# 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 !

• I'm afraid I don't understand the question after the edit. Are you confused about what would be the practical difference between European options and American or Bermudan options? Jun 30, 2019 at 15:25
• @BobJansen : Here's the confusion : a stream of intermediary contingent flows is equivalent to a single flow paid at maturity, with the proper capitalization factor. So, is the distinction American vs European relevant only to "legal" aspects of the contract (when to exercise) or is it more prescriptive on the nature of the contingent flows (dependence on a single fixing for instance). Jun 30, 2019 at 16:06

• I know these statements. The exercise was not my first concern (one may think that exercise is systematic, with a payoff equal to $(S_T - K)^+$). My question was, if one exercise his option at expiry $T$, can the payout upon exercise depend on the whole path of the underlying, such as $(max(S_1, S_2, ..., S_T) - K)^+$ because this obviously falls within the scope of the first definition (exercise upon expiry). Jun 29, 2019 at 9:56