Comparing Path Dependence Through Option Pricing Methods
Summary
The document compares lookback and Asian options, whose payoffs depend respectively on an extremum and an average over the underlying price path. It challenges the idea that one can simply rank these contracts as more or less path-dependent. The accepted answer says that degree is not well-defined; full-path versus single-point dependence can help describe memory needs in Monte Carlo, but is less useful as a general measure.
Instead, it recommends comparing the most efficient available pricing approaches and their asymptotic computational costs. The examples range from closed-form formulas for some contracts to quadrature, grid methods, and Monte Carlo for others. In practice, easier-to-implement Monte Carlo may be chosen even when another approach has better asymptotic efficiency; control variates and related techniques can also help. The classifications are illustrative, not exhaustive, and the document notes that practitioner methods and implementation effort matter.
Key ideas
- Lookback payoffs use path extrema, while Asian payoffs use path averages.
- There is no universally defined ranking of options by degree of path dependence.
- The most efficient pricing method and its computational cost offer a practical comparison.
- Monte Carlo may be used for convenience even when a more efficient method exists.
- Control variates and other practitioner techniques can improve pricing approaches.
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Full text
# Is Lookback option more path-dependent than an Asian option
# 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?
## Answer by Brian B (score 2)
https://quant.stackexchange.com/a/77395
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.
## Answer by KaiSqDist (score 1)
https://quant.stackexchange.com/a/77049
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.