In BS option pricing, why is the drift rate of GBM equal to risk free rate for all stocks in risk neutral?

Question

Can the drift rate μ depend on specific stock ? If not what is the rationale for the discounted Stock price to be a martingale ?

\begin{align} & dS_t/S_t = \mu dt + \sigma dW_t \end{align}

Thanks

Answer 1

Your SDE describes the evolution of the stock price under the physical measure $\mathbb{P}$. However, the BSM model is developed under the risk neutral measure $\mathbb{Q}$. Without digging into results of measure theory and the Girsanov theorem of change of measure, you intuitively "change the weight" of the trajectories such that on average the rate of change is $r dt$. In fact, you define a new probability measure. In the risk neutral world, investors are not compensated for the excess risk and all risk premiums diminish. Back in your question, under the $\mathbb{P}$-Measure the stock can have its own idiosyncratic drift parameter $\mu$, but under the $\mathbb{Q}$-Measure, the drift parameter should be $r$. Hence, under this measure the discounted process with numaire the money market account is a martingale. Then, steps are straightforward.

Answer 2

An easy way to think about it:

In the risk-neutral world ($\mathbb{Q}$), investors don’t care about risk and don’t pay you more or less than they pay for a risk-free investment. They don’t really see a difference. Hence, all assets have the same return, the risk-free rate $r$.

Of course, in the real world ($\mathbb{P}$), people do care and a stock has a different return $\mu$. There is a risk premium investors pay. And of course, different stocks may have different drift rates.