Complete Guide to the Black-Scholes Model

Do you ever wonder how hedge funds make money regardless of whether the market goes up or down?

They often use sophisticated strategies and tools like the Black-Scholes model to mitigate risk and hedge their positions constantly. Through this, they have a short-term risk-neutral portfolio, protecting them against massive price movements. Yet at the same time, they’re profiting from the volatility of the securities held in their portfolio. 🧐

Sound too good to be true? Well, it’s not.

The Black-Scholes Model, or the Black-Scholes-Merton (BSM) model, is an options pricing model widely used by market participants like hedge funds to determine the theoretical fair value of an options contract (along with other information) about their relation to the underlying asset.

They then use that information to make their 4D-chess trading moves that always seem to print money and allow them to charge hefty fees to their clients. In theory, a retail trader can use the same tools to get the same results. However, there are some complications. 

First, some limitations come with the model, and its results are not always perfect. Second, it is tough for retail investors to find inefficiencies in the options market. And third, the formula derived from the Black-Scholes equation described in the model looks like this:

But don’t worry! You won’t have to learn another language to understand how the Black-Scholes model works. The concepts are pretty easy to grasp if we get past the math. 

This article takes a very accessible overview of the Black-Scholes model, how it works, what the Black-Scholes formula looks like, how professionals and retail traders use it, and its possible limitations. 

What is the Black-Scholes Model? 📚

Economists Fisher Black and Myron Scholes developed the thesis of the model. They published it in their 1973 paper, “The Pricing of Options and Corporate Liabilities.” Still, it was Robert C. Merton who coined the term in his article “Theory of Rational Option Pricing” and expanded on it. Thus, it’s often also referred to as the Black-Scholes-Merton (BSM) model.

It was revolutionary at the time of publication because it was the first model that allowed traders to predict the value of options at expiration. So instead of just relying on good old “trust and instincts,” the world now had a tool to measure if option contracts were undervalued by the market or overvalued. Additionally, the BSM model allowed traders to hedge their portfolios with more precision than ever seen before.

In 1997, the Nobel Memorial Prize in Economic Sciences was awarded to Myron Scholes and Robert Merton. Unfortunately, as Fisher Black had passed away in 1995, he was not eligible to receive the award for the model that bears his name as Nobel Prizes are not awarded posthumously.

The BSM model has been used by professional traders like hedge funds for decades now for various purposes but most famously for a strategy known as “delta hedging,” which makes their portfolio’s risk profile as close to neutral as possible.

Quick Example: Delta Hedging 👨‍🏫

An option’s delta refers to the correlation between its price and the underlying asset price. 

Call options have a positive delta value (moves up with the underlying asset’s price), and put options have a negative delta value (move down as the price rises). The underlying asset will always have a delta of 1 since it is 100% correlated to its price action.

A fundamental delta hedging strategy can be explained with the following example:

You own 1,000 shares and think it will be very valuable within the following year. Yet, you also think there might be a chance the price will tank in the next few months before going back up again (maybe the CEO posted something on Twitter and is about to get canceled, or maybe it’s a pharma play, but the short-term results will be massively affected by a trial result.) 📅

Since every share has a delta of 1 (i.e., 100% correlation to the underlying asset), your position will have a total delta of 1000 (1 x 1000) based on 1,000 shares. 

In this case, your holdings are entirely reliant on the price action of the underlying asset—if the stock’s price moves up by $1, your portfolio will move up by $1,000, but conversely, if the stock price goes down $1, you lose $1,000 too. 

To establish a risk-neutral portfolio, we need the delta to be 0. In this case, we can do it by buying put options that have a negative delta.

A Closer Look at Delta Hedging 👇

So at this point, you would need to ensure you’re using a trusted options trading broker with the required tools and research capabilities. Once that box is checked, let’s say the put options on the stock have a delta (or volatility) of -0.5 (for every $1 gain in the stock’s price, the option’s value goes down $0.5, and vice versa). You buy 20 put options (each contract representing 100 shares) and the total delta of the options is -1,000 (20 x 100 x (-0.5)). 

