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Consider Black-Scholes (B, S) market model. Let $r = 0$ (hence, $B_t ≡ 1$), $S_0 = 0 $.
Stock price is described by $dS_t = σS_tdW_t$.
Find the price of the option that pays $(S_T^3 - S_T^2 )_+ = max(S_T^3 - S_T^2, 0)$.

Any help on how exactly to apply Girsanov here?

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  • $\begingroup$ Apparently it is different, because $S_t$ is a martingale, but $S_t^2$ is not. $\endgroup$
    – Kyle
    Jul 19, 2021 at 9:40
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    $\begingroup$ yes, $S_t$ is a martingale but I don’t see why the standard approach wouldn’t work just because $r=0$. $\endgroup$
    – Kevin
    Jul 19, 2021 at 9:48
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    $\begingroup$ I also think that the link provided by Kevin gives you the approach: 1) Split up the expectation using the indicator function. 2) Use the measure where $S_t^3$ is numeraire for the first part of the expectation. 3) Use the measure where $S_t^2$ is numeraire for the second part of the expectation. $\endgroup$
    – mmencke
    Jul 19, 2021 at 10:14
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    – cigien
    Jul 19, 2021 at 23:57

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