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Exercise 11 of chapter 8 of Mark Joshi's "The concepts and practice of mathematical finance", asks to compare prices of an American and a European digital (binary) calls when out-of-the-money. The options have same strike and expiry, and are cast on an asset following Brownian motion with no interest rates.

Before peeking at the solution, I had argued that due to the fact we are out-of-the-money, American optionality plays less of a role and the two would be worth about the same.

However, the answer to the exercise states that, given $r = 0$ and the symmetry of paths of a Brownian motion, when the American call pays off there is a 50% chance the European also will, so the latter is worth about half the former.

Now, I don't understand the logic of this answer: since we have no knowledge of how bad out-of-the-money we are, I'll concede there is a 50% chance that the asset will end up being worth more than its current value, yes, and also that the value of the European digital is its current probability to end up in-the-money, but how can it be justified that the European is worth half of the American digital?

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Lets assume the price of the underlying equals the strike at some point prior to expiry. Then the probability of the price being still greater or equal the strike at expiry is 0.5.

So the probability of the European option paying out is exactly half of the probability for the American option.

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  • $\begingroup$ We are out-of-the-money now, but in case we were in, the American immediately yields its payout, while the European has a 50% chance of ending up in-the-money, given the symmetry of a driftless Brownian motion. Hence the European is worth 50% the American no matter if we are in or out of the money. Have I understood it correctly? $\endgroup$
    – Giogre
    Apr 10, 2021 at 20:17
  • $\begingroup$ I think yes, but let me add that that to be in-the-money at some future time, when we were initially out-of-the-money, we need to be at-the-money at some point as well due to continuity. And at this point we can apply the symmetry of the driftless Brownian motion. $\endgroup$
    – Robert Schlichtner
    Apr 10, 2021 at 20:35
  • $\begingroup$ Yes you are right in specifying, I should have written 'at-the-money' instead of 'in-the-money' in my comment above, the American binary option can be exercised immediately as soon as at-the-money is reached. $\endgroup$
    – Giogre
    Apr 10, 2021 at 20:45

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