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I have been told most real options are American.

However, this isn't really true. Markets are closed at times, there are delays in transactions, or the owner of the option might be sleeping, or just otherwise not keeping up with the markets. So real life behavior is very discrete.

So why are we so much interested in pricing options where you can exercise at any point in $[0, T$]? Especially when that problem is very hard?

Is it not far more realistic to be interested in Bermudan options with e.g. daily-monitoring (252 days per year)?

Bermudan options can also be far more easily priced than Americans. Just use backwards induction with a stepsize $\Delta = 1/252$.

An American price would be obtained if the stepsize $\Delta \rightarrow 0$, but that is very hard to calculate in practice.

So if Bermudan options are more realistic and easier to price, why care about American options that seem mostly a unrealistic mathematical construct?

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  • $\begingroup$ what's the difference in value of an american option and a dayly callable bermudan option? $\endgroup$ – will Mar 1 at 11:50
  • $\begingroup$ @will American option would be worth slightly more since you can call it during the entire day, but a Bermudan can only be called on a specific time during each day. $\endgroup$ – efwofo Mar 1 at 15:01
  • $\begingroup$ Would you pay more for one that can be called at any point? How much would you give me to buy the continuously callable option and sell me the daily callable counterpart? $\endgroup$ – will Mar 1 at 17:14
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Working in discrete time or continuous time is mostly a matter of convenience. What most people do in some field of finance or economics is suggestive of what tends to be easier, though it's a kind of rule of thumb.

Off the top of my head, CT has the convenience of easily handling uneven time steps and allowing easy aggregation. It also handles changing distributions seemlessly, which is why a lot of progress in macroeconomics as of lately happened in CT (because you try to deal with an entire distribution of income, wealth, etc. and it changes over time).

Now, you could pick a very small $\Delta$ and see for yourself what happens when you follow your own advice. In practice, you'll discretize the damn thing anyways.

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