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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 !

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  • $\begingroup$ 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? $\endgroup$
    – Bob Jansen
    Jun 30, 2019 at 15:25
  • $\begingroup$ @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). $\endgroup$
    – Aguel
    Jun 30, 2019 at 16:06

2 Answers 2

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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).

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  • $\begingroup$ Can you explain in more detail, perhaps with an example? You're stating that the payoff can be path-dependent, but not explaining how that would work. $\endgroup$ Jun 29, 2019 at 19:42
  • $\begingroup$ Consider a payoff that depends on n values of the spot price before a specific date and blends all those values by means of a function. That payoff will depend on the specific path of the spot price. By function you can simply think of a simple average, like the case of asian options --> this was the example $\endgroup$
    – Vitomir
    Jun 30, 2019 at 11:01
  • $\begingroup$ That's circular reasoning. How can a payoff depend on the spot price except on the exercise date? Are you talking about a contract that specifically defines its value as based on that blend? $\endgroup$ Jun 30, 2019 at 14:32
  • $\begingroup$ Am I the only one thinking that the last comment shows lack of preparation on the matter of option pricing coupled with quite some degree of arrogance? $\endgroup$
    – Vitomir
    Jun 30, 2019 at 14:56
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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.

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  • $\begingroup$ 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). $\endgroup$
    – Aguel
    Jun 29, 2019 at 9:56
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    $\begingroup$ @Aguel Yes. European refers to the fact that it can only be exercised at expiry. The payoff can be anything. For example the average price over a time frame vs the strike. Of course these are “exotic” options and are not as liquid when compared to the basic exchange traded options. $\endgroup$
    – AlRacoon
    Jun 29, 2019 at 10:06
  • $\begingroup$ @AlRacoon : I updated my question to show what confuses me in this definition : as long as we admit path-dependance, every payoff may be written as a single flow paid at maturity (ignoring counterparty credit risk). $\endgroup$
    – Aguel
    Jun 29, 2019 at 14:55

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