## Formula to Calculate Implied Volatility Formula?

Implied volatility is one of the important parameters and a vital component of the Black-Scholes model which is an option pricing model that shall give the option’s market price or market value. Implied volatility formula shall depict where the volatility of the underlying in question should be in the future and how the marketplace sees them.

When one does reverse engineering in the black and Scholes formula, not to calculate the value of option value, but one takes input such as the market price of the option, which shall be the intrinsic value of the opportunity. Then one has to work backward and then calculate the volatility. The volatility which is implied in the price of the option is thus called the implied volatilityImplied VolatilityImplied Volatility refers to the metric that is used in order to know the likelihood of the changes in the prices of the given security as per the point of view of the market. It is calculated by putting the market price of the option in the Black-Scholes model.read more.

**C = SN (d**

_{1}) – N (d_{2}) Ke^{-rt} You are free to use this image on your website, templates etc, Please provide us with an attribution linkHow to Provide Attribution?Article Link to be Hyperlinked

For eg:

Source: Implied Volatility Formula (wallstreetmojo.com)

Where,

- C is the Option Premium
- S is the price of the stock
- K is the Strike PriceStrike PriceExercise price or strike price refers to the price at which the underlying stock is purchased or sold by the persons trading in the options of calls & puts available in the derivative trading. Thus, the exercise price is a term used in the derivative market.read more
- r is the risk-free rateRisk-free RateA risk-free rate is the minimum rate of return expected on investment with zero risks by the investor. It is the government bonds of well-developed countries, either US treasury bonds or German government bonds. Although, it does not exist because every investment has a certain amount of risk.read more
- t is the time to maturity
- e is the exponential term

**NOTE:**

One has to work backward in the above formula to calculate implied volatility.

### Calculation of the Implied Volatility (Step by Step)

The calculation of implied volatility can be done in the following steps:

**Gathered the inputs of the Black and Scholes model, such as the Market Price of the underlying, which could be stock, the market price of the option, the strike price of the underlying, the time to expire, and the risk-free rate.****Now, one has to input the above data in the Black and Scholes Model.****Once the above steps are completed, one needs to start doing an iterative search by trial and error.****One can also do interpolationInterpolationInterpolation is the mathematical procedure applied to derive value in between two points having a prescribed value. It approximates the value of a given function at a given set of discrete points. It can be applied in estimating varied concepts of cost, mathematics, statistics.read more, which could be near to the implied volatility, and by doing this, one can get approximate nearby implied volatility.****This is not simple to calculate as it requires care at every stage to compute the same.**

### Examples

#### Example #1

**Assume that at the money call price is 3.23, the market price of the underlying is 83.11, and the strike price of the underlying is 80. There is only one day left for the expiration, assuming that the risk-free rate is 0.25%. Based on the given information, you are required to calculate the implied volatility.**

**Solution**

We can use the below Black and Scholes formula to calculate approximate Implied Volatility.

Use the below-given data for the calculation of implied volatility.

- Call Option Value: 3.23
- Stock Price: 83.11
- Strike Price: 80.00
- Risk Free Rate:0.25%

**C= SN (d _{1}) – N (d_{2}) Ke ^{-rt}**

3.23 = 83.11 x N(d1) – N(d2) x 80 x e^{-0.25%*1}

Using iterative and trial and error method, we can try calculating at Implied Volatility say at 0.3 where the value shall be 3.113, and at 0.60, the value shall be 3.24. Hence the Vol lies in between 30% and 60%.

**Trial and Error Method – Call Price at 30%**

=$83.11*e^{(-0.00%*0.0027)})*0.99260-$80.00*e^{(-0.25%*0.0027)}*0.99227

**=$3.11374**

**Trial and Error Method – Call Price at 60%**

**=$83.11*e**^{(-0.00%*0.0027)})*0.89071-$80.00*e^{(-0.25%*0.0027)}*0.88472**=$3.24995**

Now we can use the interpolation method, to calculate the implied volatility at which it shall exist:

**= 30% + (3.23 – 3.11374)/ (3.24995 – 3.11374) x (60% – 30%)****=55.61%**

Therefore, the implied Vol shall be 55.61%.

