Black–Scholes Model Explained: Formula, Examples, and Limits

Editorial responsibility: Oliver Sinz · Last verified: October 27, 2026

The Black–Scholes model, also called the Black–Scholes–Merton model, gives the theoretical value of a European-style option from six inputs: the underlying price, the strike price, the time to expiration, the risk-free interest rate, the dividend yield, and the volatility of the underlying’s returns. Its central idea is that, under the model’s assumptions, an option can be replicated by continuously trading the underlying stock and a risk-free asset, so its value must equal the cost of that replication, whatever anyone expects the stock to do.

That result rests on strong assumptions, and most of what a trader needs to know about the model is where those assumptions hold and where they fail. This guide presents the call and put formulas, including the dividend-yield version, defines each input with its units, and works one valuation through every intermediate step. It then tests the replication argument with a hedging simulation, takes the assumptions one at a time, and shows why model values and executable option prices are different things.

All prices, volatilities, and scenarios in this article are hypothetical model calculations, not market data. Nothing here is a recommendation to trade.

What the Black–Scholes model is

Fischer Black and Myron Scholes published the model in 1973 (Black and Scholes, 1973). In the same year, Robert Merton published a more general derivation and extended the model, among other things, to stocks paying a continuous dividend yield (Merton, 1973). Listed options trading began on the newly opened Chicago Board Options Exchange in April 1973 (Cboe). In 1997, Merton and Scholes received the Nobel Memorial Prize in Economic Sciences for the method; Black had died in 1995 (Nobel Prize press release).

Three points frame the rest of this guide:

• It values European options. The formula assumes the option can be exercised only at expiration. The right to exercise American-style options early is a separate problem with no general closed-form solution.
• Its output is a theoretical value, not a market price. The Options Industry Council makes the same distinction: market forces, not formulas, set actual premiums, and pricing models are better suited to exploring hypothetical scenarios than to forecasting prices (OIC: Black-Scholes Formula).
• The most important input cannot be observed. Volatility over the option’s remaining life is unknown when the option trades. In practice, traders often run the model in reverse to find the volatility implied by an option’s market price.

The Black–Scholes formula

For a European call (C) and put (P) on an underlying with a continuous dividend yield q:

C = S × e^(−qT) × N(d1) − K × e^(−rT) × N(d2)

P = K × e^(−rT) × N(−d2) − S × e^(−qT) × N(−d1)

where

d1 = [ln(S ÷ K) + (r − q + σ²/2) × T] ÷ (σ × √T) and d2 = d1 − σ × √T

N(x) is the cumulative standard normal distribution: the probability that a standard normal random variable is at most x. With q = 0, these are the original Black–Scholes formulas; the q terms are Merton’s dividend-yield extension.

The inputs, their units, and what they mean

Two conventions trip up hand calculations and calculators alike. All rates and volatilities enter as decimals: 25% volatility is 0.25. The rate r and the yield q are continuously compounded annual rates, so a rate quoted with annual compounding, R, converts to r = ln(1 + R), and 4.0% continuously compounded equals about 4.08% compounded annually. Volatility is not a compounded rate at all; it is the annualized standard deviation of log returns. Time enters in years, and the choice between calendar days and trading days changes the answer, as the input checks later in this guide show.

What is missing matters as much as what is there: the formula has no input for the stock’s expected return. Two traders with opposite views of a stock, using the same six inputs, compute the same value. The section on replication explains why.

The same formula in forward-price terms

Dividends enter the formula only through the forward price of the underlying, F = S × e^((r − q)T), and interest rates through the forward price and the discount factor. Substituting gives

C = e^(−rT) × [F × N(d1) − K × N(d2)], with d1 = [ln(F ÷ K) + σ²T/2] ÷ (σ√T)

This is essentially the form Black (1976) used for options on futures and forward contracts. It makes each input’s role explicit. Higher rates raise call values and lower put values largely because they raise the forward price; dividends do the opposite because they lower it. Volatility and time set how widely the forward’s possible outcomes are dispersed, and e^(−rT) discounts the result to today.

A worked Black–Scholes example

Hypothetical inputs: stock XYZ at $100.00, a European call and put with a $105 strike, 45 calendar days to expiration, 25% volatility, a 4.0% continuously compounded risk-free rate, and no dividends.

