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The Black-Scholes-Merton formula for a European call option price is $C = S_0 N(d_1) - X e^{-rT} N(d_2)$. In this formula, $S_0$ is the current stock price, $X$ is the strike price, $r$ is the risk-free interest rate, $T$ is the time to maturity, $N(\cdot)$ is the cumulative distribution function of the standard normal distribution, and $d_1$ and $d_2$ are intermediate variables.

Conditions: The option is a European call option.; The underlying asset pays no dividends during the life of the option.

The Black-Scholes-Merton formula for a European call option price is $C = S_0 N(d_1) - X e^{-rT} N(d_2)$. In this formula, $S_0$ is the current stock price, $X$ is the strike price, $r$ is the risk-free interest rate, $T$ is the time to maturity, $N(\cdot)$ is the cumulative distribution function of the standard normal distribution, and $d_1$ and $d_2$ are intermediate variables.

Conditions: The option is a European call option.; The underlying asset pays no dividends during the life of the option.

The intermediate variable $d_1$ is calculated using the formula $d_1 = \frac{\ln(S_0/X) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$. The intermediate variable $d_2$ is not computed from scratch; instead, it is derived directly from $d_1$ using the relationship $d_2 = d_1 - \sigma\sqrt{T}$.

Conditions: $S_0 > 0$ and $X > 0$ to evaluate the natural logarithm $\ln(S_0/X)$.; $\sigma > 0$ to serve as a valid denominator.; $T > 0$ for the square root $\sqrt{T}$ to be meaningful.

The intermediate variable $d_1$ is calculated using the formula $d_1 = \frac{\ln(S_0/X) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$. The intermediate variable $d_2$ is not computed from scratch; instead, it is derived directly from $d_1$ using the relationship $d_2 = d_1 - \sigma\sqrt{T}$.

Conditions: $S_0 > 0$ and $X > 0$ to evaluate the natural logarithm $\ln(S_0/X)$.; $\sigma > 0$ to serve as a valid denominator.; $T > 0$ for the square root $\sqrt{T}$ to be meaningful.

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.

The derivation of the Black-Scholes-Merton formula is not required in this segment because its mathematical derivation is relatively complex. The video explicitly states that learners only need to memorize the formula and understand how to substitute values into it to compute option prices, rather than mastering the underlying proof.

Conditions: The context is the introductory learning segment for the Black-Scholes-Merton formula.; The focus is on practical application and memorization rather than theoretical proof.

The derivation of the Black-Scholes-Merton formula is not required in this segment because its mathematical derivation is relatively complex. The video explicitly states that learners only need to memorize the formula and understand how to substitute values into it to compute option prices, rather than mastering the underlying proof.

Conditions: The context is the introductory learning segment for the Black-Scholes-Merton formula.; The focus is on practical application and memorization rather than theoretical proof.