#### Example #2

**Stock XYZ has been trading at $119. Mr. A has purchased the call option at $3, which has 12 days remaining to expire. The choice had a strike price of $117, and you can assume the risk-free rate at 0.50%. Mr. A, a trader, wants to compute the implied volatility based on the above information given to you.**

**Solution**

We can use the below Black and Scholes formula to compute approximate Implied Volatility.

Use the below-given data for the calculation of implied volatility.

- Call Option Value: 3.00
- Stock Price: 119.00
- Strike Price: 117.00
- Risk Free Rate: 0.50%
- Time to Expire: 12.00

**C= SN (d _{1}) – N (d_{2}) Ke ^{-rt}**

3.00 = 119 x N(d1) – N(d2) x 117 x e^{-0.25%*12/365 }

Using iterative and trial and error method, we can try calculating at Implied Volatility say at 0.21 where the value shall be 2.97 and at 0.22 the value shall be 3.05, hence the vol lies in between 21% and 22%.

**Trial and Error Method – Call Price at 21%**

- =$119.00*e
^{(-0.00%*0.0329)})*0.68028-$117*e^{(-0.50%*0.0329)}*0.66655 **=$2.97986**

**Trial and Error Method – Call price at 22%**

- =$119.00*e
^{(-0.00%*0.0329)})*0.67327-$117*e^{(-0.50%*0.0329)}*0.65876 **=$3.05734**

Now we can use the interpolation method to calculate the implied volatility at which it shall exist:

**= 21% + (3. – 2.97986) /(3.05734 – 2.97986)x (22% – 21%)****=21.260%**

** **Therefore, the implied Vol shall be 21.26%

#### Example #3

**Assume the stock price of Kindle is $450, and its call option is available at $45 for the strike price of $410 with a risk-free rate of 2%, and there are three months to the expiry for the same. Based on the above information, you are required to compute implied volatility.**

**Solution:**

We can use the below Black and Scholes formula to compute approximate Implied Volatility.

Use the below-given data for the calculation of implied volatility.

- Call Option Value: 45.00
- Stock Price: 450.00
- Strike Price: 410.00
- Risk Free Rate: 2.00%
- Time to Expire: 90.00

**C= SN (d _{1}) – N (d_{2}) Ke ^{-rt}**

45.00** **= 450 x N(d1) – N(d2) x 410 x e^{-2.00% *(2*30/365)}

Using iterative and trial and error method, we can try calculating at Implied Volatility say at 0.18 where the value shall be 44.66 and at 0.19 the value shall be 45.14, hence the vol lies in between 18% and 19%.

**Trial and Error Method – Call Price at 18%**

- =$450.00*e
^{(-0.00%*0.2466)})*0.87314-$410*e^{(-2.00%*0.2466)}*0.85360 **=$44.66054**

**Trial and Error Method – Call Price at 19%**

- =$450.00*e
^{(-0.00%*0.2466)})*0.86129-$410*e^{(-2.00%*0.2466)}*0.83935 **=$45.14028**

Now we can use interpolation method, to calculate the implied volatility at which it shall exist:

**= 18.00% + (45.00 – 44.66054) / (45.14028– 44.66054) x (19% – 18%)****=18.7076**

** **Therefore, the implied Vol shall be 18.7076%.

Refer to the above given excel sheet for detailed calculation.

### Relevance and Uses

Being forward-looking implied volatility, it shall aid one to gauge the sentiment about the volatility of the market or a stock. However, it has to be noted that the implied volatility will not forecast in which the direction an option is leaning towards. This implied volatility can be used to compare with historical volatility, and hence decisions can be made based on those cases. This could be the measure of risk that the trader is putting into.

### Recommended Articles

This has been a guide to the Implied Volatility Formula. Here we discuss the calculation of implied volatility along with practical examples and a downloadable excel template. You can learn more about derivatives from the following articles –

Narinder Kumar says

Thanks for the clarification of IV concept with Excel sheet

Dheeraj Vaidya says

Thanks for your kind words!