Gamma of a Lookback Option

Question

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 1

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.