What Is the Black-Scholes Model? Formula and How It Prices Options

Search Fintrev

Search our guides, reviews and glossary.

The Black-Scholes model, also called the Black-Scholes-Merton (BSM) model, is a mathematical model that gives a theoretical price for a European-style option, one that can be exercised only at expiration. It takes five inputs: the underlying asset’s price, the strike price, the time to expiration, the risk-free interest rate and the volatility of the underlying. From these it produces a closed-form value for a call or a put. Traders also run the formula in reverse, starting from an option’s market price to find the volatility that price implies. That reverse use is the main way the model works in day-to-day options markets.

What Is the Black-Scholes Model?

The Black-Scholes model is a framework for valuing options, contracts that give the right but not the obligation to buy or sell an asset at a set price by a set date. Before it, traders priced options by rule of thumb or by estimating the expected payoff, which required a view on where the asset was headed. Black-Scholes removed that requirement. If a trader can continuously hedge an option with the underlying asset and a risk-free loan, the option’s price must equal the cost of that hedge, and the stock’s expected return drops out of the formula.

The term covers two related things:

• The Black-Scholes equation is a partial differential equation that any derivative on the asset must satisfy under the model’s assumptions.
• The Black-Scholes formula is the specific solution of that equation for a European call or put.

When traders say “Black-Scholes,” they usually mean the formula. The equation matters because other derivatives can be priced by solving it with different payoff conditions.

History of the model

Fischer Black and Myron Scholes published the original result in 1973 in “The Pricing of Options and Corporate Liabilities” in the Journal of Political Economy. Robert C. Merton published “Theory of Rational Option Pricing” the same year. He extended the mathematics and added the treatment of dividends, which is why the model is often called Black-Scholes-Merton.

The timing mattered. The Chicago Board Options Exchange began listing standardized stock options in 1973, and the formula gave the new market a common way to value contracts.

Scholes and Merton received the Nobel Memorial Prize in Economic Sciences in 1997. Black had died in 1995. The prize is not awarded posthumously, though the Nobel committee acknowledged his role.

How the Black-Scholes Model Works

The model starts from an assumption about how prices move. The underlying asset follows geometric Brownian motion, a random walk in which percentage changes are normally distributed with constant drift and constant volatility. Over any horizon, the price itself is therefore lognormally distributed. It can rise without limit but cannot fall below zero.

From there the argument runs in three steps:

1. Combine the option with a carefully sized position in the underlying so that small price moves in one offset the other. This creates a portfolio that is momentarily risk-free.
2. A risk-free portfolio must earn the risk-free rate, or there would be an arbitrage opportunity.
3. Writing that condition mathematically gives the Black-Scholes equation. Solving it for a European payoff gives the formula.

The hedge has to be adjusted continuously as the price moves. The model assumes this can be done for free.

Black-Scholes assumptions

The standard model assumes:

• The option is European and can be exercised only at expiration.
• No dividends are paid during the option’s life (the Merton extension relaxes this).
• There are no transaction costs or taxes, and trading is continuous.
• The risk-free rate and the volatility of the underlying are known and constant for the life of the option.
• Returns are normally distributed, so prices are lognormal.
• Short selling is allowed with full use of the proceeds, and there are no riskless arbitrage opportunities.

Calculating the Black-Scholes model

For a non-dividend-paying asset, the European call price C and put price P are:

C = S × N(d1) - K × e^(-rT) × N(d2)
P = K × e^(-rT) × N(-d2) - S × N(-d1)

d1 = [ ln(S / K) + (r + σ² / 2) × T ] / (σ × √T)
d2 = d1 - σ × √T

Where:

• S is the current price of the underlying asset.
• K is the strike price.
• T is the time to expiration in years.
• r is the continuously compounded risk-free rate.
• σ (sigma) is the annualized volatility of the underlying’s returns.
• N(x) is the cumulative standard normal distribution, the probability that a standard normal variable is less than x.
• e^(-rT) is the discount factor that converts the strike into present value.

