Price a European call and put, step by step. The formulas, the six inputs, and a fully worked numerical example — with a live calculator to experiment with each stage.
→ Skip straight to the full Black-Scholes calculatorThe Black-Scholes model is a closed-form formula for pricing European-style call and put options, published in 1973 by Fischer Black and Myron Scholes with foundational mathematical work by Robert Merton. More than five decades later, it remains the default model for option pricing and the foundation of virtually every business valuation that involves optionality — from 409A fair value opinions to ASC 718 stock compensation expense to Chaffe DLOM analyses.
The insight behind the model is that the fair price of an option can be expressed as a combination of two probabilities — the probability that the stock finishes above the strike, and a risk-neutral probability adjustment — discounted back to today at the risk-free rate. The formula turns six observable or estimable inputs into a single number: what a rational, no-arbitrage price must be.
This tutorial walks through the full computation. You will see the formula, understand each of its six inputs, compute the intermediate values d₁ and d₂ by hand, apply the cumulative normal distribution N(·), and arrive at call and put prices — cross-checked against the live calculator widget below.
The call and put pricing formulas with continuous dividend yield q are:
Each term has an economic interpretation. In the call formula, S · e−qT · N(d₁) is the present value of receiving the stock if the option is exercised, weighted by a delta-like probability. K · e−rT · N(d₂) is the present value of paying the strike, weighted by the risk-neutral probability N(d₂) that the option finishes in-the-money. The call price is simply the difference.
S = current stock price. K = strike price. T = time to expiration in years. r = continuously-compounded risk-free rate. σ = volatility (standard deviation of log returns, annualized). q = continuous dividend yield. N(·) = cumulative standard normal CDF. n(·) = standard normal PDF (bell curve). e = Euler's number ≈ 2.71828.
Black-Scholes turns exactly six inputs into a price. Four are market-observable, one is a structural term of the contract, and one is estimated from historical data or peer-company analysis.
| Symbol | Input | Meaning & Typical Source |
|---|---|---|
| S | Stock Price | Current price of the underlying. Observable for public stocks; estimated via OPM or hybrid approach for private companies. |
| K | Strike Price | The fixed price at which the option holder can buy (call) or sell (put). A contractual term — fully known. |
| T | Time to Expiry | Years until expiration. For 409A work, use the expected option term (typically 5–10 years) rather than the contractual maximum. |
| r | Risk-Free Rate | Continuously-compounded annualized rate. Use the Treasury yield matching the expected term (e.g., 5-year Treasury for T = 5). |
| σ | Volatility | Annualized standard deviation of log returns. For private companies, estimate from a guideline public company peer set. The most sensitive input. |
| q | Dividend Yield | Continuous dividend yield. Often 0 for growth-stage companies; for dividend-paying stocks, use the trailing or expected forward yield. |
Calls rise with S, T, σ, and r; fall with K and q. Puts are the mirror image for S and K (higher S hurts puts, higher K helps them) but similar for T, σ (both raise put values too — buyers love uncertainty regardless of direction). For a complete breakdown of each sensitivity, see the Option Greeks explainer.
Before you can evaluate the call or put formula, you need the two intermediate quantities d₁ and d₂. They are not the final probabilities — they are the inputs to the cumulative normal distribution that produces those probabilities.
Walk through the arithmetic with the canonical test case — a one-year at-the-money call on a non-dividend stock, 20% volatility, 5% risk-free rate:
Numerator of d₁: ln(100/100) + (0.05 − 0 + ½ × 0.20²) × 1 = 0 + (0.05 + 0.02) × 1 = 0.07
Denominator of d₁: σ · √T = 0.20 × 1 = 0.20
d₁ = 0.07 / 0.20 = 0.3500
d₂ = d₁ − σ√T = 0.3500 − 0.20 = 0.1500
N(d₂) is the risk-neutral probability that the call expires in-the-money. N(d₁) is a related probability, adjusted by a factor that captures the expected stock value conditional on exercise. The call price is the expected payoff expressed as the difference between these two probability-weighted pieces, each discounted to today.
With d₁ = 0.3500 and d₂ = 0.1500 in hand, evaluate the cumulative standard normal at each. This is the classic z-table lookup from statistics, expressed to four decimal places:
The final step is substitution into the call and put formulas. Compute the discount factors first, then plug in:
e−qT = e0 = 1.0000 | e−rT = e−0.05 ≈ 0.9512
C = 100 × 1.0000 × 0.6368 − 100 × 0.9512 × 0.5596
C = 63.68 − 53.23 = $10.45
P = 100 × 0.9512 × 0.4404 − 100 × 1.0000 × 0.3632
P = 41.89 − 36.32 = $5.57
Parity says C − P = S · e−qT − K · e−rT. Plugging in: 10.45 − 5.57 = $4.88 on the left, and 100 − 95.12 = $4.88 on the right. The equation balances to the cent — a consistency check that the two prices are internally coherent.
Here is the complete calculation assembled in a single view. If you can reproduce these numbers, you can price any European option with Black-Scholes.
Leave the sliders below at their defaults (S=100, K=100, T=1, σ=20%) and compare the widget's d₁, d₂, N(d₁), N(d₂), Call, and Put values to the hand-computed numbers above. They should agree to the fourth decimal for d₁/d₂/N values and to the cent for the prices.
The elegance of the closed-form Black-Scholes solution comes from a set of simplifying assumptions. Understanding each one is essential for applying the model correctly in practice — especially in valuation contexts where the conclusion must be defensible to auditors and counterparties.
European exercise. The model assumes exercise only at expiration. American-style options, which allow earlier exercise, require numerical methods. For dividend-paying stocks and deep in-the-money puts, the early-exercise premium can be material and Black-Scholes will under-price those contracts.
Constant volatility. The model assumes σ is a single fixed number over the life of the option. Real markets exhibit a volatility smile (implied volatility varies with strike) and a volatility term structure (it varies with expiration). Stochastic-volatility models like Heston relax this assumption at significant analytical cost.
Log-normal stock prices, no jumps. Returns are assumed to follow a continuous geometric Brownian motion with no gaps. Real markets exhibit fat tails and overnight jumps from earnings, M&A announcements, and macro events. Merton's jump-diffusion extension adds a jump component; Black-Scholes is the no-jump baseline.
No transaction costs, continuous trading, unlimited shorting. The no-arbitrage derivation requires perfectly frictionless markets. Real-world replication involves bid-ask spreads, borrowing costs, and finite position sizing. For valuation purposes these are usually second-order relative to input-estimation uncertainty.
In practice, the dominant source of error in a Black-Scholes valuation is almost never the model's assumptions — it is the estimation of its inputs. A private-company volatility assumption based on a guideline public company peer set has standard error measured in percentage points, not basis points. A DLOM Chaffe estimate with a 25% σ input differs from one using 35% by hundreds of basis points of discount. That is why auditors scrutinize input support (guideline company selection, expected-term methodology, risk-free rate tenor choice) far more than they question the Black-Scholes model itself.
The full calculator handles both sides simultaneously, solves for implied volatility from a market price, draws payoff diagrams, and checks put-call parity. Everything shown above, with the Greeks too.
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