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There are two stocks: $S_t$ and $P_t$ $$dS_t = S_t(\mu dt + \sigma dB_t)$$ $$dP_t = P_t((\mu + \varepsilon) dt + \sigma dB_t)$$ Is there any risk-neutral measure? My thoughts are pretty simple: $μ$ is for the physical measure, so there's no risk-neutral measure. Please shed light on this question.

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Well there can definitely be a risk-neutral measure but only one of the processes is a tradable. For instance, consider the total return process of a stock $S_t$ $$ dS_t = \mu S_t dt + \sigma S_t dW_t $$ and its price return $$ dP_t = (\mu - q)P_t dt + \sigma P_t dW_t, \quad P_0 := S_0 $$ Under the risk -neutral measure you have exactly the same SDEs with $r$ replacing $\mu$, but only one of them is tradable. In this example the total return process is tradable, the price return is not.

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  • $\begingroup$ Thanks for your answer, so the method is to modify dPt and there's one unique risk-neutral measure? This is actually a multiple-choice question with: (a) There exists a unique risk-neutral measure. (b) There does not exist a risk-neutral measure. (c) There does not exist an arbitrage. (d) There exists more than one risk-neutral measure. $\endgroup$
    – nearhome
    May 17, 2023 at 13:54
  • $\begingroup$ Just feeling weird because I think risk-neutral measure means no arbitrage, then (c) actually includes (a) (d) $\endgroup$
    – nearhome
    May 17, 2023 at 13:57
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    $\begingroup$ Which is proof that multiple choice should be forbidden in the (applied) sciences. I have no clue what the assumptions are for the multiple choice test. Is it stated that $S_t$ and $P_t$ are both tradable? If so that would lead to arbitrage since one of them would have drift $r$ and the other $r-q$ and yet both are tradable. I'm not going through all the other possibilities, it all depends on assumptions, and I'll leave that to you. $\endgroup$
    – Frido
    May 17, 2023 at 14:02
  • $\begingroup$ Thanks, they are both tradable. I could leave the multiple choice behind, but still curious about the definition of having a risk-neutral measure. So how can I tell if there's a unique measure or more than one measure? $\endgroup$
    – nearhome
    May 17, 2023 at 14:08
  • $\begingroup$ Good question. Short answer: there is a unique risk-neutral measure if and only if all possible claims on a process is traded (and therefore replicable), no bid-ask and no transaction costs, and infinite liquidity. Clearly too much to ask for, hence unique risk-neutral measure will always be wrt to a certain set of instruments and assumptions. I think there could be quite a few questions on this topic asked already. Otherwise post a new question maybe. $\endgroup$
    – Frido
    May 17, 2023 at 14:18

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