The Trillion Dollar Equation - Black-Scholes Model

Mathematics and Random Walks Transform Finance and Risk Management

Derivatives, a multi-trillion dollar industry, originate from physics and mathematics, with the discovery of atoms, heat transfer, and blackjack strategies. The Medallion Investment Fund, founded by mathematics professor Jim Simons in 1988, demonstrates the power of mathematical models, delivering 66% annual returns for 30 years. In contrast, Sir Isaac Newton, a renowned physicist, lost a significant portion of his wealth investing in the South Sea Company due to his inability to predict market behavior, famously admitting, "I can calculate the motions of the heavenly bodies, but not the madness of people."

Louis Bachelier, born in 1870, pioneered the use of mathematics to model financial markets. At the Paris Stock Exchange, Bachelier observed the trading of options, which date back to ancient Greece, and found that no one had developed a consistent method for pricing them. He applied his knowledge of probability and proposed a PhD thesis on the topic. Bachelier's primary insight was that stock prices, influenced by numerous factors, are virtually impossible to predict accurately. Thus, over the long term, stock prices follow a random walk, moving up or down based on the flip of a coin. This randomness is a hallmark of efficient markets, making it difficult for traders to predict prices and earn consistent profits.

Bachelier's Efficient Market Hypothesis posits that if you and others could predict the stock market, you would trade based on those predictions today, driving up (or down) the stock price, which would eliminate the predictive power of your initial prediction for tomorrow. This concept can be visualized using a Galton board, which simulates the random walk of stock prices and the unpredictability of individual ball paths, while still maintaining a predictable pattern when considering all balls together—a normal distribution. Bachelier applied this concept to stock prices, arguing that each time step represents an additional layer of pegs, with the stock price able to move only a little up or down in the short term but capable of a wider range after more time.

From Card Counting to Option Pricing: Ed Thorpe's Innovations in Finance - He kept beating the house, and they kept changing the rules.

Ed Thorp received his Ph.D. in mathematics from the University of California, Los Angeles in 1958, and worked at the Massachusetts Institute of Technology (MIT) from 1959 to 1961. He was a professor of mathematics from 1961 to 1965 at New Mexico State University, and then joined the University of California, Irvine where he was a professor of mathematics from 1965 to 1977 and a professor of mathematics and finance from 1977 to 1982.

Edward O Thorp, a young physics graduate, became famous for inventing card counting in blackjack in the 1950s. By keeping track of the cards played, he could adjust his bets based on the odds, making a fortune before casinos caught on and changed the rules. Thorpe then wrote a book "Beat the Dealer" on his technique, which I and a lot of other MIT grads read and went to the NJ casinos to get rich. They made a movie "21" about them.

He founded Princeton Newport Partners (PNP), in 1974, which he stated to be the world's first market neutral hedge fund making money in both up and down markets. The company was a pioneer in quantitative trading techniques, profiting from mispricings in derivatives, and later statistical arbitrage, which involved trading a large number of stocks for short-term returns. PNP achieved an annualized rate of return of 20 percent after fees for over two decades, without a single down quarter, until becoming embroiled in the junk bond schemes of Michael Milken's circle at Drexel Burnham Lambert. Thorp and other principals at PNP were eventually cleared of wrongdoing, but the financial burdens imposed by the ensuing Racketeer Influenced and Corrupt Organizations Act investigation forced PNP to liquidate. His circle of associates later regrouped as TGS Management, with focus on statistical arbitrage.

Thorpe's innovations laid the foundation for the Black-Scholes-Merton option pricing model, which revolutionized finance by offering an explicit formula for pricing options based on various factors, such as volatility and time. The model has been widely adopted in the finance industry, enabling more efficient risk management for businesses, governments, and individuals. It also sparked the growth of the derivatives market, which today is valued at hundreds of trillions of dollars, and has been instrumental in the creation of new financial products like exchange traded options, credit default swaps, and securitized debt.

