TT110

The call's eventual strike is 80; the put's is 120. Those are prices selected by the contract's rule, rather than a strike agreed numerically in advance.

Four monitored prices 100, 80, 120 and 110 USD per unit produce a minimum of 80 and maximum of 120. Floating call cash payoff is 110 minus 80 equals USD 30; floating put is 120 minus 110 equals USD 10. Both pay at expiry for exactly one underlying unit.
Four monitored prices 100, 80, 120 and 110 USD per unit produce a minimum of 80 and maximum of 120. Floating call cash payoff is 110 minus 80 equals USD 30; floating put is 120 minus 110 equals USD 10. Both pay at expiry for exactly one underlying unit.
Original synthetic example. Filled dots are contractual observations, equally spaced by observation order; connectors do not specify between-date prices. The undiscounted cash payments at expiry are USD 30 for the call and USD 10 for the put, each for one underlying unit, before premium.

The attractive phrase “buy at the low” describes the call's payoff calculation. In these cash-settled examples it is not an instruction to buy stock at an earlier date, an early-exercise decision, or permission to choose an arbitrary quantity after seeing the path.

Why there is no positive-part sign

Let D\mathcal{D} be the contractual observation set, with expiry T∈DT\in\mathcal{D}. Define the monitored minimum and maximum by

mD=min⁡t∈DSt,MD=max⁡t∈DSt.m_{\mathcal{D}}=\min_{t\in\mathcal{D}}S_t, \qquad M_{\mathcal{D}}=\max_{t\in\mathcal{D}}S_t.
The terminal payoffs, in cash for one underlying unit, are
CT=ST−mD,PT=MD−ST.C_T=S_T-m_{\mathcal{D}}, \qquad P_T=M_{\mathcal{D}}-S_T.

Because expiry counts, mD≤ST≤MDm_{\mathcal{D}}\le S_T\le M_{\mathcal{D}}. Both amounts are already nonnegative: wrapping them in max⁡(⋅,0)\max(\cdot,0) changes nothing. This reasoning relies on the stated observation window. A variant whose extrema window ends before expiry needs its own payoff terms.

“Floating” also does not mean the payoff is guaranteed to be positive. A call pays zero when expiry is a monitored minimum; a put pays zero when expiry is a monitored maximum. Both pay zero on a constant path.

As a useful ledger check, buying one of each with the same monitoring and settlement terms gives combined terminal cash

CT+PT=MD−mD.C_T+P_T=M_{\mathcal{D}}-m_{\mathcal{D}}.

The two terminal-price terms cancel. In the example, USD 30 plus USD 10 equals the monitored range of USD 40. This is a sum of two contractual payoffs, before either premium; it does not state a trading profit.

The schedule changes the strike

If only the start and expiry dates counted on the same supplied path, the minimum would be 100 and the maximum 110. The floating call would pay USD 10 and the put USD 0. The intermediate dip and peak would have no contractual effect.

For nested observation sets with the same terminal price, adding observations can only lower the minimum or raise the maximum. It therefore cannot reduce either floating lookback payoff. Continuous monitoring takes extrema over the entire monitoring interval; those may lie beyond any finite set of sampled levels. The four observations above alone cannot reveal the continuous minimum or maximum.

This distinction matters in Monte Carlo valuation. Sampling a continuous contract only on a grid misses possible lows and highs, tending to understate both floating payoffs. That is a monitoring effect, separate from random sampling error or an approximation to the price process. A discretely monitored contract, by contrast, should use its specified observation dates. Broadie, Glasserman and Kou distinguish the contractual schedule from a numerical valuation grid.

From payoff to valuation

A fixed-strike lookback option makes a different comparison: its call compares the monitored maximum with an advance strike, and its put compares that strike with the monitored minimum. Swapping only the word “fixed” for “floating” reverses which extremum belongs to each call or put.

To value an outstanding floating call, the modeller needs its minimum recorded so far as well as current spot and the remaining contract terms; a floating put needs its recorded maximum. The past extremum remains relevant even after the price moves away from it. A valuation model must then account for future prices, future eligible extrema and discounting. None of the cash amounts above is today's option price.

The owned lesson Lookback Options — Extrema Payoffs and Reflection develops the payoff taxonomy and running-extremum state variables. Its public page is a preview; the full lesson requires membership.

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