Understanding Black-Scholes Merton closed form formula for Vanilla Call Option

Call (premium) =  S₀ N(d₁)  -  Ke^-r(T-t) N(d₂)

To understand this equation we need to start with payoff of call option at maturity.

Max( S_T – K, 0)  =>  If at maturity Stock price(S_T) ends above Strike(K), Options buyer gets Stock worth S_T and pays Strike K for it.

The value of a derivative contract is expected present value of its future cash flows discounted at risk-free rate (under risk-neutral pricing framework). Hence, above equation can be divided into two parts.

1. A) S₀ N(d₁): Present value of expected Stock price received by the buyer of Option.
2. B) Ke^-r(T-t) N(d₂): Present value of expected cash flow (Strike) paid by buyer.

Starting with B as it is easy to understand:

B represents expected present value of Strike that is paid to obtain underlying by buyer in case call option matures in the money.

As stated above there are two important terms to consider: present value and expected value.

Present value: As Strike K is constant, so under risk neutral frame-work,  discounting rate of risk-free rate is used to calculate present value. Ke^-r(T-t)

Expected value: Strike is only paid when option matures ITM. So to calculate the expected present value of cash-flow, present value mentioned as above needs to be multiplied by the probability of Option ending in the money, which is N(d₂).

*Important point: Here is that N(d₂) is not actual probability but probability under the risk-neutral framework. To get actual probability all required is to replace risk-free interest rate with actual expected return on the stock.

Coming back to part A:

As similar to discussed above, A is expected present value of Stock received by the buyer. Unlike part B, explanation of constituent terms of A is not straight forward. S₀ represents the present value of stock and hence we might think that similar to part B, to get expected value we need to multiply the present value of Stock, S₀ with the probability of Option maturing ITM. But in part A S₀ is multiplied by N(d₁). Two probability terms used in part A “N(d₁)” and Part B “N(d₂)” are different. This is because N(d₁) is not a pure probability that option will mature ITM but is: Conditional expected  present value of the unit value of the stock given stock ends in ITM. Which when multiplied by S₀ simply gives: Conditional expected present value of the stock given stock ends in ITM.

E[S_T/given(S_T>K)]e^-r(T-t)

Why part A “S₀ N(d₁)” could not be simplified as part B?

In part B cash-flow in the case of Option maturing ITM was constant “K”. Thus, the two terms present value and probability could be separated. While in the case of part A option buyer receives stock whose value at maturity is not fixed and linked to probability distribution and hence probability term couldn’t be completely separated from expected value.

Conclusion:

S₀ N(d₁) – Expected present value of what buyer receives as Stock.

Ke^-r(T-t) N(d₂): Expected present value of what buyer have to pay as Strike.

N(d₂): Probability of option maturing ITM (under risk-neutral framework).

N(d₁): Conditional expected  present value of a unit value of the stock given stock ends in ITM. And N(d₁) is also Delta of an option.