**First, a bit of theory. What is volatility ?**

In mathematical finance, volatility is a measure of “irregularity” of a price movement. It is important to note that volatility does not measure the magnitude of the price movement per se (a stock can go up every day with a constant speed and have zero (historical) volatility), but rather the magnitude of the change of the velocity of the price movement (if a stock moves up and down and up and down 10-15% a day, then its volatility is high even though its performance over a 1-month period may be zero). In other words, *volatility is a measure of second variation* (and not first variation)

The mathematical formula for calculating volatility (*historical volatility*, to be more precise) is as follows:

Assume that you have a time series A_n (for example, A_n = the closing price of the SP500 index at day n), and you want to calculate its volatility for the period A_N, A_{N+1}, …, A_{N+20} (the 20-day historical volatility of day N+20).

First you calculate the speeds of change:

X_i = log (A_{N+i}/A_{N+i-1}) (i=1,…,20)

Then you calculate the standard deviation of the sequence X_i:

sigma = sqrt[ sum (M-X_i)^2 / (20-1)]

where M is the mean value of X_1,…,X_20

By definition, sigma (or sigma times C, where C is certain constant) is called the volatility of A_n (more precisely, 20-day historical volatility at time N+20). In case of daily stock prices, the constant C is by convention the square root of the number of trading days in a year: C = sqrt(262), i.e. the volatility is annualized.

**Volatility and prices of options**

Volatility is one of the main variables which determine the price of the options of a stock, via the Black-Scholes formula or refined formulas (the other main variables and parameters are: the strike price, the price of the stock, time to exipiry, and inerest rate). However, the volatility used here must be the volatility of the future, and not the past. Though the future volatility is related to the past volatility, it can be very different.

Since no one knows what the future volatility will be, Black-Scholes formula does not give us a direct way to calculate the price of options (but it provides a coherence in options pricing for options of different strike prices and expiry dates). However, this formula can be used (via the inverse function theorem) to calculate the volatility once you know the option prices ! Volatility calculated in this way is called *implied volatility*.

**Example: Huron (Nasdaq: HURN)**

The following information about HURN and its historical and implied volatility is taken from the website ivolatility.com (the date of these data is 22/Aug/2009):

Price |
Change (%) | 52 wk High | 52 wk Low | Stock volume | Avg. options volume | Avg. options open interest |

19.00 | -0.48 (-2.46%) | 65.63 09/02/2008 | 11.99 08/12/2009 | 2,377,580 | 18,250 | 50,240 |

You may notice that the historical volatility of HURN was relatively low a month ago (about 22-25%, depending on how many days you use to calculate it), but currently it is extremely high (due to a crash that HURN suffered just 3 weeks ago: it lost 70% in one day).

There is a strong correlation between historical and implied volatility. When the historical volatility is low then the implied volatility is usually also low, and when the historical volatility shoots up, then the implied volatility also jumps up (people are expecting the “aftershocks” of an “earthquake”). When the historical volatility is low, then the implied volatility is usually higher then the historical volatility, but when the historical volatility is very high, then the implied volatility is lower than the historical volatility. The following volatility chart will illustrate this point:

**Playing with decreasing implied volatility**

Look at the right side of the above chart. You’ll notice that, over the last week, while the historical volatility is at the highs, the implied volatility drops very significantly, from about 130% to about 85%.

This is a typical situation which happens to many stocks: after a short period of very high implied volatility (after some very important news which change the outlook of the stock completely, or when some very big news are expected) the volatility decreases when no more big news are on the horizon and the trading volume dries up.

What does this decrease in implied volatility do to options prices ? A lot ! A change of about 10% in implied volatility may imply a change of 10-20% in the price of near-the-money options (all the other things being equal). When the implied volalility drops from 130% to 85% during the last 4 days (from Tuesday 18 to Friday 21), the prices of some next-month near-the-money options lose nearly 50% even though the stock price has not change much in total !

When you are expecting the implied volatility to drop, the sensible thing to do with options is to SELL them. You still have to choose which options to sell: call or put, what strike, what expiry dates, … Not getting into the details, let me just say that if you think that the stock has a very strong support and has an non-zero chance of moving up big, then it is safer to sell puts than calls. That’s what I did with HURN.

1st trade: Sold HURN Put September 17.5 on Aug 18 at 13h41 (NY time) for 2.00 (HURN price at that time: 18.50)

2nd trade: Sold HURN Put September 17.5 on Aug 19 at 10h42 for 1.30 (HURN price at that time: 19.90)

Current price of this put: 1.00-1.10 (current HURN price: 19.00)

Notice that even though the stock went down 90 cents from where I sold my put in the second trade on Wednesday Au 19, the price of my put went down 20-30 cents too, making my trade a winner. (Usually when the price goes down, you would expect the puts to go up, but in this case both puts and calls go down !)

The above trades are conservative (the puts are out of the money, and I wouldn’t mind buying HURN at 17.5) and relatively low-risk. But of course, there are no risk-free trades, and even when you have a very good analysis of the market, you still need a very good risk management in order to survive. Always remember that risk management is the most important thing in trading. More about that later.

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