Black-Scholes Model

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 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 prices 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.
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Black-Scholes Model

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. 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."

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.

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 on the underlying asset are log-normally distributed.
  • The option is European and can only be exercised at expiration.

Alternatively, for the pricing of the more commonly traded American-style options, firms will use a binomial or trinomial model or the Bjerksund-Stensland model.

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:

C = S t N ( d 1 ) K e r t N ( d 2 ) where: d 1 = l n S t K + ( r + σ v 2 2 )   t σ s   t and d 2 = d 1 σ s   t 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 \begin{aligned} &C = S_t N(d _1) - K e ^{-rt} N(d _2)\\ &\textbf{where:}\\ &d_1 = \frac{ln\frac{S_t}{K} + (r+ \frac{\sigma ^{2} _v}{2}) \ t}{\sigma_s \ \sqrt{t}}\\ &\text{and}\\ &d_2 = d _1 - \sigma_s \ \sqrt{t}\\ &\textbf{where:}\\ &C = \text{Call option price}\\ &S = \text{Current stock (or other underlying) price}\\ &K = \text{Strike price}\\ &r = \text{Risk-free interest rate}\\ &t = \text{Time to maturity}\\ &N = \text{A normal distribution}\\ \end{aligned} C=StN(d1)KertN(d2)where:d1=σs tlnKSt+(r+2σv2) tandd2=d1σs twhere:C=Call option priceS=Current stock (or other underlying) priceK=Strike pricer=Risk-free interest ratet=Time to maturityN=A normal distribution

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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.

Drawbacks of the Black-Scholes Model

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. Moreover, the model assumes dividends and risk-free rates are constant, but this may not be true in reality. The model also assumes volatility remains constant over the option's life, which is not the case because volatility fluctuates with the level of supply and demand.

Additionally, the other assumptions—that there are no transaction costs or taxes; that the risk-free interest rate is constant for all maturities; that short selling of securities with use of proceeds is permitted; and that there are no risk-less arbitrage opportunities—can lead to prices that deviate from the real world's.

What Does the Black-Scholes Model Do?

Black-Scholes, also known as Black-Scholes-Merton (BSM), was the first widely used model for option pricing. Based on the assumption 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, and factoring in other important variables, the equation derives the price of a European-style call option. 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 Black-Scholes model makes certain assumptions. Chief among them is that the option is European and can only be exercised at expiration. Other assumptions are that no dividends are paid out during the life of the option; that market movements cannot be predicted; that no transaction costs in buying the option; that risk-free rate and volatility of the underlying are known and constant; and that the returns on the underlying asset are log-normally distributed.

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.

Article Sources

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  1. Fischer Black and Myron Scholes, "The Pricing of Options and Corporate Liabilities." Journal of Political Economy, Volume 81, No. 3, 1974, Pages 637-654 

  2. Robert C. Merton, "Theory of Rational Option Pricing." The Bell Journal of Economics and Management Science, Volume 4, No. 1, 1973, Pages 141-183.
  3. The Nobel Prize. "The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997: Robert C. Merton Myron Scholes." Accessed Sept. 2, 2021.