When you wake up in the morning, do you know what the weather will be like today? Will it be sunny or will it rain? Just by walking outside, you would have a hard time making a prediction. This is because when you stand on the ground you cannot see beyond the horizon. You cannot see into the future. The sky may be blue where you are, but just over the horizon there may be a big thunderstorm headed your way. How can you decide if you should plan your outdoor picnic for later in the day?

The Black-Scholes Option Pricing Model

The Greeks is derived from the Black-Scholes Theoretical Options Pricing Model (1973). The model is a form of an improved version of a previous model developed by A. James Boness in his Ph.D. thesis at the University of Chicago. Since then, the Black-Scholes Option Pricing Model has been the subject of much attention. The original work continued to be improved by other scholars; in 1973, Robert Merton improved the assumption of no dividends. In 1976, Jonathan Edwards "Jon" Ingersoll, Jr. went one step further and improved the assumption of no taxes or transaction costs. In 1976, Merton responded by removing the restriction of constant interest rates.

Black scholes model’s assumptions explained:

The First Assumption

Stock pays no dividends during the option contracts life (this assumption is imperfect because most companies pay dividends; a common way of adjusting the model for this situation is to subtract the discounted value of a future dividend from the stock price).

The Second Assumption

European exercise style used; meaning, an option can be exercised on the expiration date. The American exercise term allows the option to be exercised at any time during the options life, making American options more valuable due to their greater flexibility. This imperfection is negligible because option contracts are rarely exercised before the last few days of their life (a trader that wants to exercise will consider the time value).

The Third Assumption

Markets are efficient; the market operates in continuous time manner (neither start nor end of day trading, twenty-four hours a day seven days a week).

The Fourth Assumption

No commissions this can lead to a distortion to the model’s output because usually market participants do have to pay a commission buying and/or selling options contracts.

The Fifth Assumption

Interest rates remain constant until the end of the option’s life contract. In reality, U.S. Government Treasury Bills with 30 days left until maturity are usually used to represent the risk-free interest rate. Over the course of the option’s life (there are long-term options called LEAP® options that even trade for three years) interest rates change occasionally, therefore violating one of the assumptions of the model.

The Sixth Assumption

Statistically prices are lognormal distributed. This assumption suggests underlying stock prices are normally distributed.

As mentioned, the Greeks are derived from the Black-Scholes Options Pricing Model, which helps to calculate a fair market value of an option contract. Greeks are used to measure the risks of taking options contract positions.

There are five Greeks: delta, gamma, theta, vega, and rho.

Each Greek measures a different aspect of the risk in an option position, and corresponds to a parameter on which the value of an instrument is dependent.

Theta = the effects of passage of time.

Delta = the effect of changes in the underlying security price.

Gamma = the effect of changes in the underlying with respect to the rate of change of delta.

Vega = the effects of volatility.

Rho = the effects of interest rates

Developed in 1973 by Fisher Black, Robert Merton, and Myron Scholes, the Black-Scholes Option Pricing Model is still widely used today. It is regarded as one of the most accurate ways to determine a fair price of an option. The formula uses the stock’s current share price, the option strike price, time to expiration, risk-free interest rate, and volatility to calculate the value of an option. 
To better understand it, the Black-Scholes Model can be divided into two parts. The first part, SN(d₁), calculates the anticipated profit from acquiring a stock. This is found by multiplying the stock price [S] by the change in the call premium with respect to a change in the underlying stock price [N(d₁)].
The second part, Ke(-rt)N(d₂), gives the present value of paying the exercise price on the expiration day.

The difference between these two parts is the fair market value of the call option.
The Black-Scholes Option Pricing Model is based on a number of assumptions, which are as follows.

• · Markets are efficient. The stock price obeys a Geometric Brownian motion with unchanging volatility and drift.
• · It is possible to buy a fraction of a share of stock.
• The stock pays no dividends during the option’s life.
• There are no restrictions on short selling.
• · European exercise terms are used. European exercise terms state that the option can only be exercised on the expiration date. American exercise terms permit the option to be exercised at any time during the life of the option. This restriction is not a major concern because very few calls are ever exercised before the end of their life.
• · Risk-free interest rates remain both known and constant. Such rates do not exist in reality and must be approximated. In recent years, this assumption has been lifted.
• · There is no opportunity for arbitrage.
• No commissions or taxes are charged. Of course, option buyers must usually pay a commission. The fees that individual investors pay can sometimes affect the output of the model.

The Black-Scholes Option Pricing Model can be used successfully when investors recognize its limitations and use a powerful option software such as Asio Information Tools.

Another point to look at, You go inside and turn on the weather channel. And there you see the big picture. You can see if there are rainclouds headed your way. You can even see an invisible low-pressure system that is bringing warm weather. With this information, you can decide whether to go on a picnic today or plan to eat inside. 
The tools that have allowed you to see over the horizon and predict future events are the powerful computers and software programs that weather forecasters have today. They can take an enormous amount of data, process it, and, based on historical patterns, make reasonable predictions. Of course, no prediction is foolproof; that big storm headed your way may dissipate before it reaches you. But over time, these complex programs provide a high level of accuracy.
The stock market and option trading are much the same. There are historical patterns that are known and accessible to anyone. But to make sense of these patterns and to factor in a multitude of current events is impossible for one person. It is like standing in your front yard and trying to predict the weather!
Many factors influence the value of an option, including the price of the asset today and its price history in the past. Then, of course, there’s the future—whether the stock will rise or fall in price. When trading options, you need to be able to access historical information and make sense of a nearly infinite number of forces that influence option prices. A good way to do this is with powerful software from Asio Information Tools. But before the days of home computers, investors struggled to devise a simple formula that would help them calculate the future value of a stock option.