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

Conditions

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

Reasoning, step by step

1. Identify the current stock price $S_0$ and the strike price $X$.
2. Determine the risk-free interest rate $r$ and the time to maturity $T$.
3. Calculate the volatility $\sigma$ of the underlying asset.
4. Compute the intermediate variables $d_1$ and $d_2$.
5. Evaluate the standard normal cumulative distribution function $N(d_1)$ and $N(d_2)$.
6. Substitute these values into the formula $C = S_0 N(d_1) - X e^{-rT} N(d_2)$ to find the call option price.

Example

The video displays the formula $C = S_0 N(d_1) - X e^{-rT} N(d_2)$ at the top of the page and the instructor handwrites it while explaining that the call option price equals the stock price multiplied by the standard normal distribution at $d_1$, minus the discounted strike price multiplied by the standard normal distribution at $d_2$.

Common misconceptions

• Confusing the call option formula with the put option formula.
• Forgetting to discount the strike price $X$ by the factor $e^{-rT}$.
• Assuming $N(d_1)$ and $N(d_2)$ are probability densities rather than cumulative probabilities.

Watch the explanation

Explore next

• What is the Black-Scholes-Merton formula for a European put option price?
• How are the intermediate variables $d_1$ and $d_2$ calculated?
• Why is the derivation of the Black-Scholes-Merton formula not required in this segment?