Black-Scholes & The Greeks

Call option · value now

$—

at S=100, K=100, T=1y, σ=20%

The right to buy at K for the next T years. The green curve is what that right is worth now.

Δ delta—

Γ gamma—

ν vega—

Θ theta—

ρ rho—

A call option is the right — not the duty — to buy a stock at a fixed price K within the next T years. Black-Scholes (1973) is the formula that prices that right. Drag the sliders, or tap a preset, and watch the price curve, the payoff line, and all five Greeks move together.

Jump to a scenario

S — current stock price100

K — strike (the agreed buy price)100

T — time to expiry, in years1.00

r — risk-free rate0.05

σ — volatility (annualised stdev)0.20

Try this Drag S from 60 → 140. The green curve always sits above the dashed payoff — that gap is "time value." Now drag T down to 0.01 and the curve collapses onto the payoff. Time value goes to zero at expiry.

What you're looking at

The green curve is what the option is worth right now, for any spot price S. The dashed line is what it would pay at expiry: zero below K, then a line slanting up. The dot marks the current spot.

The five strips below plot the Greeks — how the price responds to small changes. Each is a slope: Δ says "if S moves +1, the option moves +Δ"; Γ says "if S moves +1, Δ itself moves +Γ"; and so on across the whole spot range.

The Black-Scholes formula

For a European option on a non-dividend-paying stock:

d₁ = (ln(S/K) + (r + σ²/2)·T) / (σ·√T)
d₂ = d₁ − σ·√T

C = S·N(d₁) − K·e^−rT·N(d₂)
P = K·e^−rT·N(−d₂) − S·N(−d₁)

where N(·) is the standard normal CDF. This sim uses the Abramowitz-Stegun 7.1.26 polynomial approximation, accurate to ~1×10⁻⁷.

Test point: S=100, K=100, r=0.05, σ=0.20, T=1 → C ≈ 10.4506, P ≈ 5.5735.

The five Greeks

Δ delta: ∂price/∂S — how much the option moves when S moves by 1. Call Δ runs 0→1, put Δ runs −1→0.

Γ gamma: ∂²price/∂S² — curvature, same for call and put. Largest when ATM and short-dated.

ν vega: ∂price/∂σ — per 1.00 (i.e. 100%) move in vol. Same for call and put. Largest when ATM and far from expiry.

Θ theta: ∂price/∂t — per year. Almost always negative for long options. You pay rent every day you hold.

ρ rho: ∂price/∂r — per 1.00 move in rates. Calls have positive ρ, puts negative.

Put-call parity

For European options on a non-dividend stock:

C − P = S − K·e^−rT

This is a no-arbitrage identity, not a model assumption — it holds regardless of which pricing model you use, as long as both the call and put exist on the same underlying with the same K and T. Try it: at S=100, K=100, r=0.05, T=1, parity says C − P = 100 − 95.123 = 4.877. The sim values should match within rounding.

What this model assumes

Constant volatility, constant rates, frictionless trading, continuous hedging, lognormal returns, no dividends, no jumps. All of these fail in the real world to varying degrees.

The model still anchors derivatives pricing because (a) its closed form is computable in microseconds, (b) it gives the right intuition for how Greeks move, and (c) it provides the calibration target for more sophisticated models. Quoting "implied volatility" instead of price is how traders communicate within the BS framework while implicitly admitting σ isn't constant.

Citations

Black, F. & 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, 4, 141–183.
Hull, J.C. Options, Futures, and Other Derivatives, 11th ed., Ch. 15 (BS), Ch. 19 (Greeks).
Abramowitz, M. & Stegun, I. (1964). Handbook of Mathematical Functions, formula 7.1.26 (normal CDF approximation).

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