For the writer (seller) of a call option, it represents an obligation to sell the underlying security at the strike price if the option is exercised. The call option writer is paid a premium for taking on the risk associated with the obligation.

There are several types of volatility e.g. historical volatility, implied volatility, the volatility index, intraday volatility, to name but four. Here we will concentrate on one of the most important measures, some would say the important measure, that is implied volatility.

Volatility, of course, is a measure of uncertainty. A high-volatility stock has a greater potential range than a low-volatility stock. But when we talk about the above types of volatility, the measure is a statistical formula that determines the one standard deviation annual distribution.
Implied volatility is the estimated volatility of a security's price. In general, implied volatility increases when the market is bearish, when investors believe that the asset's price will decline over time, and decreases when the market is bullish, when investors believe that the price will rise over time.

Example: If we have a stock trading at £100 with an implied volatility of 15 percent, the options are implying that the stock will be higher or lower by 15 percent within one standard deviation. (One standard deviation equals 68 percent in a normal distribution.) So the stock has a 68 percent probability of being between £85 and £115.

There are several ways to use volatility data to value options. The basic premise is the same with volatility as it is for stocks: Buy low and sell high. The first is to simply compare the implied volatility to the historical volatility.

The theory is that if the historical volatility is greater than the implied, then the option is cheap; if the historical is less than the implied, it is expensive. This can be done using the spot data of the day, but that often presents an incomplete picture.

Understanding Greeks

First, you should understand that the numbers given for each of the Greeks are strictly theoretical. That means the values are projected based on mathematical models. Most of the information you need to trade options - like the bid, ask and last prices, volume and open interest - is factual data received from the various option exchanges and distributed by your data service and/or brokerage firm.

But the Greeks cannot simply be looked up in your everyday option tables. They need to be calculated, and their accuracy is only as good as the model used to compute them. To get them, you will need access to a computerized solution that calculates them for you. All of the best commercial options-analysis packages will do this, and some of the better brokerage sites specializing in options (OptionVue & Optionstar) also provide this information. Naturally, you could learn the math and calculate the Greeks by hand for each option. But given the large number of options available and time constraints, that would be unrealistic. Below is a matrix that shows all the available options from December, January and April, 2005, for a stock that is currently trading at $60. It is formatted to show the market price, delta, gamma, theta, and vega for each option. As we discuss what each of the Greeks mean, you can refer to this illustration to help you understand the concepts.

Delta - Measures the exposure of option price to movement of underlying stock price. The option's delta is the rate of change of the price of the option with respect to its underlying security's price. The delta of an option ranges in value from 0 to 1 for calls (0 to -1 for puts) and reflects the increase or decrease in the price of the option in response to a 1 point movement of the underlying asset price.

As the delta can change even with very tiny movements of the underlying stock price, it may be more practical to know the up delta and down delta values. For instance, the price of a call option with delta of 0.5 may increase by 0.6 point on a 1 point increase in the underlying stock price but decrease by only 0.4 point when the underlying stock price goes down by 1 point. In this case, the up delta is 0.6 and the down delta is 0.4.

Changes in volatility and its effect on the delta

As volatility rises, the time value of the option goes up and this causes the delta of out-of-the-money options to increase and the delta of in-the-money options to decrease.
volatility-and-delta

Gamma - Measures the exposure of the option delta to the movement of the underlying stock price. The option's gamma is a measure of the rate of change of its delta. The gamma of an option is expressed as a percentage and reflects the change in the delta in response to a one point movement of the underlying stock price.

Like the delta, the gamma is constantly changing, even with tiny movements of the underlying stock price. It generally is at its peak value when the stock price is near the strike price of the option and decreases as the option goes deeper into or out of the money. Options that are very deeply into or out of the money have gamma values close to 0.

Suppose for a stock XYZ, currently trading at $47, there is a FEB 50 call option selling for $2 and let's assume it has a delta of 0.4 and a gamma of 0.1 or 10 percent. If the stock price moves up by $1 to $48, then the delta will be adjusted upwards by 10 percent from 0.4 to 0.5.

However, if the stock trades downwards by $1 to $46, then the delta will decrease by 10 percent to 0.3.

Theta - Measures the exposure of the option price to the passage of time. The option's theta is a measurement of the option's time decay. The theta measures the rate at which options lose their value, specifically the time value, as the expiration date draws nearer. Generally expressed as a negative number, the theta of an option reflects the amount by which the option's value will decrease every day.

A call option with a current price of $2 and a theta of -0.05 will experience a drop in price of $0.05 per day. So in two days' time, the price of the option should fall to $1.90.
Passage of time and its effects on the theta

Longer term options have theta of almost 0 as they do not lose value on a daily basis. Theta is higher for shorter term options, especially at-the-money options. This is pretty obvious as such options have the highest time value and thus have more premium to lose each day.

Conversely, theta goes up dramatically as options near expiration as time decay is at its greatest during that period.
volatility-and-theta

In general, options of high volatility stocks have higher theta than low volatility stocks. This is because the time value premium on these options are higher and so they have more to lose per day. The chart above illustrates the relationship between the option's theta and the volatility of the underlying security which is trading at $50 a share and have 3 months remaining to expiration.

Call Option Value Formula =IF(ISBLANK(DividendYield),EXP(-RiskFreeRate * TimeToMaturity) * StrikePrice * (1 - K5) - SpotPrice * (1 - J5),EXP(-RiskFreeRate * TimeToMaturity) * StrikePrice * NORMSDIST(-I5) - EXP(-DividendYield * TimeToMaturity) * SpotPrice * NORMSDIST(-H5))

Put Option Value Formula =IF(ISBLANK(DividendYield),EXP(-RiskFreeRate * TimeToMaturity) * StrikePrice * (1 - K5) - SpotPrice * (1 - J5),EXP(-RiskFreeRate * TimeToMaturity) * StrikePrice * NORMSDIST(-I5) - EXP(-DividendYield * TimeToMaturity) * SpotPrice * NORMSDIST(-H5))

Delta Call Formula = J5*EXP(-DividendYield * TimeToMaturity)

Delta Put Formula =(J5 - 1) * EXP(-DividendYield * TimeToMaturity)

More formulas to follow plus spreadsheets in the Cloud (Google Drive & Zoho) to be used for testing actual situations.
Interactive Spreadsheet:calculate Black Scholes & Greeks
Carry out your own calculations using this zoho interactive spreadsheet. We hope to develop case study examples as a guide using European Call Options for various companies.

To access and manipulate the zoho interactive sheet please click HERE