For the put, N(−d1) = 0.675714 and N(−d2) = 0.706617:

P = 104.4835 × 0.706617 − 100 × 0.675714 = 73.8298 − 67.5714 = $6.2584

The put is $5 in the money, so $1.26 of its value is time value. Per standard 100-share contract, the model values are about $177 for the call and $626 for the put.

Two checks are worth running on any Black–Scholes calculation:

• Put-call parity. For European options on a stock without dividends, C − P must equal S − PV(K). Here $1.7749 − $6.2584 = −$4.4835, and $100 − $104.4835 = −$4.4835.
• Precision. The call value is the difference between two much larger numbers, so rounding errors are magnified. Rounding d1 and d2 to two decimals, as printed normal tables require, gives N(−0.46) = 0.3228 and N(−0.54) = 0.2946 and a call value of about $1.50 instead of $1.77, an error of roughly 16%. Keeping four decimals in N(d1) and N(d2) reduces the error to a fraction of a cent.

What the two terms mean

The call formula has a natural reading: the value of receiving the stock if the option finishes in the money, minus the value of paying the strike in those same outcomes.

• N(d2) = 29.3% is the model’s risk-neutral probability that XYZ finishes above $105. Multiplied by the present value of the strike, it gives the value of the expected strike payment, $30.65.
• S × N(d1) = $32.43 is the present value of the shares received in those outcomes. N(d1) exceeds N(d2) because the shares are delivered precisely when the stock is high: given that XYZ finishes above $105, its average risk-neutral price is about $111.08. The call therefore pays an average of $6.08 when it pays at all, which happens with 29.3% risk-neutral probability. That works out to an expected payoff of $1.78, or $1.77 discounted at 4% for 45 days.

With no dividends, N(d1) is also the call’s delta, its sensitivity to a $1 move in the stock; with a yield, delta is e^(−qT) × N(d1). Both N(d1) and N(d2) come out of the model’s risk-adjusted pricing. Neither is a forecast of how likely the option is to pay off.

Adding dividends: Merton’s dividend-yield version

A dividend transfers value from the stock to its owners. The option holder does not receive it, so expected dividends lower the forward price, which lowers call values and raises put values. In Merton’s version, the yield q is paid continuously. For the same 105 options with q = 2%:

• S × e^(−qT) = 100 × e^(−0.02 × 0.123288) = 99.7537
• d1 = [−0.048790 + (0.04 − 0.02 + 0.03125) × 0.123288] ÷ 0.087781 = −0.4838, and d2 = −0.5716
• Call = $1.6963 and put = $6.4260. Parity now reads C − P = S × e^(−qT) − PV(K) = −$4.7297.

A continuous yield describes a broad stock index, whose many constituents pay dividends on many dates, better than an individual stock, which pays discrete amounts on known ex-dividend dates. Over a year the difference can be negligible: four $1.00 quarterly dividends, subtracted from the stock price at their present value (the “escrowed dividend” convention), give one-year values within about a cent of a 4% yield, depending on the exact ex-dates. Over 45 days, timing dominates:

The yield spreads the dividend evenly and lowers the 45-day forward price by about $0.49. An actual dividend lowers it by the full $1.00 if the ex-date falls before expiration and by nothing if it falls after. The escrowed convention is itself a modeling choice, since it applies the volatility to the stock price net of the dividend’s present value. For American-style calls, a dividend can also make early exercise worthwhile, which the formula does not capture.

Why expected return drops out: replication and risk-neutral pricing

Black and Scholes’s argument starts with a hedge. Hold one option and sell short delta shares of the stock. Over a very short interval, the option’s gain or loss from a small stock move is offset by the short shares. If trading is continuous and frictionless and volatility is known, the combined position is riskless, so, to rule out arbitrage, it must earn exactly the risk-free rate. Writing that condition out gives the Black–Scholes partial differential equation, shown here with a dividend yield:

∂V/∂t + ½σ²S² × ∂²V/∂S² + (r − q) × S × ∂V/∂S − r × V = 0

V is the option’s value. Solving the equation with the payoff at expiration as its boundary condition, max(S − K, 0) for a call, produces the formula above. The stock’s expected return never appears, because the hedge removes the option’s exposure to the stock’s direction. Whatever premium investors demand for holding the stock, the option’s value relative to the stock does not depend on it.