Physicists and Mathematicians Reveal Stock Market Patterns and Risk
The idea discussed revolves around the possibility of beating the stock market through the use of accurate models and computational power, which can help identify patterns and predict market behavior. Notably, physicists and mathematicians have often been the ones to discover these patterns and randomness in the stock market. Their contributions go beyond personal wealth, as they have provided new insights into risk and opened up whole new markets. By modeling market dynamics, they have determined the accurate price of derivatives and helped eliminate market inefficiencies. However, if all patterns in the stock market were to be discovered and understood, the market would become perfectly efficient, resulting in entirely random price movements.

Black-Scholes PDE and Its Derivation

The video discusses the Black-Scholes Partial Differential Equation (PDE) and its derivation. The Black-Scholes model is based on several assumptions, including constant short-term interest rate, no dividends, no transaction costs, no borrowing constraints, and allowance of short selling. The model was developed by Myron Scholes and Fischer Black, who were awarded the Nobel Prize in 1977 for their work.
The derivation of the Black-Scholes PDE involves pricing a derivative using replication, creating a risk-free portfolio, assuming the call option is a smooth function, and applying the Ito's calculus. The underlying asset is assumed to follow a Geometric Brownian Motion (GBM), which consists of a drift term and a diffusion term with constant volatility. The call price is dependent on the underlying and time. Taking partial derivatives with respect to each parameter and canceling out certain terms, the PDE is obtained. The PDE is then solved using the boundary conditions, which state that at the expiry, the payoff of a call option is the maximum of 0 and the difference between the underlying and the strike price. The solution yields the Black-Scholes equation for a call option, which is widely used in option pricing.

Black-Scholes Model: What It Is, How It Works, Options Formula

What Is the Black-Scholes Model?

The Black-Scholes model, also known as the Black-Scholes-Merton (BSM) model, is one of the most important concepts in modern financial theory. This mathematical equation estimates the theoretical value of derivatives based on other investment instruments, taking into account the impact of time and other risk factors. Developed in 1973, it is still regarded as one of the best ways for pricing an options contract.

Key Takeaways

The Black-Scholes model, aka the Black-Scholes-Merton (BSM) model, is a differential equation widely used to price options contracts.
The Black-Scholes model requires five input variables: the strike price of an option, the current stock price, the time to expiration, the risk-free rate, and the volatility.
Though usually accurate, the Black-Scholes model makes certain assumptions that can lead to predictions that deviate from the real-world results.
The standard BSM model is only used to price European options, as it does not take into account that American options could be exercised before the expiration date.

Investopedia / Jiaqi Zhou

History of the Black-Scholes Model

Developed in 1973 by Fischer Black, Robert Merton, and Myron Scholes, the Black-Scholes model was the first widely used mathematical method to calculate the theoretical value of an option contract, using current stock prices, expected dividends, the option's strike price, expected interest rates, time to expiration, and expected volatility.

The initial equation was introduced in Black and Scholes' 1973 paper, "The Pricing of Options and Corporate Liabilities," published in the Journal of Political Economy.1 Robert C. Merton helped edit that paper. Later that year, he published his own article, "Theory of Rational Option Pricing," in The Bell Journal of Economics and Management Science, expanding the mathematical understanding and applications of the model, and coining the term "Black–Scholes theory of options pricing."2

In 1997, Scholes and Merton were awarded the Nobel Memorial Prize in Economic Sciences for their work in finding "a new method to determine the value of derivatives." Black had passed away two years earlier, and so could not be a recipient, as Nobel Prizes are not given posthumously; however, the Nobel committee acknowledged his role in the Black-Scholes model.3

How the Black-Scholes Model Works

Black-Scholes posits that instruments, such as stock shares or futures contracts, will have a lognormal distribution of prices following a random walk with constant drift and volatility. Using this assumption and factoring in other important variables, the equation derives the price of a European-style call option.

