InvestorQ : To what extent does the change in the volatility impact the value of a Put option? If the volatility increases, does it impact all put options similarly? # To what extent does the change in the volatility impact the value of a Put option? If the volatility increases, does it impact all put options similarly? Answer 1 year ago

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.

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

 Input Data Input Data Stock Price now (P) 120 Stock Price now (P) 120 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 -0.3800 d1 -0.2597 d2 -0.4666 d2 -0.3752 Delta N(d1) Normal Cumulative Density Function 0.3520 Delta N(d1) Normal Cumulative Density Function 0.3975 Bank Loan N(d2)*PV(EX) 39.8844 Bank Loan N(d2)*PV(EX) 44.0366 Value of Put 6.8345 Value of Put 8.1490 (Note - Period is reduced to yearly decimals Change in Value 19.23%

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 an OTM put is impacted when the volatility goes up by 10 bps from 30% to 40%. An OTM put option is one where the strike price is lower than the market price. Check the table below.

 Input Data Input Data Stock Price now (P) 130 Stock Price now (P) 130 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 0.5443 d1 0.4335 d2 0.4577 d2 0.3180 Delta N(d1) Normal Cumulative Density Function 0.7069 Delta N(d1) Normal Cumulative Density Function 0.6677 Bank Loan N(d2)*PV(EX) 84.2001 Bank Loan N(d2)*PV(EX) 77.7705 Value of Put 2.1747 Value of Put 3.5065 (Note - Period is reduced to yearly decimals Change in Value 61.24%

What can you infer from a comparison of the OTM, ITM and the ATM scenarios? As you move from ITM to ATM to OTM put options, the impact on the value of the put options is much bigger. This is specifically applicable to near money option which is what we have considered in this case. But the impact is much more in the OTM puts that are near the money compared to ITM options that are near the money. As the put options keep going deep ITM or deep OTM, the relative impact on the value of the put option of a shift in the volatility keeps on gradually reducing. This is a very important aspect in understanding options trading.

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