The impact of change in volatility on the call option value will depend on whether the option is in-the money, at the money or out of the money. Even within ITM and OTM options, the impact will depend on whether the ITM is deep ITM or near ITM. Similarly, the impact on the OTM options will also depend on whether it is deep OTM or near OTM. An in-the-money option is one where the option is profitable if exercised. For example, in case of a call options (which the right to buy) the option will be in the money if the market price of the stock is more than the strike price of the contract and it will be deep in the money if the market price is substantially higher than the strike price. On the other hand, the call option will be out of the money (OTM) if the market price of the stock is lower than the strike price and deep OTM if the market price is substantially lower than the strike price. Let us look at simulated comparisons to understand this point better.

Let us first look at how the option value of a Deep ITM call is impacted when the volatility goes up by 10 bps from 30% to 40%.

 Input Data Input Data Stock Price now (P) 180.00 Stock Price now (P) 180.00 Exercise Price of Option (EX) 125.00 Exercise Price of Option (EX) 125.00 Number of periods to Exercise in years (t) 0.08333 Number of periods to Exercise in years (t) 0.08333 Compounded Risk-Free Interest Rate (rf) 5.00% Compounded Risk-Free Interest Rate (rf) 5.00% Standard Deviation (annualized s) 30.00% Standard Deviation (annualized s) 40.00% Output Data Output Data Present Value of Exercise Price (PV(EX)) 124.4803 Present Value of Exercise Price (PV(EX)) 124.4803 s*t^.5 0.0866 s*t^.5 0.1155 d1 4.3019 d1 3.2517 d2 4.2153 d2 3.1363 Delta N(d1) Normal Cumulative Density Function 1.0000 Delta N(d1) Normal Cumulative Density Function 0.9994 Bank Loan N(d2)*PV(EX) 124.4787 Bank Loan N(d2)*PV(EX) 124.3737 Value of Call 55.5198 Value of Call 55.5230 (Note - Period is reduced to yearly decimals Change in Value 0.01%

Let us now turn and look at how the option value of an ATM call is impacted when the volatility goes up by 10 bps from 30% to 40%. An ATM call option is one where the strike price and the market price are at the same level.

 Input Data Input Data Stock Price now (P) 125.00 Stock Price now (P) 125.00 Exercise Price of Option (EX) 125.00 Exercise Price of Option (EX) 125.00 Number of periods to Exercise in years (t) 0.08333 Number of periods to Exercise in years (t) 0.08333 Compounded Risk-Free Interest Rate (rf) 5.00% Compounded Risk-Free Interest Rate (rf) 5.00% Standard Deviation (annualized s) 30.00% Standard Deviation (annualized s) 40.00% Output Data Output Data Present Value of Exercise Price (PV(EX)) 124.4803 Present Value of Exercise Price (PV(EX)) 124.4803 s*t^.5 0.0866 s*t^.5 0.1155 d1 0.0914 d1 0.0938 d2 0.0048 d2 -0.0217 Delta N(d1) Normal Cumulative Density Function 0.5364 Delta N(d1) Normal Cumulative Density Function 0.5374 Bank Loan N(d2)*PV(EX) 62.4791 Bank Loan N(d2)*PV(EX) 61.1650 Value of Call 4.5732 Value of Call 6.0067 (Note - Period is reduced to yearly decimals Change in Value 31.35%

Finally, let us now turn and look at how the option value of a Deep OTM call is impacted when the volatility goes up by 10 bps from 30% to 40%. A Deep OTM call option is one where the strike price is substantially higher than the market price. Check the table below.

 Input Data Input Data Stock Price now (P) 80.00 Stock Price now (P) 80.00 Exercise Price of Option (EX) 125.00 Exercise Price of Option (EX) 125.00 Number of periods to Exercise in years (t) 0.08333 Number of periods to Exercise in years (t) 0.08333 Compounded Risk-Free Interest Rate (rf) 5.00% Compounded Risk-Free Interest Rate (rf) 5.00% Standard Deviation (annualized s) 30.00% Standard Deviation (annualized s) 40.00% Output Data Output Data Present Value of Exercise Price (PV(EX)) 124.4803 Present Value of Exercise Price (PV(EX)) 124.4803 s*t^.5 0.0866 s*t^.5 0.1155 d1 -5.0619 d1 -3.7711 d2 -5.1485 d2 -3.8866 Delta N(d1) Normal Cumulative Density Function 0.0000 Delta N(d1) Normal Cumulative Density Function 0.0001 Bank Loan N(d2)*PV(EX) 0.0000 Bank Loan N(d2)*PV(EX) 0.0063 Value of Call 0.0000 Value of Call 0.0002 (Note - Period is reduced to yearly decimals Change in Value Negligible

What can you infer from a comparison of the Deep OTM, Deep ITM and the ATM scenarios? As you move either towards Deep ITM or Deep OTM call options, the impact on the value of the call options very negligible when the volatility shifts. That is because the impact of this movement on the time value of the option is very limited. But the impact is much more in the ATM calls as well as in cases where the ITM and OTM call options are near to the money. But, as the call options keep going deep ITM or deep OTM, the relative impact on the value of the call option resulting from a shift in the volatility becomes virtually negligible. This is a very important aspect in understanding options trading.