Core Concept · Derivatives· 12 min read

Options Pricing & Greeks

An option's price is the cost of the portfolio that replicates it — no more, no less. Black-Scholes makes that idea concrete under a few assumptions, and the Greeks are simply the price's sensitivities to the things that move. Interviews want the intuition (replication, hedging, risk-neutral pricing), not a memorised PDE.

Why this matters in interviews

Pricing a vanilla and explaining its Greeks is the canonical options warm-up for trading and dev seats.
The Greeks are the language desks use to talk about risk all day.
Confusing the real-world and risk-neutral measures is the single most-watched-for error.

Core intuition

No-arbitrage + replication: if you can build a portfolio that pays exactly the option's payoff in every state, the option must cost what that portfolio costs — otherwise there's free money. Pricing is a hedging argument in disguise.

Risk-neutral pricing: under the risk-neutral measure every asset drifts at the risk-free rate, and the price is the discounted expected payoff. The real-world drift μ never enters the price — it's a forecasting quantity, not a pricing one.

The Greeks are derivatives of the price: delta w.r.t. spot, gamma w.r.t. delta, vega w.r.t. vol, theta w.r.t. time. A market maker is paid theta for carrying gamma and hedges delta to stay neutral.

Key ideas & definitions

Replication / hedging: Build the payoff from stock + bond (or other options); the price is the build cost.
Risk-neutral measure (Q): The pricing measure where drift = r; price = e^(−rT)·E_Q[payoff].
Delta (Δ): ∂Price/∂Spot. For a call, N(d1). The hedge ratio.
Gamma (Γ): ∂Δ/∂Spot. Largest at-the-money; it's the cost/benefit of re-hedging.
Vega: ∂Price/∂Vol. Largest for longer-dated, near-the-money options.
Theta (Θ): ∂Price/∂Time. Usually negative for a long option — the price of holding optionality.

Key formulas & relationships

Black-Scholes call

C = S·N(d1) − K·e^(−rT)·N(d2)

d1, d2

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

Put-call parity

C − P = S − K·e^(−rT)

Model-free: a pure no-arbitrage relation, true regardless of Black-Scholes.

Core Greeks (call)

Δ = N(d1), Γ = n(d1)/(S·σ√T), Vega = S·n(d1)·√T

n(·) is the normal pdf.

Worked reasoning

Why is gamma highest at the money, and why does a market maker care?

01Delta swings from ~0 (deep OTM) to ~1 (deep ITM); the steepest part of that S-curve is at the money.
02Gamma is the slope of delta, so it peaks ATM and decays in the wings.
03High gamma means your delta hedge goes stale fast — you re-hedge more often, paying the bid-ask each time. That cost is what theta compensates.

Common mistakes & misconceptions

Using the real-world drift μ to price instead of r (the risk-neutral drift).
Forgetting to discount the strike (the e^(−rT) factor).
Calling delta 'the probability of finishing in the money' — that's closer to N(d2), and only under Q.
Quoting vega per 100% vol move without stating the per-point convention.

In interview terms

Lead with replication/no-arbitrage — it shows you understand why the formula is true.
Keep the two measures straight: r prices, μ forecasts.
Talk about the Greeks as a hedger would: delta to stay neutral, gamma/theta as the trade-off.

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Study alongside

Concept — Volatility & the Vol Surface →Take-home — Black-Scholes Pricer →Take-home — Monte Carlo Pricer →

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