**Call (premium) = S _{0 }N(d_{1}) - Ke^{-r(T-t)} N(d_{2})**

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.

- A) S
_{0 }N(d_{1}): Present value of expected Stock price received by the buyer of Option. - B) Ke
^{-r(T-t)}N(d_{2}): 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(d_{2}).

***Important point:** Here is that N(d_{2}) 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. S_{0 }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, S_{0} with the probability of Option maturing ITM. But in part A S_{0 }is multiplied by N(d_{1}). Two probability terms used in part A “N(d_{1})” and Part B “N(d_{2})” are different. This is because N(d_{1}) 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 S_{0 }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 “S _{0 }N(d_{1})” 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:**

S_{0 }N(d_{1}) – Expected present value of what buyer receives as Stock.

Ke^{-r(T-t)} N(d_{2}): Expected present value of what buyer have to pay as Strike.

N(d_{2}): Probability of option maturing ITM (under risk-neutral framework).

N(d_{1}): Conditional expected present value of a unit value of the stock given stock ends in ITM. And N(d_{1}) is also Delta of an option.

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