Black Scholes Formula Explained

Black Scholes Explained: In this article we will explain how Black Scholes is the Theoretical Value of an Option. In financial markets, the Black-Scholes formula was derived from the mathematical Black-Scholes-Merton model. This formula was created by three economists and is widely used by traders and investors globally to calculate the theoretical price of one type of financial security.

European Options Defined

Just like American options, European options have two types, calls and puts. With European options however, investor’s right to buy or sell may only be exercised on the option’s expiration date. For example, a European call option is a contract where the seller gives the buyer the right to sell – within a time period – a number of shares at a predetermined price. While it is a right of a seller, it is not an obligation.

Where are European options traded? They are traded among individual and institutional investors, as well as professional traders. The trades may be for single or multiple contracts.

Options are referred to as derivatives for a reason; their price is the result of an underlying investment’s value. The most common underlying investments on which option prices are based on are publicly listed company equity shares. Other underlying investments (or “securities”) include:

  1. ETF’s (Exchange Traded Funds)
  2. Stock indexes
  3. Government securities
  4. Foreign currencies
  5. Commodities

Now, there are five features in an option contract:

  1. Type of the option – puts and calls
  2. Underlying security
  3. Number of shares
  4. Strike price
  5. Expiration Date

Let’s get a quick refresher on what put and call options are.

A put option is a contract that gives the buyer the right, but not the obligation, to “sell” a number of units of a security at a specified price (or the strike price) within a fixed period of time (in the case of European options, the buyer can only exercise the option on the expiry date).

A call option on the other hand gives the seller the right, but not the obligation, to “buy” a number of units of a security at a strike price within a fixed period of time (again, in the case of European options, the buyer can only exercise the option on the expiry date).

There are various differences between an American and a European option in terms of the underlying investments, exercise rights, settlement price, and trading of index options. To keep our focus on the Black-Scholes formula, a European call option’s underlying investments mostly involve major broad-based indices. Secondly, as mentioned previously the owners of European-style options exercise their right only at expiration. Lastly, European trading index options stop one day earlier, unlike an American option where trading ceases on the third Friday of the expiration month.

Now, why do European options matter? This type of option is usually traded at a discount because there is only one single opportunity to exercise the option. Unlike an American option where the holder can exercise his right at any time before expiry, a European option requires the holder to wait until maturity. For example, when you, an investor, buy a European call option on June 1, which expires on the third Friday of June – you cannot exercise your right to buy the underlying asset when its value goes higher than the strike price by the second week. You cannot take the opportunity to buy the underlying security at the strike price that’s supposedly cheaper than current market value because you can only exercise your right to buy on the expiration date. The one advantage a European option holder still possesses is his ability to sell his option without waiting for the expiration date.

Now Back To The Black-Scholes Formula

Three economists developed the formula in the 1970’s based on the principle that a stock either rises or falls in price in the same predictably unpredictable manner. This idea is still popular today, and the formula is widely used in global financial markets. The introduction of this formula paved the way for a rapid increase in options trading. Its main goal: get a theoretical price estimate of European-style options.

American economist Fischer Black studied monetary policy around 1970. He concluded that basing on the capital asset pricing model, discretionary monetary policy will not provide any good that the Keynesians (who believe there is a natural tendency of the credit markets to be unstable) expected it to do.

Canadian-American financial economist Myron Scholes was as a professor at the MIT Sloan School of Management. That’s where he met Fischer Black and began a research on asset pricing. Scholes also worked with Chicago’s Center for Research in Security Prices where he helped and analyzed their high frequency stock market’s popular database.

The third economist involved in this formula is Robert C. Merton. This American economist is also recognized for translating the science of finance into practice. He was also known for his development of a pension-management solution system and for first publishing a paper that expands the mathematical understanding of the options pricing model for which he coined the term “Black-Scholes options pricing model”.

Back then the Black-Scholes equation enabled pricing when an explicit formula wasn’t available. With the many available financial calculators online today, using the formula has been easier. Investors only have to input the needed figures to arrive at the theoretical estimated price of a European option. In essence, the formula helps investors calculate the theoretical price of European put and call options.

Purpose Of The Black-Scholes Model

Prior to the popularity of the model, investors found it difficult to assess (1) whether an option contract was priced accurately or not, and (2) whether it represented a good value or not. Naturally, investors would love to enjoy the benefits of an underpriced or overpriced underlying asset in order to take advantage of arbitrage opportunities. Because it was too risky to be in the market with unpredictable prices, there weren’t too many investors and traders that were interested in options. However, when the Black-Scholes formula was developed, there was a rise to the idea that it is possible to make a perfect hedging condition by combining option contracts and the underlying security when the contracts are priced correctly. In conclusion, the theory of the formula is that the trading price of an option can be calculated mathematically and that there is only one accurate price for an option. With the formula, the trader or investor can determine if the market price is higher or lower than its theoretical value.

