Black-Scholes: pricing European options closed-form
The formula every trading-desk candidate is expected to walk through in a first-round interview. One closed-form expression for a vanilla call — and the replicating-portfolio argument that gets you there is the interview question.
Lookang; model by Francisco Esquembre, Fu-Kwun, and Lookang β https://commons.wikimedia.org/wiki/File:Brownian_motion_large.gif · CC BY-SA 3.0 · Wikimedia Commons
Black–Scholes turns a vanilla European call or put into a single closed-form expression β no simulation, no tree, just plug-and-chug. The model bakes in lognormal stock dynamics, constant volatility, no dividends, frictionless trading (no fees, continuous hedging), and a constant risk-free rate. The recipe is mechanical: compute $d_1 = [\ln(S/K) + (r + \tfrac{1}{2}\sigma^2)T]/(\sigma\sqrt{T})$, then $d_2 = d_1 - \sigma\sqrt{T}$, then plug into $C = S\,N(d_1) - K e^{-rT} N(d_2)$ for a call (or the put analog). The hard part isn’t the formula β it’s building intuition for how each input bends the price. Pair this with the Greeks Explorer for a slider-driven visual feel.
β Intro Β· expand
Try first (productive failure)
Before the worked example: spend 60 seconds taking your best shot at this.
A guess is fine β being briefly wrong about a problem makes the explanation
land harder when you read it. This appears once per tutorial; skip
if you already know the trick.
60s
β Try first Β· expand
Worked example
Price a European call on a non-dividend-paying stock using Black–Scholes. Inputs: spot $S = \$100$, strike $K = \$100$, time to expiry $T = 0.5$ years, volatility $\sigma = 0.20$ (20% annualized), risk-free rate $r = 0.05$ (5% continuously compounded). Compute $d_1$, $d_2$, $N(d_1)$, $N(d_2)$, then the call price $C$. Round each step to 3–4 d.p.
β Worked example Β· expand
Practice 1 of 3Type a fraction, decimal, or expression β mathjs parses it.
β Practice Β· expand
Reflection
If volatility $\sigma$ doubles, the call price grows but does <em>not</em> double β why is the relationship sublinear? And in the put formula, why do we discount the strike $K$ but not the spot $S$? What does that asymmetry tell you about which side of the trade carries the financing burden in the risk-neutral measure?