That leads to a shortcut known as risk-neutral valuation (Cox and Ross, 1976). Because the answer does not depend on risk preferences, it can be computed as if investors were indifferent to risk: let the stock grow at r − q, take the expected payoff, and discount it at r. Harrison and Kreps (1979) and Harrison and Pliska (1981) later put the idea on a general footing, linking the absence of arbitrage to the existence of such risk-neutral, or “martingale,” probabilities.

The difference between risk-neutral and real-world expectations is easy to compute. Expected payoffs of the 105 call under different assumed growth rates for XYZ:

These are exact lognormal expectations; a simulation of 2 million risk-neutral price paths reproduces the first row ($1.7752, with a standard error of $0.0028, against $1.7749).

A trader who expects XYZ to grow 10% a year does expect the call to pay more. That payoff is risky in the same direction as the stock, though, so discounting it at the risk-free rate is wrong, and the correct risk-adjusted rate is unknown and shifts as the stock moves. The replication argument avoids the problem: the call is worth what it costs to build from the stock and borrowing, and that cost does not depend on anyone’s forecast. Risk-neutral probabilities are a pricing device, not a prediction.

Does replication work? A delta-hedging simulation

The model’s central claim can be tested. An option seller who receives the model value and hedges as the model prescribes should end with nothing on average, give or take a hedging error. The simulation below sells one 105 call contract for its model value of $177.49, buys the model’s delta in shares, finances the position at 4%, and resets the hedge to the model delta at the start of each of N equal intervals until expiration, the first reset being the initial purchase. The intervals divide the 45 calendar days evenly, weekends included, so “daily” means once per calendar day, not an exchange trading schedule. XYZ follows the model’s lognormal process, and each row uses 100,000 simulated price paths.

Rebalancing frequency, with the simulated process volatility equal to the 25% used to price and hedge:

The hedge works on average. Setting the stock’s growth rate to +10% or −10% instead of 4% leaves the daily-rebalancing results essentially unchanged; the expected return does not matter here either. But discrete hedging leaves a random error. Its standard deviation falls roughly in inverse proportion to the square root of the number of rebalances, so ten times as many trades cut it by about a factor of three, a pattern analyzed for discretely adjusted hedges by Boyle and Emanuel (1980). Even daily hedging leaves an error of $43 per contract, about a quarter of the premium.

Process volatility different from the 25% used to price and hedge, daily rebalancing:

Across scenarios, the writer’s average result tracks the difference between the premium received and the model value at the process volatility. Each row changes the volatility of the simulated process; it does not sort individual paths by the volatility they happened to realize. Ex ante, the Black–Scholes value is the arbitrage-free price under the model’s assumed dynamics, including its volatility. Ex post, a hedger’s actual result depends on the realized price path and volatility, the rebalancing schedule, and frictions. Even in the 25% row, where the model’s volatility matches the process exactly, individual outcomes remain random, with a standard deviation of $43 per contract.

Two frictions the model assumes away work against the hedger:

• Transaction costs. With a hypothetical cost of 0.05% of the value of each stock trade, charged on the initial share purchase and every rebalancing trade but not on closing the hedge at expiration, hedging costs average about $5 per contract with weekly rebalancing, $11 with daily rebalancing, and $32 with 10 trades a day. Selling the remaining shares at expiration would add about $1.60 in each case. More frequent hedging reduces the hedging error but keeps raising costs, so the model’s continuous hedge is out of reach in practice. Leland (1985) showed that transaction costs invalidate the frictionless replication argument and proposed a modified hedging strategy, usually implemented as an adjusted volatility input, that depends on the cost level and the rebalancing interval.
• Price gaps. If XYZ jumps from $100 to $110 before the writer can rebalance, the 0.32 delta in shares gains $324 per contract while the call’s model value rises by $536, a net loss of $212. A gap down to $90 costs $162. A short option hedged with stock loses on large moves in either direction, and rebalancing cannot help when the price moves before the hedger can trade. Merton (1976) extended option pricing to stock returns with jumps, a risk the stock alone cannot hedge away.

