# Black-Scholes Model

When investing in **binary options**, one important thing to know about is the Black-Scholes Model. Most models and techniques used today by brokers and financial analysts to analyze the value of a given stock option, like binary options, use the Black-Scholes Model. This is a model that was first developed by **Fischer Black** and **Myron Scholes** in 1973. It is the original for most of the options pricing models that are used today.

## Assumptions with the Black-Scholes Model

The actual Black-Scholes Model includes quite an extensive mathematical formula that can be easily seen online or in finance books. Here, we will explore the basic assumptions of the Black-Scholes Model and of how it is used for binary options. First, there is an assumption that the stock doesn't pay dividends during the option's life. Another assumption with the Black-Scholes Model for binary options and others is that the European exercise terms are the ones that are used. American terms often allow the option to be exercised during the life of the option, rather than only on the expiration date.

## More Assumptions with the Black-Scholes Model

Another assumption with the Black-Scholes Model is that markets are efficient. This means that people can't predict with consistency what direction the market, or individual stocks, will go. Another assumption for binary options with the Black-Scholes Model is that there is no commission charged. While market participants usually do pay a commission, the assumption with the Black-Scholes Model is that there won't be one. Two final assumptions are that interest rates will remain constant and known and that returns are normally distributed.

## The Influence of the Black-Scholes Model

This formula for binary options created quite a boom in options trading when it was created - and it even caused the creation of the Chicago Board Options Exchange. It's widely used throughout the market and has been found to be "fairly close" to the observed price of a given stock. Merton and Scholes actually received the 1997 Noble Prize in Economics for their work in this field. Black had already passed away at that time, but was mentioned as a contributor by the Swedish academy. Merton was honored, since he had published a paper as the first to expand on the mathematical understanding offered by Black and Scholes to help people to further understand the pricing model. He actually coined the term Black-Scholes option pricing model which became known as the Black-Scholes Model.

## Background information on the Black-Scholes Model

The Black-Scholes Model is one of the fundamental concepts of modern financial theory. Developed in 1973 by Fisher Black, Robert Merton and Myron Scholes, it is regarded as one of the best ways of determining fair prices of options and is still widely used today. In 1997, Merton and Scholes received Nobel Prize in Economics (The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel) for their work. Black was mentioned as a contributor by the Swedish academy, however, he did not receive the prize because of his death in 1995.

The Black-Scholes mathematical model explains that the price of heavily traded assets follow a geometric Brownian motion that looks like a smile or smirk with constant drift and volatility. When applied to a stock option, the model incorporates the constant price variation of the stock, the time value of money, the option's strike price and the time to the option's expiry. It should be noted that the "time value of money" portion of the formula was significantly altered when the United States' credit rating was lowered from AAA to AA+ by the Standard & Poors and as a result raised the interest rate of something that was regarded as a constant till August of 2011.

The model includes the Black–Scholes formula, which gives the price of European-style options. The formula was so successful that it created an inflation in options trading and thus the Chicago Board Options Exchange was created. The Black-Scholes valuation is largely used by binary options market traders. Trading experience has proven that the Black-Scholes theory is “fairly accurate” to the observed prices, although there are well-known inconsistencies such as the “option smirk”.

The **Black–Scholes Model Theory**

According to their findings, Black, Scholes and Merton believed the following:

**Market has no arbitrage ** - It is impossible to secure a risk free profit. Although there is arbitrage in certain market segments, these are not secure in the long run and relying on them violates that basic needs for Black-Scholes to work.

**Frictionless Lending** - Traders and brokers can borrow and lend cash at a known fixed risk-free interest rate, also known as frictionless lending. This aspect was put into questions in August 2011 as a result of the US Credit Downgrade which made the "risk-free" aspect of binary options a more profitable venture for the trader, and less profitable for the binary options trading houses.

**Asset Fraction-ability** - According to Black-Scholes valuation, it is possible to buy and sell any amount in whole or fractional increments as well as fractional increments that cannot be measured in physical money and including binary options short-selling.

**Fee-less** - Buying or selling should not incur any fees or costs.

**Mature** - The stock price adheres to a geometric Brownian motion with constant drift and volatility, this is typical of mature stocks and not typical of declining stocks or start-ups, where the direction of their price-curve is fairly obvious both in the short term and long term.

** Sans Dividend ** - The underlying commodity, stock, or option does not pay a dividend to its shareholders group being purchased. In cases where preferred stock is provided a dividend and common stock is NOT, then that company's common stock can be used for binary options trading, as long as it adheres to the rest of the requirements.

From these assumptions, Black and Scholes proved that it is possible to create a hedged position in binary options trading. A hedged position consists of a long position in the stock and a short position in the option, whose value will not depend on the price of the stock.

In order to make the model more efficient, several of these original concepts have been removed in updated versions of the model. Modern versions take into account, changing interest rates (Merton, 1976) transaction costs and taxes (Ingersoll, 1976), and dividend payout (Merton, 1973).