The first term, S × N(d1), is the current asset price weighted by a probability factor. The second term, the present value of the strike, K × e^(-rT), is weighted by N(d2) and subtracted from the first. Only the strike is discounted, because it is paid in the future. The share price is already a present value.

For dividend-paying assets, Merton’s version replaces S with S × e^(-qT), where q is the continuous dividend yield, and uses (r - q + σ²/2) in d1. Two other variants follow the same structure:

• Black model (1976): prices options on futures by working from the futures price.
• Garman-Kohlhagen model (1983): prices currency options by treating the foreign interest rate like a dividend yield. On a forex option, the spot exchange rate takes the place of S, the domestic rate takes the place of r and the foreign rate takes the place of q.

In practice, options calculators, trading platforms and spreadsheets compute the formula directly, since N(x) is built into most of them. What matters is knowing what each input does to the price:

Longer time usually raises option values. A deep in-the-money European put is an exception: its value can fall as time increases, because the strike payment is pushed further into the future.

Interpreting the Black-Scholes model

The pieces of the formula have practical meanings:

• N(d1) is the call’s delta, the approximate change in option value for a $1 move in the underlying. It is also the hedge ratio: the number of shares per option needed to offset small price moves. For a put, delta is N(d1) - 1.
• N(d2) is the probability, under the model’s risk-neutral assumptions, that the call finishes in the money. It is not a forecast of real-world odds, because the model replaces the asset’s actual expected return with the risk-free rate.
• The other Greeks (gamma, theta, vega and rho) come from differentiating the formula with respect to each input. They are covered in detail in option Greeks explained.

Two distinctions trip readers up. First, the model price is a theoretical value, not the price you will be quoted. The market price is set by supply and demand, and the difference between the two reflects disagreement about volatility more than anything else. Second, a volatility that a trader estimates from past price data (historical volatility) is not the same as implied volatility, which is backed out of a live option price.

Why Is the Black-Scholes Model Important?

Of the five inputs, four are observable: the asset price, the strike, the time to expiration and a benchmark interest rate. Only volatility is unknown. If you take an option’s market price and solve the formula for σ, you get the implied volatility, the volatility the market is pricing in.

This turns the formula into a translation tool. A $3 option on a $50 stock and a $12 option on a $400 stock with different strikes and expirations say little side by side, but their implied volatilities are directly comparable. That is why options are commonly quoted and analyzed in volatility terms. A trader buying a straddle is effectively betting that realized volatility will exceed the implied volatility paid for it.

Beyond pricing, the model gives desks, brokers and risk managers a single, consistent way to value positions across markets. Its Greeks support hedging and portfolio-level risk limits. Investors can test how a structure behaves under different volatility or price assumptions before committing capital. Later models such as binomial trees, local volatility and stochastic volatility models were built to fix its specific weaknesses, and they are still benchmarked against it.

Example of the Black-Scholes Model

Hypothetical example for illustration only. A stock trades at $100 and pays no dividend. Consider a European call option and put with a $100 strike, six months to expiration (T = 0.5), a 5% risk-free rate and 20% annual volatility.

Step 1: Calculate d1 and d2.

• ln(100 / 100) = 0
• (r + σ²/2) × T = (0.05 + 0.02) × 0.5 = 0.035
• σ × √T = 0.20 × 0.7071 = 0.1414
• d1 = 0.035 / 0.1414 = 0.2475
• d2 = 0.2475 - 0.1414 = 0.1061

Step 2: Look up the normal probabilities.

• N(d1) = N(0.2475) ≈ 0.5977
• N(d2) = N(0.1061) ≈ 0.5422

Step 3: Discount the strike.

• K × e^(-0.05 × 0.5) = 100 × 0.9753 = $97.53

Step 4: Price the call.

