Option Exercise Style and Path-Dependent Payoffs
Summary
The document distinguishes when an option may be exercised from what determines its payoff. A European option permits exercise only at expiry, an American option permits exercise over the eligible period, and a Bermudan option permits exercise on specified dates. These exercise rules are separate from whether the payoff depends only on the terminal underlying price or also on earlier prices along the path.
It uses the possibility of a path-dependent payoff, such as one based on several observed prices, to resolve confusion about European options. An Asian option is given as an example of a payoff that can depend on past prices while still following a European exercise rule. The discussion also notes that rewriting early exercise as a cash flow at maturity does not make the original option European in the exercise sense. It offers conceptual definitions rather than a pricing or hedging method, and does not develop the mathematical conditions for static replication.
Key ideas
- Exercise style specifies when the holder may exercise an option.
- A European option can be exercised only at expiry, while American exercise is available during the eligible period.
- A Bermudan option permits exercise on specified dates.
- Payoff dependence on the price path is distinct from the option’s exercise style.
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Full text
# Definition of an European Option
# Definition of an European Option
I'm a bit confused after reading an article from Henry-Labordere. He was giving an example of an European option whose payoff may depend on the whole path of the underlying : $f(S_{T_1}, S_{T_2}, ...,S_{T_n})$. I thought that by definition, an European option's payoff at maturity can only depend on the final price of the underlying, $S_{T_n}$.
Could somebody help me to find the exact mathematical definition of an European, American and path dependent options ?
Here are some details of what confuses me:
For the sake of argument, I'll suppose rates are zero, and consider for simplicity a bermudan call with a single early exercise date $T_{ex} < T$, and write the exercise condition as $1_{ex} = f(S_1,...,S_{T_{ex}})$ (which encodes the comparison $S_{T_{ex}}-K > C_{T_{ex}}$ where $C_{T_{ex}}$ is the continuation value). This payoff can then be viewed as an "European" option paying $1_{ex}(S_{T_{ex}}-K)^+ + (1-1_{ex})(S_T-K)^+$ at maturity $T$
Obviously with multiple exercise dates, one would arrive to a single flow paid at maturity. Meaning every american ou bermudean could be written as an european option.
Moreover, one can replicate any "European payoff" with vanillas options (Breeden-Litzenberger formula) and and I know that pathdep payoff cannot be hedged statically by vanillas !
I think I have something wrong but can't see what !
## Answer by Vitomir (score 2)
https://quant.stackexchange.com/a/46367
What can be path-dependent is the payoff. The characteristic European, American etc. refers to the moment when to use the optionality. In that moment you will receive a payoff. That payoff can be path-dependent, i.e. depend on previous values of the spot prices (see asian options for instance).
## Answer by Chris (score 0)
https://quant.stackexchange.com/a/46366
A european option is able to be exercised only at expiry, an american option is able to be exercised between purchase and expiry, and a bermudan option is able to be exercised at intervals in between.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.