The Black-Scholes equation requires five variables. These inputs are volatility, the price of the underlying asset, the strike price of the option, the time until expiration of the option, and the risk-free interest rate. With these variables, it is theoretically possible for options sellers to set rational prices for the options that they are selling.

Furthermore, the model predicts that the price of heavily traded assets follows a geometric Brownian motion 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.

Black-Scholes Assumptions

The Black-Scholes model makes certain assumptions:

No dividends are paid out during the life of the option.
Markets are random (i.e., market movements cannot be predicted).
There are no transaction costs in buying the option.
The risk-free rate and volatility of the underlying asset are known and constant.
The returns of the underlying asset are normally distributed.
The option is European and can only be exercised at expiration.

While the original Black-Scholes model didn't consider the effects of dividends paid during the life of the option, the model is frequently adapted to account for dividends by determining the ex-dividend date value of the underlying stock. The model is also modified by many option-selling market makers to account for the effect of options that can be exercised before expiration.

The Black-Scholes Model Formula

The mathematics involved in the formula are complicated and can be intimidating. Fortunately, you don't need to know or even understand the math to use Black-Scholes modeling in your own strategies. Options traders have access to a variety of online options calculators, and many of today's trading platforms boast robust options analysis tools, including indicators and spreadsheets that perform the calculations and output the options pricing values.

The Black-Scholes call option formula is calculated by multiplying the stock price by the cumulative standard normal probability distribution function. Thereafter, the net present value (NPV) of the strike price multiplied by the cumulative standard normal distribution is subtracted from the resulting value of the previous calculation.

In mathematical notation:

$\begin{matrix} C = S N (d_{1}) - K e^{- r t} N (d_{2}) \\ where: \\ d_{1} = \frac{l n_{K}^{S} + (r + \frac{σ_{v}^{2}}{2}) t}{σ_{s} \sqrt{t}} \\ and \\ d_{2} = d_{1} - σ_{s} \sqrt{t} \\ and where: \\ C = Call option price \\ S = Current stock (or other underlying) price \\ K = Strike price \\ r = Risk-free interest rate \\ t = Time to maturity \\ N = A normal distribution \end{matrix}$

Black, Scholes, Merton. © KhanAcademy

Volatility Skew

Black-Scholes assumes stock prices follow a lognormal distribution because asset prices cannot be negative (they are bounded by zero).

Often, asset prices are observed to have significant right skewness and some degree of kurtosis (fat tails). This means high-risk downward moves often happen more often in the market than a normal distribution predicts.

The assumption of lognormal underlying asset prices should show that implied volatilities are similar for each strike price according to the Black-Scholes model. However, since the market crash of 1987, implied volatilities for at-the-money options have been lower than those further out of the money or far in the money. The reason for this phenomenon is the market is pricing in a greater likelihood of a high volatility move to the downside in the markets.

This has led to the presence of the volatility skew. When the implied volatilities for options with the same expiration date are mapped out on a graph, a smile or skew shape can be seen. Thus, the Black-Scholes model is not efficient for calculating implied volatility.

The Black-Scholes model is often contrasted against the binominal model or a Monte Carlo simulation.

Benefits of the Black-Scholes Model

The Black-Scholes model has been successfully implemented and used by many financial professionals due to the variety of benefits it has to offer. Some of these benefits are listed below.

Provides a Framework: The Black-Scholes model provides a theoretical framework for pricing options. This allows investors and traders to determine the fair price of an option using a structured, defined methodology that has been tried and tested.
Allows for Risk Management: By knowing the theoretical value of an option, investors can use the Black-Scholes model to manage their risk exposure to different assets. The Black-Scholes model is therefore useful to investors not only in evaluating potential returns but understanding portfolio weakness and deficient investment areas.
Allows for Portfolio Optimization: The Black-Scholes model can be used to optimize portfolios by providing a measure of the expected returns and risks associated with different options. This allows investors to make smarter choices better aligned with their risk tolerance and pursuit of profit.
Enhances Market Efficiency: The Black-Scholes model has led to greater market efficiency and transparency as traders and investors are better able to price and trade options. This simplifies the pricing process as there is greater implicit understanding of how prices are derived.

