volatility input for black scholes formula

Question

I am not a mathematician but want to try and understand the BS model for option pricing. I get the intuitive sense of it but am unable to figure out calculation of volatility (as an input). Some online sources indicate taking a time series of log returns of the underlying asset and calc mean and SD and use that. But if my option has and expiry of $T+1$ and $T+2$ months, I am pretty sure I can't use the same volatility input. So is there a rule of thumb / papers that indicate how many historical data points are needed for options of different maturities (and the same strike price)? kindly let me know. Appreciate it!

Answer 1

The best authority I have seen on this stuff is Natenberg: Option Volatility and Pricing. I can't do much better than check my copy. He says: "Note that there are a variety of ways to calculate historical volatility, but most methods depend on choosing two parameters, the historical period over which volatility is to be calculated, and the time interval between successive price changes.

The historical period may be ten days, six months, five years, or any period the trader chooses. Longer periods tend to yield an average or characteristic volatility, while shorter periods may reveal unusual extremes in volatility. To become fully familiar with the volatility characteristics of a contract, a trader may have to examine a wide variety of historical time periods.

Next, the trader must decide what intervals to use between price changes. Should he use daily price changes? weekly changes? monthly changes? Or perhaps he ought to consider some unusual interval, perhaps every other day, or every week and a half. Surprisingly, the interval which is chosen does not seem to greatly affect the result. Although a contract may make large daily moves, yet finish a week unchanged, this is by far the exception. A contract which is volatile from day to day is likely to be volatile from week to week, or month to month."

So what happens in practise is weighting a series of volatilities over different time periods, as volatility exhibits serial correlation. To paraphrase the book:

For example, suppose we have the following historical volatility data on a certain underlying instrument:

• last 30 days: 24%
• last 60 days: 20%
• last 120 days: 18%
• last 250 days: 18%

Certainly we would like as much volatility data as possible. But if this is the only data available how might we use it to make a forecast? One method might be to take the average volatility over the periods which we have:

• (24% + 20% + 18% + 18%) / 4 = 20.0%

However, since the 24% over the past 30 days is more current than the other data, perhaps it should play a greater role in a forecast

• (40% * 24%) + (20% * 20%) + (20% * 18%) + (20% * 18%) = 20.8%

Further, the volatility over the past 60 days ought to be more important than that of the last 120 days, and the last 120 more important than the last 250 and so on. So we can factor that in using a regressive weighting. For example

• (40% * 24%) + (30% * 20%) + (20% * 18%) + (10% * 18%) = 21.0%

The serial correlation is used such that if the volatility on a contract over the last four weeks was 15% then the volatility over the next four weeks is more likely to be close to 15% rather than far away. Once we realise this we give different weights to different past volatility time periods. This has led theoreticians into the ARCH and GARCH models. The book goes on:

Once we have historical volatility then you take another measure for the implied volatility already priced into the market. You might weight implied volatility anywhere between 25% to 75%. For example, suppose a trader has made a current volatility forecast of 20% based on historical data and the implied volatility is currently 24%. If the trader decides to give the implied volatility 75% of the weight, his final forecast will be:

• (75% * 24%) + (25% * 20%) = 23%

A PRACTICAL APPROACH

No matter how painstaking a trader's method he is likely to find that his volatility forecasts are often incorrect, and sometimes to a large degree. Given this difficulty, many traders find it easier to take a more general approach. Rather than asking what the correct volatility is, a trader might instead ask, given the current volatility climate, what's the right strategy? Rather than trying to forecast an exact volatility, a trader will try to pick a strategy that best fits the volatility conditions in the marketplace. To do this a trader will want to consider several factors:

1. What is the long term mean volatility of the underlying contract?
2. What has been the recent historical volatility in relation to the mean volatility?
3. What is the trend in recent historical volatility?
4. Where is implied volatility and what is its trend?
5. Are we dealing with options of longer or shorter duration?
6. How stable does the volatility tend to be?