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1

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.

2

No. For example, consider a call option struck on a stock with current price \$150, strike price \$100, 1 year to maturity, 0% risk free rate, 5% dividend yield and 40% implied volatility. The intrinsic value is \$50 but the price of the call is \$47.52.

4

You could but there are difficulties associated with this approach. The main one is that $\tau$ is stochastic, ie it is different for different paths of $S$, so the standard Black-Scholes formula does not apply. For example some $\tau$s you need to check are of the form $\tau =\inf\{t : S(t) <B\}$ in which case you need to value a barrier option with ...

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