# Is there a risk-neutral measure if there are two stocks with different drift terms?

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.

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.
• 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. May 17, 2023 at 14:02