In the simple Merton structural credit risk model, the physical default probability is given by:
$$ DD_p = \frac{\ln(A / D) + (\mu -0.5\sigma^2)T}{\sigma \sqrt{T}} $$
$$ P=N(-DD_p) $$
Assuming that the physical distance-to-default(DD) and P are already provided, one needs to convert the physical default probability to risk-neutral probability in order to derive a fair value spread.
The "standard" way for this conversion in the literature seems to be the following: $$ Q=N[N^{-1}(P) + \lambda R \sqrt T] $$ where $\lambda$ is the market Sharpe ratio and R is the correlation of market and asset returns.
Essentially, the physical probability is transformed to a point in the CDF, a risk premium is added, then the sum is transformed back to arrive at the risk-neutral default probability.
My question is: why can we not simply plug in the risk-free rate(say the Treasury rate) into the distance-to-default formula since all assets have the same drift(r) under the risk-neutral measure?
$$ DD_q = \frac{\ln(A / D) + (r -0.5\sigma^2)T}{\sigma \sqrt{T}} $$
$$ Q=N(-DD_q) $$
Can someone point out what is wrong with my naive thinking? There must be a good reason why the literature is taking the more circuitous approach above.