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The put option formula can be memorized by comparing it structurally with the call option formula. Starting from the call formula $C = S_0 N(d_1) - X e^{-rT} N(d_2)$, you swap the positions of the two terms and negate the arguments inside the standard normal distribution function $N(\cdot)$.

Conditions: You already know the Black-Scholes-Merton formula for the call option.; This is a mnemonic rule for memorization, not a rigorous mathematical proof.

The put option formula can be memorized by comparing it structurally with the call option formula. Starting from the call formula $C = S_0 N(d_1) - X e^{-rT} N(d_2)$, you swap the positions of the two terms and negate the arguments inside the standard normal distribution function $N(\cdot)$.

Conditions: You already know the Black-Scholes-Merton formula for the call option.; This is a mnemonic rule for memorization, not a rigorous mathematical proof.