$\mathbb{P}$ and $\mathbb{Q}$ probability measure/distribution interpretations

Question

I'm trying to understand probability distributions implied from market prices and was reading through this reference explaining the interpretation of $N(d_1)$ and $N(d_2)$ in the log-normal vol Black-Scholes model.

I have two sets of questions:

1. If I buy a call at strike $K$ and write a call at strike $K+\Delta K$, can I back into the risk-neutral probability of the underlying rising to $[K, K + \Delta K]$ (which I would work out as $N(d_2)$) from observed vol and market prices? Is this algebraic rearrangement of the Black-Scholes equation meaningful?

2. In the context of interest rates, could there be a similar interpretation using normal vol (to allow for negative rates) to back into the probability of interest rates rising to $[X\%, (X + \Delta X)\%]$? If yes, is it possible to convert the implied distribution under $\mathbb{Q}$ to a distribution under $\mathbb{P}$ (i.e. does the other direction of Girsanov make sense)?

I'm unfamiliar with quant finance and am just digging in, so any questions asking for clarification on a certain part (preferably with a reference) would be much appreciated as well...

Thanks.

Answer 1

The price of a [K,K+dK] call spread informs you about the risk neutral probability of the underlying being above K. (Not in the interval (K,K+dK)). This is true regardless of any assumed distribution , lognormal or normal. Hence true for options on any asset (stocks, interest rates etc).

No derivative price can tell you anything about P.