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At least two ways to price this: Use Carr-Madan Use $S^2$ as a (power) numeraire, in which case you can price the payoff $(S_T - 1)_+$ under the power numeraire measure. EDIT: Put-call symmetry. Maybe I can get another -1 for my answer. Is the purpose of answering questions here to do homework for someone else or to stimulate further study and generate ...

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You can compute expectation of drifted processes as well and derive same pricing formulas,but usually its more complicated (compare derivation of Black Scholes using martinglaes and through PDE. PDE proof ,where drift is explicit, is much longer) With martingale representations you have more analytical mathematical tools/formulas available (e.g. barrier ...

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for an intuitive answer, if we start with a vanilla call as our base, then with an up & out call, we would like the underlying to go up in price yes. But as the price increases, we also increase the probability of kicking out and losing our payout - so we don't want it to go up too much. If the barrier is so far away that the probability of reaching it ...

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Standard call options are trivially more expensive than up/down and out call options. However, for high strikes, down and out options will very likely never be knocked out, therefore their prices should be close to standard call options. For low strikes, down and out call options are almost worthless, therefore , the down and out call options curve price ...

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