# Is Lookback option more path-dependent than an Asian option

Lookback option:

• Path dependency comes from taken the extremum over the whole trajectory.
• It is equivalent to a continuous barrier option which can be statically replicated which makes the continuous path dependency vanishes.

On the other hand, Asian option:

• Path dependency comes from taken the average over the whole trajectory.
• The smoothing effect of the averaging dampens the path dependency.

I read in the dynamic hedging (taleb) book that Asian are strong path dependent while Lookback are weak path dependent. But theoretically, both options depend on the joint probability law of the trajectory and should both be strongly path dependent?

I would feel that Asians are more path-dependent than Lookback options, because the payoff of the option depends on more of the path than that of the Lookback, which only relies on a certain point (extremum) of the path.

• But you can't figure out the extremum by looking at only certain points, you have to look at all points. Oct 18, 2023 at 11:55
• @bigInner Yes, the extremum is determined by looking across all points, but these points are not used in deducing the final payoff. For example, if I have a strike of 100, and the spot price went straight up to an extremum to 150 and went back down, I would have a lookback call of terminal payoff 150 - 100 = 50. However, if instead of a "straight up" path to 150, I have many "zig zags" to 150 and went back down, the terminal payoff would still be 150 - 100 = 50 i.e. the extremum is the same. However, for an Asian option, the terminal payoff would be different. Hope this helps! Oct 18, 2023 at 13:15

What types of options are "more" path dependent than others is not well-defined. The cited distinction -- full-path dependence versus single-point dependence -- is somewhat useful in the sense of estimating memory usage of a Monte Carlo pricer, but not so useful beyond that. Even American-exercise options are path-dependent, after all.

What I think is useful is the distinction according to most (asymptotically) efficient pricing scheme, such as

• $$O(1)$$ A closed-form option pricing formula exists (B-S)
• European
• Geometric-average asian
• Continuous lookback
• $$O( N^{-4})$$ A 1-d quadrature (e.g. Simpson's rule) is needed (European stochastic vola)
• $$O( N^{-1})$$ A 1-d grid scheme will suffice (American)
• $$O( N^{-1} + M^{-1})$$ A 2-d grid scheme is needed (Arithmetic asian, Bermudan swaption)
• $$O( N^{-\frac{1}{2}})$$ A 1-d Monte Carlo scheme is needed (Some exotics)
• $$O(K N^{-\frac{1}{2}})$$ Multidimensional quadrature / Monte Carlo scheme is needed (Basket, CDO tranche protection)

This leaves out a lot of practitioner tricks. These include control variates, moment-matching for basket options, quasirandom sequences for quadrature, SABR-type approximations for rates, and LSMC for convertibles and tricky exotics.

Note it is very common, even for practitioners, to take advantage of the convenience of Monte Carlo and employ it in cases where it is not the most asymptotically efficient scheme. One can do Monte Carlo in a couple lines of code, but for quasirandom sequences et cetera there is more machinery involved. This is especially true of asian options, where the grid scheme is much more of a pain to write.