Option Party Black Scholes Explanation

The Black-Scholes Formula Illustrated

The Black-Scholes Model calculates the theoretical price of an option using six factors:

  1. Whether the option is a call or a put.
  2. Current stock price.
  3. Strike price.
  4. Volatility of the underlying security.
  5. Time remaining until maturity.
  6. Risk free interest rate.

The modern formula looks like this for calls:

Black Scholes Call Formula

And like this for puts:

Black Scholes Put Formula

  • C stands for Call Premium.
  • P stands for Put Premium.
  • S is the stock price.
  • K is the option strike price.
  • T – t is the time remaining until expiration (or maturity).
  • r is the risk free interest rate.
  • e represents the irrational number that’s often called Euler’s number.
  • N is cumulative standard normal distribution.

As you can see, the formula is composed of two parts.

  1. Black Scholes Formula Part 1
    • Multiplies the stock price by the sensitivity in the call premium of the change in the underlying price.
  2. Black Scholes Formula Part 2
    • Represents the current value of paying the exercise price of the option on expiration day.

Subtracting the two parts of the equation provides us the value of the option.

Now, how do we get the values of d1 and d2?

Black Scholes Put Formula

Where:

  • S is the stock price.
  • K is the option strike price.
  • T – t is the time remaining until expiration (or maturity).
  • r is the risk free interest rate.
  • In is the natural logarithm.
  • σ is the annual volatility of the stock.

The volatility of the stock price is one of the most important factors for option pricing. The reason for that is because the option price is easily affected by volatility changes. Since volatility is not easy to determine, it is often estimated. However, there are online tools you can utilize; like volatility calculators that automatically retrieve historical or implied volatility data.

Investors should note that very often the formula is used to derive the implied volatility (σ). Since investors already see the market price of the option and all other variables, they can input all variable into the formula except the implied volatility. Then the unknown variable plug becomes implied volatility. And the way investors check if the price is fairly valued is if their estimate of volatility equals to the implied volatility which the formula provides.

The formula is obviously intimidating and looks complex. Luckily, we don’t have to go through the arduous process of calculating the option price manually because we are now equipped with numerous calculator tools on the Web that do the job for us. As long as you have the figures for the variables that the formula requires, you can quickly get the price of your put or call option.

Applying The Black-Scholes Pricing Model Into Practice

The Black-Scholes model has indeed played a crucial role in the way we interact with the financial markets. Like we said earlier, option trading became more alive after the model was introduced by the three economists. If it did not make any difference, then you wouldn’t even bother reading this post. Nowadays, option trading is fully established. The model is still widely used by traders today.

We are not expected to be fully dependent on it. Just like any financial model, it has advantages as well as limitations. A good advice for investors is to do their own research. Weigh both pros and cons of any financial model that you contemplate on using as you trade or invest in the market. Use the model to assist you in assessing whether or not a possible trade is worth it and if you agree with the implied volatility the formula calculates.

What are the things that you should know before using this model?

  1. You don’t really have to memorize the formula, or even understand it in great detail. Besides, who has the time to dig deeper into the model? Rather than manually calculating the values yourself, you can find a Black-Scholes model calculating tool online to save time. It’s easier and quicker.
  2. The model has underlying assumptions and implications you should know about.
    • Again, the option can only be exercised upon expiration.
    • The price of the underlying security may unpredictably go up or down, while the model assumes returns are normally distributed.
    • The underlying security pays no dividends.
    • The current market interest rate during the period of the contract remains constant.
    • Another thing is that “immediate benefits” are neither gained by the buyer nor the seller.
  3. Looking at the formula, we see that there are a few relationships involved.
    • First, the time left until the expiration date to the value of the call or put option. Simply put, the value of the option will be higher (in most cases) when you have more time before the expiration date. That’s a time value element. However, since the option can only be exercised on the expiry date, there are rare cases when longer term European options may be cheaper than shorter term ones. This is because investors would be happy to exercise the option sooner.
    • Second, the higher the volatility, the more expensive the option price will be.
    • The more that the option is in-the-money the more expensive it is (i.e. the higher S – K is for calls or K – S for puts).
    • Higher market interest rates translate to more expensive option prices.

Final Words:

The Black-Scholes Model is just one of the many models you can use to calculate the theoretical value of an option. You can use it anytime as you practice your strategies in the financial markets both as a trader and an investor.

Some advantages of this model include:

  1. Perhaps the best advantage this model provides investors is speed, because it allows you to calculate various prices of European put and call options in a short time span.
  2. It lets investors make better informed investment decisions when it comes to investing in European options.
  3. Another advantage is historical; this model, effectively, brought options trading to life.

The limitations of this model include:

  1. That assumptions are not always realistic (example: interest rates may not stay the same throughout the option period).
  2. A lot of stocks pay dividends, so the formula cannot be used for those.
  3. Can only be used for European options.
  4. The underlying stock’s returns are assumed to be normally distributed, which is not always the case.

We wish you the best with your strategies should you decide to use the Black-Scholes Model in your wealth management practices.