Streamlines Pricing: On a similar note, the Black-Scholes model is widely accepted and used by practitioners in the financial industry. This allows for greater consistency and comparability across different markets and jurisdictions.

Limitations of the Black-Scholes Model

Though the Black-Scholes model is widely use, there are still some drawbacks to the model; some of the drawbacks are listed below.

Limits Usefulness: As stated previously, the Black-Scholes model is only used to price European options and does not take into account that U.S. options could be exercised before the expiration date.
Lacks Cashflow Flexibility: The model assumes dividends and risk-free rates are constant, but this may not be true in reality. Therefore, the Black-Scholes model may lack the ability to truly reflect the accurate future cashflow of an investment due to model rigidity.
Assumes Constant Volatility: The model also assumes volatility remains constant over the option's life. In reality, this is often not the case because volatility fluctuates with the level of supply and demand.
Misleads Other Assumptions: The Black-Scholes model also leverages other assumptions. These assumptions include that there are no transaction costs or taxes, the risk-free interest rate is constant for all maturities, short selling of securities with use of proceeds is permitted, and there are no risk-less arbitrage opportunities. Each of these assumptions can lead to prices that deviate from actual results.

Benefits

Acts as a stable framework that can be used using a defined method.
Allows investors to mitigate risk by better understanding exposure
May be used to devise the best strategies for creating a portfolio based on an investor's preferences.
Streamlines and improves efficient calculating and reporting of figures

Limitations

Does not take into consideration all types of options
May lack cashflow flexibility based on the future projections of a security
May make inaccurate assumptions about future stable volatility
Relies on a number of other assumptions that may not materialize into the actual price of the security

What Does the Black-Scholes Model Do?

The Black-Scholes model, also known as Black-Scholes-Merton (BSM), was the first widely used model for option pricing. Based on certain assumptions about the behavior of asset prices, the equation calculates the price of a European-style call option based on known variables like the current price, maturity date, and strike price. It does so by subtracting the net present value (NPV) of the strike price multiplied by the cumulative standard normal distribution from the product of the stock price and the cumulative standard normal probability distribution function.

What Are the Inputs for Black-Scholes Model?

The inputs for the Black-Scholes equation are volatility, the price of the underlying asset, the strike price of the option, the time until expiration of the option, and the risk-free interest rate. With these variables, it is theoretically possible for options sellers to set rational prices for the options that they are selling.

What Assumptions Does Black-Scholes Model Make?

The original Black-Scholes model assumes that the option is a European-style option and can only be exercised at expiration. It also assumes that no dividends are paid out during the life of the option; that market movements cannot be predicted; that there are no transaction costs in buying the option; that the risk-free rate and volatility of the underlying are known and constant; and that the prices of the underlying asset follow a log-normal distribution.

What Are the Limitations of the Black-Scholes Model?

The Black-Scholes model is only used to price European options and does not take into account that American options could be exercised before the expiration date. Moreover, the model assumes dividends, volatility, and risk-free rates remain constant over the option's life.

Not taking into account taxes, commissions or trading costs or taxes can also lead to valuations that deviate from real-world results.

The Bottom Line

The Black-Scholes model is a mathematical model used to calculate the fair price or theoretical value. It provides a way to calculate the theoretical value of an option by taking into account the current price of the underlying asset, the strike price of the option, the time remaining until expiration, the risk-free interest rate, and the volatility of the underlying asset. The Black-Scholes model has had a profound impact on finance and has led to the development of a wide range of derivative products such as futures, swaps, and options.

Correction—July 10, 2022: This article has been edited to clarify the assumptions that asset prices follow a log-normal distribution, while returns are normally distributed.

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