Now you have a portfolio that is delta neutral. If the underlying asset’s price keeps going up, you would still maintain a net profitable position (profit from the stock holdings – costs of put contracts used to hedge). If the price tanks in the short term, your put options should be able to cover the loss in the stock holdings and protect you against the loss. 🛡

The BSM model helps us understand many exciting things about an options contract, such as its expected value at the time of expiration and the underlying asset’s implied volatility (more on this in the following sections). 

Suppose you’re worried about not executing or understanding the math behind it. In that case, there are plenty of Black-Scholes calculators that do the calculations for you within a few seconds, and they’re available for free on the internet. However, as you’re about to learn in the following section, the BSM model doesn’t exactly work perfectly. 

Also, just FYI, you’ll probably never actually beat Warren Buffet’s annualized returns with a quadratic equation solver you found on the internet. 

How Does the Black-Scholes Model Work? 🏗

Let’s say you just entered the Matrix, and Morpheus is walking you through for the first time. But, instead of wearing dark sunglasses and cool leather, he’s wearing thick-rimmed glasses, a business suit, and a mild-mannered smile. 

As he walks you through the new world, he tells you to imagine a stock market invented by the machines strictly based on the hypothetical market described by the BSM model (even they were impressed). 

You can trade only three things in this stock market: one risk-free asset that offers a guaranteed but low rate of return, a risky asset with uncertain price movements, and a European-style put option, i.e., a derivative of the risky asset. 📊

This market would be very limited in an ideal world, but in a world run by robots who use humans as batteries, you go along with it.

At this point, nerd Morpheus turns to you and tells you he’s found a way to hack the system, he has figured out how stocks and options are valued by the market precisely. As long as the market is consistent with some assumptions, he realized it is possible to determine the derivative’s payout at a specific date in the future, as long as you know some parameters of the underlying asset leading up to that date. 

He effectively tells you that even though we have no idea if the stock will go up or down, as long as we observe it regularly, we can estimate the payout value of the option on a future date. Essentially, we can peer into the Matrix.

Thus, as long as we constantly revise our portfolio’s delta based on the asset’s changing parameters, we can maintain a neutral delta portfolio and make a profit regardless of the direction in which the stock’s price moves. And that’s how the machines will lose. 

The Assumptions of the Black-Scholes Model

The Black-Scholes Model describes several assumptions about this theoretical stock market. It is essential to know these assumptions as they also limit the applications of the model and, by extension, its usefulness.

Here are the assumptions that the model makes of the hypothetical market:

• ☑ The asset with uncertain price movements is not a dividend-paying stock.
• ☑ The price movements are random.
• ☑ There are no transaction costs (and no front-running either).
• ☑ The risk-free rate asset offers a fixed and known rate of return, and the underlying asset’s volatility is known. 
• ☑ A derivative instrument is a European-style option (i.e., it can be exercised only on the expiration date.)*

* The BSM model has been expanded to price American-style options too. The process and thinking are mainly the same, but the formula differs slightly to account for the differences in instrument type. 

Black-Scholes Model Formula 📝

The Black-Scholes equation is a partial differential equation, which is objectively scarier than just ordinary equations. However, it does something remarkable—it simulates the price movement of an asset over time after taking in some information about the hypothetical market.

At this point, you realize what nerd Morpheus was talking about. If you know today’s information about the market, you can simulate the future price movements using the Black-Scholes equation. 

It looks like this:

And produces simulations like this one:

The equation essentially accounts for randomness (using calculus) and also for several observable aspects of the hypothetical market. The Black-Scholes formula is derived from the equation and essentially tells us the price at the end of the time period. The equation essentially spits out the entire dataset while the formula spits out the last row.

When solved with certain bounds, the formula is derived from the model. It looks like this for valuing call options:

When it comes to determining the value of put options, the formula is then used as follows:

If you want to get into the nitty-gritty of things and the math behind the formula, there are several good resources, but I’m afraid it would very quickly escape this article’s scope. Although for the curious-minded, we do recommend taking a deeper dive.

Coming back to the point, what’s important is that the Black-Scholes formula can quite accurately predict the future price of an option. This is nothing short of playing a video game on god mode in financial markets. 

