To what extent does the change in the volatility impact the value of a Put option? If the volatility increases, does it impact Deep ITM and Deep OTM put options similarly?
Answer 
Answer
The impact of change in volatility on the put option value will depend on whether the option is in-the money, at the money or out of the money. An in-the-money option is one where the option is profitable if exercised. For example, in case of a put options (which the right to sell) the option will be in the money if the market price of the stock is less than the strike price of the contract. On the other hand, the put option will be out of the money (OTM) if the market price of the stock is higher than the strike price. Let us look at simulated comparisons to understand this point better. However, here we are looking at deep ITM and deep OTM puts. A deep ITM put option will be one where the market price is substantially lower than the strike price. A deep OTM put option will be one where the market price is substantially higher than the strike price of the contract.
Let us first look at how the option value of a Deep ITM put is impacted when the volatility goes up by 10 bps from 30% to 40%.
Input Data
Input Data
Stock Price now (P)
90
Stock Price now (P)
90
Exercise Price of Option (EX)
125
Exercise Price of Option (EX)
125
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^0.5
0.0866
s*t^0.5
0.1155
d1
-3.7018
d1
-2.7511
d2
-3.7884
d2
-2.8666
Delta N(d1) Normal Cumulative Density Function
0.0001
Delta N(d1) Normal Cumulative Density Function
0.0030
Bank Loan N(d2)*PV(EX)
0.0094
Bank Loan N(d2)*PV(EX)
0.2583
Value of Put
34.4804
Value of Put
34.4893
(Note - Period is reduced to yearly decimals
Change in Value
0.03%
Let us now turn and look at how the option value of an ATM put is impacted when the volatility goes up by 10 bps from 30% to 40%. An ATM put 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
Stock Price now (P)
125
Exercise Price of Option (EX)
125
Exercise Price of Option (EX)
125
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^0.5
0.0866
s*t^0.5
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 Put
4.0535
Value of Put
5.4869
(Note - Period is reduced to yearly decimals
Change in Value
35.36%
Finally, let us now turn and look at how the option value of a Deep OTM put is impacted when the volatility goes up by 10 bps from 30% to 40%. A Deep OTM put option is one where the strike price is substantially lower than the market price. Check the table below.
Input Data
Input Data
Stock Price now (P)
200
Stock Price now (P)
200
Exercise Price of Option (EX)
125
Exercise Price of Option (EX)
125
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^0.5
0.0866
s*t^0.5
0.1155
d1
5.5185
d1
4.1642
d2
5.4319
d2
4.0487
Delta N(d1) Normal Cumulative Density Function
1.0000
Delta N(d1) Normal Cumulative Density Function
1.0000
Bank Loan N(d2)*PV(EX)
124.4802
Bank Loan N(d2)*PV(EX)
124.4770
Value of Put
0.0000
Value of Put
0.0001
(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 put options, the impact on the value of the put options is very negligible when the volatility shifts upward. That is because the impact of this volatility rise on the time value of the option is very limited. But the impact is much more in the ATM puts as well as in cases where the ITM and OTM put options are near to the money. But, as the put options keep going deep ITM or deep OTM, the relative impact on the value of the put option resulting from a rise in the volatility becomes virtually negligible. This is a very important aspect in understanding options trading.