• C = 100 × 0.5977 - 97.53 × 0.5422 = 59.77 - 52.88 ≈ $6.89

Step 5: Price the put. Using N(-d1) = 0.4023 and N(-d2) = 0.4578:

• P = 97.53 × 0.4578 - 100 × 0.4023 = 44.65 - 40.23 ≈ $4.42

The put can be checked with put-call parity, which holds for European options regardless of model: C - P = S - K × e^(-rT). Here 6.89 - 4.42 = 2.47, and 100 - 97.53 = 2.47.

Reading the results:

• The call’s delta of about 0.60 means it gains roughly $0.60 for a $1 rise in the stock, at least initially.
• The model assigns about a 54% risk-neutral probability that the call expires in the money.
• If volatility is raised to 30% with everything else unchanged, the call rises to about $9.6.

That last point shows the model’s sensitivity to volatility: a 10-point change in the one unobservable input moves the price by roughly 40%.

Limitations of the Black-Scholes Model

The formula is exact only inside its assumptions. Each broken assumption pushes real prices away from model prices in a specific direction.

Constant volatility and the volatility skew

If prices were truly lognormal with one constant volatility, every option on the same asset and expiration would show the same implied volatility at every strike. Markets have not behaved that way, particularly since the October 1987 crash. Across strikes for a single expiration, the line is not flat:

• In equity markets, out-of-the-money puts typically carry higher implied volatility than at-the-money options. This produces a downward-sloping skew.
• In some markets, both deep in-the-money and far out-of-the-money options carry higher implied volatility than at-the-money options. This produces a smile.

The skew reflects what the model leaves out. Real returns have fat tails, so large moves, especially sharp declines, occur more often than a normal distribution predicts. Demand for downside protection also bids up put prices relative to the model. As a result, no single volatility can price every strike. Traders instead use a volatility surface, a grid of implied volatilities by strike and expiration, and feed the matching value into the formula for each contract.

Early exercise and American options

The standard formula ignores early exercise, so it does not correctly value American options, which include most U.S. single-stock options. Early exercise can matter for calls just before a dividend and for deep in-the-money puts. Firms typically price American options with binomial or trinomial trees or approximations such as the Bjerksund-Stensland model. Market makers also adapt Black-Scholes itself for early exercise. Index options such as SPX are European-style, which is one reason the formula fits them more naturally than it fits single stocks.

Constant rates and dividends

The model treats the risk-free rate as constant for all maturities and dividends as either absent or a smooth known yield. Real rates change, and real dividends are discrete payments on specific dates. The usual fix for dividends is to adjust the stock price for dividends expected before expiration, which works for short-dated options but grows less reliable over longer horizons.

Frictionless, continuous trading

The hedging argument requires adjusting the hedge continuously at no cost. In practice hedges are rebalanced at intervals, every trade pays a spread or commission, and taxes apply. Prices can also gap overnight or on news, so a delta hedge can lose money in a single jump. The model price is therefore a reference point, not a guaranteed cost of replication.

Black-Scholes vs. Other Option Pricing Models

The main alternatives solve the same pricing problem with different trade-offs:

• Binomial model. This model splits time into steps where the price moves up or down, then works backward from expiration. Because the option’s value can be checked at every node, it handles early exercise and discrete dividends. As the number of steps grows, the European binomial price converges to the Black-Scholes price.
• Monte Carlo simulation. This method generates thousands of random price paths and averages the discounted payoffs. It is slower but suited to path-dependent contracts, such as options whose payoff depends on an average price.
• Stochastic volatility models (such as Heston). These let volatility itself move randomly, which allows them to reproduce the skew that Black-Scholes misses.

Black-Scholes remains the fastest method and the shared language: even when another model sets the price, the result is often quoted back as a Black-Scholes implied volatility.

The Bottom Line

The model matters most to options traders, risk managers and forex traders valuing currency options through the Garman-Kohlhagen variant. Treat its output as a theoretical reference, since real markets show volatility skews, fat tails, discrete dividends and costly hedging. Its lasting practical value is as a shared language, letting traders compare options across assets and strikes in implied volatility terms.