### Historical Volatility

**Historical volatility** is a measure of how much the stock price fluctuated during a given time period (in past). It is referred to as the asset’s actual or realized volatility.

Traders make use of historical volatility to estimate the future movement, but there is a chance that the future volatility could deviate from the expected value as the factors influencing the price could change. Major fundamental changes could cause the asset price to stray away from the expected historical volatility.

**Calculation**

Historical statistical volatility is calculated as follows:

1) Daily returns are calculated as return = natural log of (t/t1). Where t is the closing price of the present day and t1 is the closing price one day prior.

2) Then standard deviation of these returns is calculated for the desired time period.

3) The standard deviation value is then annualized by multiplying by the square root of 356.

**Example**

Let us calculate the Historical volatility for Nifty futures for a 10 day period.

Symbol | Date | Expiry | Close | Daily Returns |

NIFTY | 5-Jan-18 | 25-Jan-18 | 10573.2 | |

NIFTY | 8-Jan-18 | 25-Jan-18 | 10631.4 | 0.61% |

NIFTY | 9-Jan-18 | 25-Jan-18 | 10646.9 | 0.13% |

NIFTY | 10-Jan-18 | 25-Jan-18 | 10637.05 | -0.05% |

NIFTY | 11-Jan-18 | 25-Jan-18 | 10654.05 | 0.18% |

NIFTY | 12-Jan-18 | 25-Jan-18 | 10686.35 | 0.28% |

NIFTY | 15-Jan-18 | 25-Jan-18 | 10743.3 | 0.56% |

NIFTY | 16-Jan-18 | 25-Jan-18 | 10709.55 | -0.38% |

NIFTY | 17-Jan-18 | 25-Jan-18 | 10791.8 | 0.82% |

NIFTY | 18-Jan-18 | 25-Jan-18 | 10810.8 | 0.26% |

Daily Volatility | 0.36% |

10 day Volatility | 1.15% |

**Step 1. Daily Returns**

The daily returns are calculated using the excel formula = LN (10631.4/10573.2) = 0.61%.

The daily returns are similarly calculated for all the 10 days.

**Step 2. Daily Volatility**

The daily volatility is calculated using the standard deviation function.

Daily volatility = STDEV (0.61%:0.26%) = 0.36%.

**Step 3. 10 day volatility**

10 day volatility = 0.36%*Square root of (10) = 1.15%.

Therefore, over a 10 day period we could expect movement of 1.15% in Nifty Futures in either direction.

To compute the annual volatility, some traders take the square root of 252 (number of trading days in a year), while others take square root of 365 (calendar days in a year).

On the NSE website, daily volatility is multiplied by the square root of 365 to compute the annual volatility.

The daily and annualized volatility for all the F&O stocks is readily available on the NSE website.

## Interpretation of Historical Volatility

- Historical Volatility does not measure direction; it just measures how much the securities price is deviating from its average.
- When a security’s Historical Volatility is rising, or higher than normal, it means prices are moving up and down farther/more quickly than usual and is a sign that something is likely to happen, or has already happened, regarding the underlying security.
- When a security’s Historical Volatility is falling, it implies that that the uncertainty regarding the security has reduced and things are returning back to usual.

Volatility tends to surge when there is heavy price fluctuation in the market. In the above example, Nifty Futures witnessed a correction from 10400 levels all the way till 10050, which was followed by a pullback in the markets back to 10400 levels again. During this phase, volatility increased from 7.6 all the way till 11.9. This was followed by a period of sideways consolidation in the markets during which the volatility cooled from levels of 11.9 to 6.8.

## Implied Volatility

Implied volatility is the expected magnitude of a stock’s future price changes, as implied by the stock’s option prices. Implied volatility is represented as an annualized percentage.

If market participants are willing to pay a high price for options, then that implies they are expecting major movements in the stock price or implied volatility in the near term. On the other hand, if there is no heavy demand for options and trades aren’t willing to pay much for options, then it indicates that market is not expecting significant price movement. Implied volatility is just a way to describe the size of the market’s expectations for stock price movements.

Consider the following stocks and their respective option prices (options with 23 days to expiration):

Stock | 35 Call Price | 30 Put Price | Implied Volatility |

NHPC(33.5) | Rs. 0.7 | Rs. 0.20 | 40.53% |

RCOM(34) | Rs. 2.9 | Rs. 1.45 | 104.49% |

As we can see, both stocks are of nearly the same price. However, same strikes for each stock have different prices. In the case of RCOM, the call and put prices are much higher than NHPC’s options. The reason of this being the 63.96 % difference in implied volatility between the two options. This indicates that the traders expect heavy price fluctuation in the price of RCOM compared to NHPC. Many professional traders consider buying or selling options based on the implied volatility.

Stocks with a high I.V tend to witness major price swings hence the risk-reward ratio is also higher for traders in such stocks. RCOM is clearly the more risky of the two and more rewarding as well.

**Difference between Historical and Implied volatility**

Historical Volatility | Implied Volatility |

Historical volatility is calculated from the previous price movement in the stock. | Implied volatility is derived from option pricing model. |

Can be calculated for any stock | Can be calculated only for stocks that trade in options segment |

Not a reliable estimate of future volatility as the factors influencing price could change. | Better estimate of the future volatility of the stock, takes into consideration the current scenario i.e. based on present demand and supply in options |