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Option Trading Math Part 1 - Delta and Theta

The ancient Greeks are justly praised for inventing much of elementary mathematics. But it was left to moderns to create the tools that help options traders quantify risk and calculate prices. Chief among these tools are several quantities known fondly as The Greeks: delta, theta, gamma and vega.

While the underlying mathematics is heavy going, the basic concepts are simple and can be used by any trader to help measure risk and maximize profits.

The Greeks are based on factors that common sense would suggest affect the price of an option. The determinants are the underlying asset's market price, the option strike price, the time left to expiration, volatility and short-term interest rates. All these pieces of data are readily available and it's clear why they would affect an option's value.

Take the strike price for example. That's the contractually specified price at which the asset, say a stock, would have to be bought or sold if the option were exercised.

Suppose MSFT (Microsoft) were selling at $28 per share and the option considered was a June 31 call. (Note: the '31' refers to the strike price, not the date on which the option expires.) This option is 'out-of-the-money' since the strike price is higher than the current market price.

Clearly, the price of the option itself (the 'premium') will be affected by just how far out-of-the-money the option is. One measure of this difference is the first Greek: delta.

Not a simple difference, the delta is a ratio which compares the change in price of the asset to the change in price of the option. For example, if the delta in the above example were 0.7, for every $1 rise in MSFT the call option can be expected to increase by 70 cents ($0.70).

A trader doesn't need to know how to calculate it, only how to use it. (Any good options trading software will show all four Greeks, along with price, expiration, etc.) Delta tends to increase the closer the option is to expiration for those close to in-the-money. Delta is also affected by changes in implied volatility. (The latter is also frequently provided by trading software.)

Theta measures what is sometimes referred to as the 'time decay' of an option. Since all have an expiration date, and since the less time left the less likely the market price will move in a desired direction, theta is a measure of risk and value.

Suppose that MSFT June 31 call were priced at $3 and the theta were 0.5. Then, in theory, the value of the option would drop by 50 cents ($0.50) per day.

As expiration nears, the price for a premium can be expected to decline at a faster rate. An option with, say, two days left is losing value quicker than one with three months remaining. That change is reflected in the value of theta.



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