Understanding Lookback Option Gamma Through Knock-Out Calls
Summary
The discussion makes the gamma behavior of a fixed-strike maximum lookback call easier to visualize by approximating its payoff with a strip of knock-out calls. Each option in the strip has a different barrier, and a rebate is paid when the underlying reaches that barrier. As the asset establishes new highs, additional calls are knocked out and their rebates accumulate, approximating the lookback payoff.
The explanation then links the strip to gamma: a live knock-out call can have positive gamma, while an option that has already knocked out no longer contributes gamma. Options nearer their relevant strike or barrier tend to have more gamma. After the underlying rises and knocks out several calls, a subsequent decline leaves fewer live options in the strip and therefore less aggregate gamma. This is a conceptual approximation, not a full pricing derivation; it does not quantify gamma over spot and time or establish continuity of the resulting profile.
Key ideas
- A fixed-strike maximum lookback call can be approximated by a strip of knock-out calls with rebates.
- As the underlying makes new highs, more calls in the strip knock out and rebates accumulate.
- A live knock-out option can contribute positive gamma, while a knocked-out option contributes none.
- After an advance knocks out options, a later decline can leave the remaining strip with less gamma.
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Full text
# Gamma of a Lookback Option # Gamma of a Lookback Option From this book, http://docs.finance.free.fr/Options/Exotic_Options_Trading.pdf, it states that > The gamma profile of a Max lookback option becomes intuitive when viewing it as a ladder option. Indeed, as long as the stock goes up there will be gamma on the lookback option and the gamma will decrease quickly when the stock goes down, as the options below have already knocked out and therefore have no gamma on them any more. I just cannot see how this is intuitive nor why there will be gamma on the lookback option as long as the stock goes up. What I am missing here? I cant picture how this gamma would change with spot and time, moreover I could not be sure through this explanation above the curves would even be continuous. ## Answer by mbison (score 2, accepted) https://quant.stackexchange.com/a/33431 In the book of De Weert he approximates the price with a strip of knock-outs. For example the lookback call with fixed strike pays the (max(S) - K)+, is approximated by a strip of knockout calls with a rebate. So whenever the stock sets a new high, another call knocks out and you receive your rebate. In his example the rebates are 1 cent apart. So if the stock moves from $\$45$ to $\$50$, you will have collected $\$5$ in rebates. Anyways: - do you agree that a knock-out call while it is a live is long gamma? - do you agree that a barrier option that is knocked out has no gamma? - do you agree that a barrier option has more gamma when spot is closer to the strike? Assuming you agree with the 3 bullets above. you can see that once the stock goes up and knocks out a few barrier options. and then spot drops, the strip of barriers will have less gamma.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.