Pricing American Asian Options with Rolling Averages
Summary
The document asks how to price an American-style Asian option whose exercise payoff uses the average underlying price over a recent rolling window. It explains why early exercise and path dependence complicate standard approaches: Monte Carlo and tree methods are presented as unsuitable, while a single running-average state variable does not capture the information needed to update a rolling average. These limitations motivate the search for alternative numerical methods.
The response suggests several approximations and techniques: draw on methods for convertible-bond soft calls, use Longstaff–Schwartz least-squares Monte Carlo, approximate the averaging window with fewer observations at wider intervals, or incorporate a Brownian-bridge-based exercise probability into a PDE scheme. It also notes that ignoring the exercise feature may introduce little error far from expiry, when substantial variance remains. These are practitioner suggestions rather than a worked comparison; the document gives no numerical tests or implementation details, and the approximations may need validation for a specific contract.
Key ideas
- A rolling average depends on prices entering and leaving the window, so its current value alone may not summarize the path needed for future updates.
- Early exercise makes pricing a path-dependent Asian option more difficult than pricing a European-style contract.
- Least-squares Monte Carlo is proposed as one possible approach, though the response describes it as slow and difficult to tune.
- A reduced number of averaging observations at wider intervals can make a higher-dimensional PDE approximation more manageable.
- A Brownian-bridge exercise probability can be incorporated into a PDE scheme as a practical approximation.
Tags
Full text
# How to price the American style Asian option with recent N day average
# How to price the American style Asian option with recent N day average
How to price the `American style Asian option with recent N day average`, for example, we exercise at `t` day, then the payment is $$\Psi(t) = \dfrac{1}{N}\sum\limits^t_{i=t - N+1}S_i$$ Since the `early exercise` and `path dependence`, we can not use the `Monte Carlo simulation` and `tree method`. And since the average is not from the begin day to today, we are not allowed to use the addition variable: $$I_j = \sum\limits^j_{i=j - N+1}S_i$$ namely, we can't use the `PDE` method. This is because, we don't know $I_{j+1}$ just from $I_j$ and $S_{j+1}.$
I only know above three methods to price the option. Is there any reference or advanced method to price such option?
## Answer by Brian B (score 2, accepted)
https://quant.stackexchange.com/a/33516
In convertible bond pricing there is something similar called a "soft call" with similar properties so you might want to search for literature on them. The main difference is that soft calls are an exercise condition rather than an exercise price.
One key point is that, if expiration $t$ is distant, very little error is introduced by ignoring the softness. This is because there is so much variance remaining that paths exceeding the exercise price tend to do so by a lot, and for a long while.
Otherwise we are in the realm of "tricks", among which I count Longstaff Schwartz (LS). In practice LS is not used much because it is so slow and tricky to metaparameterize.
One good trick is to pretend N is a much smaller number (say 2 or 3) at greater intervals (like 10 day periods). Then you can add a couple dimensions $I^n_j$ to your PDE solver (often on a reasonably small grid).
Another trick used by practitioners is to associate any given level of $S$ with a probability of exercise based on the brownian bridge. This is easy to accommodate in a PDE scheme.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.