Options Theory, Visually: Payoffs, Black–Scholes, and the Greeks
The Black–Scholes formula, C = S·N(d₁) − K e−rT·N(d₂), is the kind of result that's easy to memorise and hard to reconstruct: two cumulative normals that "fall out of the math" with no obvious reason for there being two. This note rebuilds the formula from the payoff upward, attaching every piece to a concrete mechanism — so the two terms have a meaning, not just a place in the equation.
Here's the whole story in one sentence, and everything below just unpacks it:
📈 The bent payoff: calls, puts, and P/L
Owning the stock is a straight line: gain a dollar when it rises, lose one when it falls. An option bends that line at the strike K. A call is the right to buy at K; a put is the right to sell at K. At expiry only the intrinsic value survives:
You pay a premium up front, so your profit & loss is the payoff minus what you paid (and flipped if you're the seller). Toggle the contract, drag the strike, the premium, and the spot — the solid line is payoff at expiry, the dashed line is your P/L:
⏳ Before expiry: intrinsic value plus time value
Before maturity an option is worth more than its intrinsic payoff, because the future is still uncertain — there's time for the bet to come good. The live price is a smooth curve sitting above the kinked payoff, and the gap is the time value:
Drag time-to-expiry T toward zero and watch the curve melt down onto the hockey stick — the time value bleeds away. That decay is theta, and it accelerates near expiry:
🌪️ Volatility is the whole game
Here is the engine room. Under the risk-neutral measure the stock drifts at the risk-free rate and its terminal price is log-normal. The option is worth the discounted average payoff over that distribution:
Now the punchline. The payoff is convex — clipped at zero on the downside, open on the upside — so spreading the distribution wider adds expected payoff (big gains aren't cancelled by big losses, because the losses are capped). Drag σ: the bell fattens, the shaded expected-payoff mass grows, and the price climbs. The readout computes e−rTE[payoff] numerically — and it lands on the Black–Scholes price below.
🧮 Black–Scholes in closed form
Where does the formula come from? Build a portfolio of the option and −Δ shares so the random moves cancel. A hedged, riskless portfolio must earn the risk-free rate — that no-arbitrage argument turns into the Black–Scholes PDE:
Solve it with the terminal condition V(S, T) = payoff(S) and the discounted-expectation integral collapses into two cumulative normals:
d1 = [ ln(S/K) + (r + σ²/2)T ] / (σ√T), d2 = d1 − σ√T
Read it as the bet itself: N(d2) is the risk-neutral probability the call finishes in the money; N(d1) is that probability re-weighted by how much stock you'd be holding. Drive all five inputs and watch the price and the curve respond:
🔢 The Greeks: slopes, curvature, and decay
The Greeks are just partial derivatives of the price — its sensitivities to each input. Two of them are right there in the picture: delta is the slope of the price curve, and gamma is its curvature. The tangent line below is delta; how fast that tangent rotates as you move spot is gamma.
Θ = ∂V/∂t = −S·N'(d1)σ/(2√T) − rK e−rTN(d2) · ρ = ∂V/∂r = K T e−rTN(d2)
Slide spot, volatility, and time. Δ a call's delta runs 0→1; Γ gamma peaks at the money; Θ theta is negative (time hurts the owner); 𝜈 vega is largest at the money and with more time left.
🎲 Proof by simulation: Monte-Carlo onto Black–Scholes
If the price really is a risk-neutral expectation, we should be able to just sample it. Draw thousands of terminal prices ST from the log-normal, take each discounted payoff, and average. By the law of large numbers that running average must crawl onto the closed-form Black–Scholes price. Press start and watch the noisy estimate home in on the dashed line: