Interview Question at HAP Capital - Delta of the option

intuit2k2thanks edtkh.. Your formula just seems to be %OTM * premium... any intuition for this?

I have kind of updated the formula. Bear in mind this is purely an estimate and that this estimate would not work for a case where the Strike=Stock Price though..

Let's consider an ITM call for the following for illustration: A) Strike = 34, Stock Price = 104 and Delta = 0.95. Then Premium at this point would be 1.41. B) Strike = 34, Stock Price = 100 and Delta = 0.92 Then Premium at this point would be 1.39.

Since Case A is more in the money than Case B, the delta would necessarily be closer to 1 in the former than the latter (delta for call tends to 1.00 as its moneyness increases). Similarly, the premium for Case A would also be higher than that for Case B.

In the event that the strike > stock (ie. OTM call), you'd get a negative value for (Stock Price - Strike Price). However, you could still pretty much extrapolate by using the above formula after switching your delta.

Let's consider an OTM call for the following for illustration: C) Strike = 34, Stock Price = 14 and Delta = 0.25. Then Premium at this point would be 0.525. D) Strike = 34, Stock Price = 10 and Delta = 0.15 Then Premium at this point would be 0.354.

In Case C, your actual Delta is actually 1-0.25=0.75 (we compute using the delta from the perspective of an ITM put instead). By taking the absolute value in difference between Strike and Stock (ie.34-14=20), Premium in Case C = 0.525.

Similarly for Case D, your actual Delta would be 1-0.15=0.85. Premium in Case D would then yield 0.354.

Take note that when Strike > Stock Price, you would necessarily need to compute the premium by using the ITM put delta as opposed to the OTM call delta (given). Since the "put delta + call delta = 1" holds at all times (put-call parity), this can be done easily.

Clearly, this would then make sense as the premium increases as the stock price increases...