The Black–Scholes assumptions, one by one

Black and Scholes stated their “ideal conditions” explicitly, and Merton’s version relaxed several of them. The table groups them by what they describe.

Where the model breaks down

Volatility skew and fat tails

On October 19, 1987, the Dow Jones Industrial Average fell 22.6% in a single day (Federal Reserve History). Under a lognormal model with 20% annual volatility, a one-day fall of that size is roughly a 20-standard-deviation event, which the model treats as practically impossible. Since that crash, S&P 500 index options have priced in a persistent skew, with higher implied volatilities at lower strikes (Rubinstein, 1994). Sociologist Donald MacKenzie traces how the gap between Black–Scholes values and Chicago option prices narrowed during the 1970s, as the market changed in ways that made the model’s assumptions more accurate, and how the fit deteriorated after the crash (MacKenzie and Millo, 2003; MacKenzie, 2003). The fit was never perfect: studying trades and quotes from 1976 to 1978, Rubinstein (1985) found statistically significant departures from the model’s prices, some of which reversed direction over time.

The cost of a skew to a single-volatility model is easy to see. Take a hypothetical 45-day chain whose implied volatility is 30% at the 90 strike, 25% at 100, and 20% at 110. Priced at 25%, the 90 put is worth $0.41 and the 110 call $0.72; at their own implied volatilities, they are worth $0.76 and $0.34. One σ cannot match both sides of the chain.

Researchers responded with models that relax the constant-volatility, continuous-path assumptions, including jumps (Merton, 1976), stochastic volatility (Heston, 1993), and implied trees fitted to the whole skew (Rubinstein, 1994). Testing several such models on S&P 500 options, Bakshi, Cao, and Chen (1997) found stochastic volatility and jumps important for pricing, and stochastic volatility alone the most useful addition for hedging. Rather than abandon Black–Scholes, the market kept it as a quoting convention: options are routinely quoted in implied volatility, and the skew records where the single-volatility assumption fails.

Early exercise

The formula prices European exercise only. With non-negative interest rates, an American call on a stock that pays no dividends before expiration should not be exercised early, so it is worth the same as the European call and the formula applies (Merton, 1973). American puts, and calls facing dividends, need numerical methods such as binomial trees (Cox, Ross, and Rubinstein, 1979); OIC notes that American-style equity options are typically priced with a binomial model for this reason (OIC). For the example, a binomial tree values an American 105 put at $6.35 against $6.26 for the European put. For European options, the same kind of tree converges to Black–Scholes: with about 5,000 steps it values the call within two hundredths of a cent of $1.7749.

Theoretical values versus executable prices

The model’s $1.77 is a single number. A market quote is two: a hypothetical $1.70 bid and $1.85 offer would bracket the model value, but a buyer would pay $1.85 and a seller would receive $1.70, before commissions and fees. Different defensible inputs, such as another implied volatility, a dividend forecast, a different rate, or a trading-day count, produce different theoretical values for the same contract. And a model value is not a forecast of profit. An option bought at exactly its Black–Scholes value can gain or lose depending on how the stock and its volatility actually behave, as the simulation showed.

Using Black–Scholes in practice

Building a Black–Scholes calculator in a spreadsheet

With cells named S, K, T, r, q, and sigma (rates, yield, and volatility as decimals; T in years):

d1:   =(LN(S/K)+(r-q+sigma^2/2)*T)/(sigma*SQRT(T))
d2:   =d1-sigma*SQRT(T)
Call: =S*EXP(-q*T)*NORM.S.DIST(d1,TRUE)-K*EXP(-r*T)*NORM.S.DIST(d2,TRUE)
Put:  =K*EXP(-r*T)*NORM.S.DIST(-d2,TRUE)-S*EXP(-q*T)*NORM.S.DIST(-d1,TRUE)

In Excel, NORM.S.DIST with the second argument TRUE returns the cumulative standard normal distribution (Microsoft Support). With the example’s inputs (100, 105, 45/365, 0.04, 0, 0.25), the sheet should return 1.7749 for the call and 6.2584 for the put.

