I saw this interesting argument that you can use the value of a put spread to find the approximate market implied probability of underlying finishing below the mid point of the two strikes, assuming the distance of strikes and DTE are reasonably small.

Namely, for two put options on the same underlying with strikes $K_2>K_1$ and stock price at expiry $S_T$, we have

$$ \frac{V(K_2)-V(K_1)}{K_2-K_1}\approx P(S_T<\frac{K_2+K_1}{2}) $$

In other words, the odds you get for buying a put spread equals to the probability of stock finishing below mid of strikes.

My question is how to derive this result, and does it applies to call spreads as well.

I saw this interesting argument that you can use the value of a put spread to find the approximate market implied probability of underlying finishing below the mid point of the two strikes, assuming the distance of strikes and DTE are reasonably small.

Namely, for two put options on the same underlying with strikes $K_2>K_1$ and stock price at expiry $S_T$, we have

$$ \frac{P(S_t,K_2)-P(S_t,K_1)}{(K_2-K_1)-\big(P(S_t,K_2)-P(S_t,K_1)\big)}\approx P(S_T<\frac{K_2+K_1}{2}) $$

In other words, the odds you get for buying a put spread equals to the probability of stock finishing below mid of strikes.

My question is how to derive this result, and does it applies to call spreads as well.