I would like to ask whether there is an intuition for the drift of price processes under the Stock numeraire.
I find it intuitive that the martingale measure under the Money Market numeraire induces the drift "r" to all price processes (via the appropriate change of measure): with the money market compounding continuously at rate "r", all prices need to drift at this rate "r", otherwise the price processes discounted by the money market numeraire would not be martingales (i.e. any price process that wouldn't drift at "r" would give rise to arbitrage between Spot and Forwards, i.e. there would be miss-pricing of Forwards under the money market numeraire if the price process didn't drift at "r").
Same holds for the Discount bond numeraire under deterministic rates (because the Bond numeraire under deterministic rates turns out to be the money market numeraire scaled by a constant).
However, I haven't managed to build similar reasoning for the Stock price numeraire.
We know that the Stock price process under the Stock numeraire is:
\begin{align*} \frac{dS}{S} &= rdt + \sigma dW_t\\ &=\big(r+\sigma^2\big)dt + \sigma d \widehat{W}_t. \end{align*}
Above, $W_t$ is a standard Brownian motion under the risk-neutral measure associated with the Money market numeraire, whilst $\widehat{W_t}$ is a standard Brownian motion under the pricing measure associated with the Stock numeraire.
Why does the Stock numeraire induce the drift:
\begin{align*} &\big(r+\sigma^2\big) \end{align*}
Why would (intuitively) being able to borrow at the rate of the stock mean that price processes must have this drift?
Thank you so much,