If you know the option’s fair value at its expiry date, you can always know if the current market is overvaluing it. 

Example of a Black-Scholes Calculation 🧮

So, let’s look at how we can use the BSM model practically. In this example, we try to determine if a European-style option’s payout is undervalued or overvalued by the market. For this, let’s look at ADS.DE, the ticker for Adidas AG. 

At the time of writing, its price is 237.75 EUR. Currently, we can see that a 1-month call option with a strike price of 245 EUR is valued at 5.09 EUR. At first glance, we are not sure if this is a good deal since we barely know anything about the underlying stock (personally speaking.)

However, regardless of our level of knowledge, we can use the BSM model to help us check if this option is overvalued by the market currently or undervalued.

To calculate the future value of an option with the BSM model, we need first to find out the things we can observe—strike price, time to expiration, underlying asset’s price, volatility, dividend yield, and the risk-free return rate. 

For Adidas AG, we find the following information:

Strike-price: 245 EUR

Time to Expiration: 1 month

Underlying Asset’s Price: 237 EUR

Volatility: 26.46% (this information is publically available)

Dividend Yield: 1.26% (the original BSM model did not account for dividend but later variations were developed to account for it too)

Risk-Free Return Rate: -0.08 (Germany’s Risk-Free Return Rate (10 Year Bond Yield))*

*Germany has a negative risk-free return rate currently so we assume the risk-free return to be 0 in this case.

Once we have all the parameters, we can simply plug them into an application like Wolfram Alpha’s BSM calculator and get the fair value of the contract on expiration. Once solved, we get the following results:

Value: 3.88

delta: 0.339

gamma: 0.02

vega: 24.850

theta: -38.897

rho: 6.283

According to the BSM model, the fair value of the option on expiry should be EUR 3.88. Since the call option’s cost was 5.09 EUR, we can make an assumption that it is currently probably overvalued. 

The Uses of the Black-Scholes Model 📘

The BSM model is one of the most critical financial models ever created. It has been extended, expanded upon, dissected, and studied for decades by some of the finest financial minds. 

Its application has also varied as more experiments were conducted. However, the most important uses for the model are the following:

Pricing Options 🎛

The Black-Scholes Model can be extended to price more than just European options. Changing it a little and accounting for other factors makes it possible to find the theoretical fair value of an American-style option, binary option, and even dividend-paying instruments.

Finding Implied Volatility 🔍

Another important use of the Black-Scholes Model is finding the implied volatility for a stock. The BSM model uses observable parameters and then determines the fair value of an option. In the process of doing so, it stimulates the asset’s volatility over time. Thus, we can reverse engineer the model by inputting the option’s current market value, and the model should simulate the implied volatility that the market is currently pricing in.

Implied volatility should not be confused with historical volatility; the latter measures volatility an asset has experienced in the past while the former tells us what kind of volatility the market expects from the stock. So naturally, the information can be beneficial to traders in the context of the option and in estimating the underlying asset’s short-term movements.

The Limitations of the Black-Scholes Model

As stated above, the Black-Scholes model does not come without limitations. For example, initially published, the basic model only works when certain assumptions are held in a market. These assumptions, however, are not found in reality. Thus, the results estimated by the model can often be imprecise. 

Some of the limitations of the BSM model include:

• ☑ It does not account for trading costs and fees.
• ☑ It assumes the risk-free return to be consistent throughout the period.
• ☑ It cannot account for wild swings and follows a random-walk simulation model for prices.
• ☑ It does not account for taxes.

We should also note that the BSM model was initially conceived in 1973—a year before Richard Nixon resigned as the US President on live TV due to the watergate scandal, and only four years after the Apollo 11 moon landing. 

Since then, the world has changed a lot, and we have more computing power on a smartphone than the best computers back then. More advanced models include the Heston model, a stochastic volatility model used to price options.

Conclusion 🏁

Math is hard, but it can be profitable. And it’s not even that hard if you made it this far. The BSM model cannot predict the future 100% accurately, but it is a beautiful application of mathematics in finance. 

While it might not start printing money immediately, learning more about these theories helps us get a better understanding of the markets and also get better at making those 4D chess moves occasionally. ♟

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