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I would like to get the price of an option which pays at time T the minimum between the logarithm of (S(1,T) / S(1,0)) and the logarithm of (S(2,T) / S(2,0), with the following processes:

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(The two brownians motions are not correlated).

I decided to use the martingale approach for this problem. By choosing the risk-less asset as the numeraire, I know I have to use the Radom-Nikodym derivative but I am a little bit stuck. Does anybody have some hints to give regarding this problem?

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    $\begingroup$ $\min(S_1, S_2) = S_1 + \min(S_2-S_1, 0) = S_1 - \max(S_1-S_2, 0) = S_1 - S_2\max(S_1/S_2-1, 0)$. You can use $S_2$ as the numeraire. $\endgroup$ – Gordon Apr 17 '18 at 15:18
  • $\begingroup$ Thank you for your advise. But how to handle the logarithm? Does it not change anything ? $\endgroup$ – Skyly83 Apr 20 '18 at 16:05
  • $\begingroup$ It certainly will. You can use $S_2$ as the numeraire. $\endgroup$ – Gordon Aug 13 '18 at 16:45

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