Input checks

Volatility dominates the result; the other inputs matter less but are easy to get wrong:

A five-point volatility error moves this call’s value by about a third, far more than any other input choice here. That is one reason traders quote and compare options in implied volatility, using the formula to translate between volatility and price.

Key takeaways

• The Black–Scholes–Merton model values European options from six inputs. With a continuous dividend yield, C = S e^(−qT) N(d1) − K e^(−rT) N(d2), and the put follows by symmetry.
• Inputs enter as decimals and time in years. The rate and dividend yield are continuously compounded annual rates; volatility is the annualized standard deviation of log returns. Volatility is the hardest input to pin down, and it dominates the result.
• The stock’s expected return does not appear, because in the model the option can be replicated by continuously adjusting a position in the stock and borrowing. Risk-neutral probabilities such as N(d2) are pricing devices, not forecasts.
• Dividends act through the forward price, and rates through the forward price and discounting. A continuous yield suits indexes and long horizons; for single stocks over short horizons, the timing of each ex-dividend date matters.
• In practice, replication is approximate. Discrete hedging leaves errors, transaction costs grow with hedging frequency, price gaps cannot be hedged with stock, and a hedged position’s actual result depends on the realized price path and volatility, even when the option was priced consistently with the model ex ante.
• Market prices depart from the model systematically, most visibly in the volatility skew since 1987. The model survives mainly as a quoting convention and a benchmark.
• A Black–Scholes value is a theoretical number for a given set of inputs, not an executable price or an expected profit.

Primary sources

• Black, F., and Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637–654.
• Merton, R. C. (1973). Theory of Rational Option Pricing. Bell Journal of Economics and Management Science, 4(1), 141–183.
• Black, F. (1976). The Pricing of Commodity Contracts. Journal of Financial Economics, 3(1–2), 167–179.
• Cox, J. C., and Ross, S. A. (1976). The Valuation of Options for Alternative Stochastic Processes. Journal of Financial Economics, 3(1–2), 145–166.
• Merton, R. C. (1976). Option Pricing When Underlying Stock Returns Are Discontinuous. Journal of Financial Economics, 3(1–2), 125–144.
• Harrison, J. M., and Kreps, D. M. (1979). Martingales and Arbitrage in Multiperiod Securities Markets. Journal of Economic Theory, 20(3), 381–408.
• Cox, J. C., Ross, S. A., and Rubinstein, M. (1979). Option Pricing: A Simplified Approach. Journal of Financial Economics, 7(3), 229–263.
• Boyle, P. P., and Emanuel, D. (1980). Discretely Adjusted Option Hedges. Journal of Financial Economics, 8(3), 259–282.
• Harrison, J. M., and Pliska, S. R. (1981). Martingales and Stochastic Integrals in the Theory of Continuous Trading. Stochastic Processes and their Applications, 11(3), 215–260.
• Leland, H. E. (1985). Option Pricing and Replication with Transactions Costs. Journal of Finance, 40(5), 1283–1301.
• Rubinstein, M. (1985). Nonparametric Tests of Alternative Option Pricing Models Using All Reported Trades and Quotes on the 30 Most Active CBOE Option Classes from August 23, 1976 through August 31, 1978. Journal of Finance, 40(2), 455–480.
• Heston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Review of Financial Studies, 6(2), 327–343.
• Rubinstein, M. (1994). Implied Binomial Trees. Journal of Finance, 49(3), 771–818.
• Bakshi, G., Cao, C., and Chen, Z. (1997). Empirical Performance of Alternative Option Pricing Models. Journal of Finance, 52(5), 2003–2049.
• MacKenzie, D., and Millo, Y. (2003). Constructing a Market, Performing Theory: The Historical Sociology of a Financial Derivatives Exchange. American Journal of Sociology, 109(1), 107–145.
• MacKenzie, D. (2003). An Equation and Its Worlds: Bricolage, Exemplars, Disunity and Performativity in Financial Economics. Social Studies of Science, 33(6), 831–868.
• Nobel Prize Outreach: The Prize in Economic Sciences 1997, press release
• Options Industry Council: Black-Scholes Formula · Options Exercise FAQ
• Cboe: The Creation of Listed Options at Cboe
• Federal Reserve History: Stock Market Crash of 1987
• Microsoft Support: NORM.S.DIST function