European call
C = S_0 N(d1) - K exp(-rT) N(d2)The call is the difference between a delta-weighted spot claim and a discounted strike claim.
Derivatives
Black–Scholes Option Pricing
The textbook formula for European options in an idealised world with continuous hedging and constant volatility.
Mental model
Black–Scholes prices a European option by asking what a continuously rebalanced replicating hedge must cost.
Black–Scholes assumes the underlying follows geometric Brownian motion, trading is frictionless, and positions can be re-hedged continuously.
Under these assumptions, European option prices can be written in closed form and all standard Greeks (Δ, Γ, Vega, Θ, Rho) are available analytically.
Explore the mechanism
How do spot, strike, time, rates, and volatility combine to set a European option's no-arbitrage value?
Black–Scholes price and Greeks with continuous dividend yield. The chart shows price vs spot S; the gap to the intrinsic line is the time value. The shaded region is where the option finishes in the money at expiry.
Price: 9.4134d₁ = 0.250 · d₂ = 0.050P(ITM, risk-neutral) ≈ 52.0%
Delta
0.5987
Gamma
1.933e-2
Vega
38.6668
Theta (per year)
-5.3804
Rho
50.4572
Notes
Vega is per 1.0 volatility (not per 1%). Theta is shown per year; you can mentally scale it to days or months. Delta is the hedge ratio: units of underlying per option.
Numbers
Intrinsic value at S = 100: 0.0000
Time value ≈ 9.4134
Moneyness S / K ≈ 1.000
Put–call parity C − P ≈ 2.9554; S e−qT − K e−rT ≈ 2.9554 (diff ≈ 7.11e-15)
Interpretation
The orange dashed line is the intrinsic value you would get by exercising now. The blue line is the Black–Scholes price. The red bracket at the current S is the time value: the extra premium for uncertainty and time to expiry.
The shaded band shows where the option ends up in the money at expiry. Under the risk-neutral measure, d₂ encodes both the forward moneyness and volatility; N(d₂) (or N(−d₂) for puts) is the probability of finishing in the money.
Practitioners read this as: price = intrinsic + time value, with delta as the hedge ratio and gamma/vega telling you how fragile that hedge is. Put–call parity is the consistency check tying it all together: if it fails badly in real markets, there is either a data problem or an arbitrage story.
UNDERSTAND
What to notice
In practice, Black–Scholes is more a language than a belief: markets quote implied volatilities from observed prices rather than believing volatility is constant.
Control-style view: the Black–Scholes hedge is a continuous feedback law (delta hedging) that keeps the portfolio locally insensitive to small price movements.
The quantitative structure
Notation
S_0- spot price
K- strike
T- time to expiry
r- continuously compounded risk-free rate
sigma- constant volatility
Black-Scholes variables
d1 = [ln(S_0/K) + (r + 0.5 sigma^2)T] / (sigma sqrt(T)); d2 = d1 - sigma sqrt(T)Moneyness, time, rates, and volatility are compressed into two normal scores.
This model assumes
- The underlying follows GBM with constant volatility.
- Trading is frictionless and continuous, and the European option can be replicated.
- Rates are deterministic and the basic formula assumes no dividends.
A small example
An at-the-money call
Spot and strike are both 100, expiry is one year, the rate is 0%, and volatility is 20%.
- d1 = 0.10 and d2 = -0.10.
- Using standard normal values, N(d1) is about 0.5398 and N(d2) about 0.4602.
- Call value is 100 * (0.5398 - 0.4602) = 7.96.
ResultThe Black–Scholes call value is approximately 7.96.
Where it appears
- benchmarking European option prices
- quoting implied volatility
- computing analytic Greeks for validation
Where the intuition breaks
What this view leaves out
- Constant volatility, continuous trading, frictionless hedging, and continuous paths are approximations; observed smiles and hedging gaps expose their limits.
Common mistakes
“Black–Scholes is accurate because market returns are normally distributed.”
Not quite.The formula follows from a specific dynamic-hedging model; market prices commonly violate its constant-volatility implication.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Black–Scholes Option Pricing
Black–Scholes Option PricingYOU ARE HERE
Built onArbitrage and No-ArbitrageBlack–Scholes valuation rests on dynamic replication and the exclusion of arbitrage.
Built onGeometric Brownian Motion (GBM)The classical Black–Scholes model assumes GBM dynamics for the underlying asset.
Built onRisk-Neutral Measure (Intuition)Black–Scholes prices the discounted derivative payoff under risk-neutral dynamics.
Foundation ofGreeks and Risk FactorsBlack–Scholes provides the canonical analytical setting for understanding option Greeks.
Foundation ofImplied Volatility Smile and SurfaceImplied volatility is obtained by inverting a benchmark option-pricing formula such as Black–Scholes across strikes and maturities.
Extended byStochastic Volatility (Heston) and Characteristic FunctionsHeston extends the constant-volatility Black–Scholes framework by making variance stochastic.
Built on · Arbitrage and No-Arbitrage
Black–Scholes valuation rests on dynamic replication and the exclusion of arbitrage.
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