# Options Trading: A Mathematical View

Providing a trading edge is clearly an important objective of every trading plan. However, it pays to remember that first and foremost, trading is all about managing risk. When you trade options, you're dealing with the risks of three variables: price movement of the underlying asset, changes in volatility and time decay.

To measure and effectively manage risk when trading options, we need a computerized option-pricing program that calculates and displays the "Greeks" for each option. The Greeks (Delta, Gamma, Theta and Vega) are parameters that describe how an option is expected to behave. These are the vital signs of an option. They are dynamically interrelated and change as time to expiration draws closer, as underlying asset price moves higher and lower, and as volatility expectations change.

The Delta of an option is the factor that shows how much the option will change in price as the underlying asset changes in price. Gamma shows how much the option's Delta will change as the underlying asset changes in price. Theta shows how much value the option loses each day. And Vega shows how much the option's price is affected by changes in implied volatility.

Professional floor traders implement trading approaches on the basis of these Greeks. As a retail options trader, you can do the same. Once you specify your price and volatility expectations, you know which options strategy to initiate.

For example, when implied volatility is high, the strategy of selling option premium is logical, because implied volatility tends to revert toward its long-term average. Therefore, you capture profits as expensive option premiums decline to more reasonable levels. In addition, time decay (Theta) works in your favor with each passing day you hold your short options.

The risks of this strategy are that option premiums become more expensive if implied volatility increases (Vega risk) and your position becomes unbalanced if the underlying asset moves (Delta risk and Gamma risk). The important point, however, is that a computer can quantify these risk factors, and you can manage the relative risk among them by adjusting your positions.

Understanding the relative impact of the Greeks on positions you hold is indispensable. Below, I have listed the more salient mathematical relationships of these Greek variables:

- The Delta of an at-the-money option is about 50%. Out-of-the-money options have smaller Deltas, and they decrease the further out-of-the-money you go. In-the-money options have greater Deltas and they increase the further in-the-money you go. Call Deltas are positive and put Deltas are negative.
When you sell options, Theta is negative and Gamma is positive. This means you make money through time decay, but price movement is undesirable. So the profits you're trying to earn through option time decay when you sell puts and calls may never be realized if the stock moves quickly in price. Also, rallies in price of the underlying asset will cause your overall position to become increasingly Delta-short and to lose money. Conversely, declines in the price of the underlying asset will cause your position to become increasingly Delta-long and to lose money.
When you buy options, Theta is negative and Gamma is positive. This means you lose money through time decay, but price movement is desirable. So profits you're attempting to earn through volatile moves of the underlying stock may never be realized if time decay causes losses. Also, rallies in price result in your position becoming increasingly Delta-long, and declines result in your position becoming increasingly Delta-short.
Theta and Gamma increase as you get close to expiration, and they're greatest for at-the-money options. This means the stakes grow if you're short at-the-money, because either the put or call can easily become in-the-money and move point for point with the equity. You can't adjust quickly enough to accommodate such a situation.
When you sell options, Vega is negative. This means that if implied volatility increases, your position will lose money, and if it decreases, your position will make money. When you buy options, Vega is positive, so increases in implied volatility are profitable and decreases are unprofitable.
Vega is greatest for options far from expiration. Vega becomes less of a factor, while Theta and Gamma become more significant as options approach expiration.

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The Greeks covered in this article are even more important if the options trader plans on adjusting his or her original position. Practice following these mathematical relationships and applying these trading principles to help achieve some degree of consistency in your trading results.

Happy trading.

By Jeff Neal, contributing writer and options strategist at

Optionetics.com. Email him at

jeff@optionetics.com.