How can the Black-Scholes-Merton put option formula be memorized from the call option formula?

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.

Reasoning, step by step

1. Write down the call option formula: $C = S_0 N(d_1) - X e^{-rT} N(d_2)$.
2. Swap the order of the two terms: place the discounted strike price term first and the stock price term second.
3. Change the sign of the arguments inside the $N(\cdot)$ function from $d_1$ and $d_2$ to $-d_1$ and $-d_2$.
4. Combine these changes to form the put option formula: $P = X e^{-rT} N(-d_2) - S_0 N(-d_1)$.
5. Verify the formula against the standard definition.

Example

The instructor explains that for the put option P, there is a pattern consistent with the call option C: the two terms swap positions, and the arguments inside N all become their opposites. The screen simultaneously displays both formulas to highlight this structural contrast.

Common misconceptions

• Forgetting to swap the positions of the two terms.
• Negating the terms themselves instead of just the arguments inside the $N(\cdot)$ function.
• Assuming this mnemonic rule proves the put-call parity or the formula's validity.

Watch the explanation

Explore next

• What is the Black-Scholes-Merton formula for a European put option price?
• What is the Black-Scholes-Merton formula for a European call option price?