Module 01 · Derivatives

Option Pricing Laboratory

Two canonical routes to the arbitrage-free price of a European option — the Black-Scholes-Merton closed form and the Cox-Ross-Rubinstein binomial lattice — implemented side by side, with every Greek computed live in your browser.

Contract & Market

Spot S 100

Strike K 100

Maturity T (years) 1.00

Risk-free rate r 5.0%

Volatility σ 20.0%

Dividend yield q 0.0%

Binomial lattice

CRR steps N 200

American style is priced on the lattice only; the closed form applies to European exercise.

Black-Scholes (European)—

CRR Binomial—

Difference—

Delta Δ—

Gamma Γ—

Vega (per 1% vol)—

Theta (per day)—

Rho (per 1% rate)—

Intrinsic value—

Greeks are Black-Scholes analytical values for the European contract.

Option value against the underlying price at several times to maturity, holding the current strike, rate, volatility, and yield fixed. The dashed line is intrinsic value.

Price as a function of volatility (vega is the slope). Optionality is, in a precise sense, a long position in uncertainty.

The closed form

Black-Scholes-Merton

Under the model's assumptions, the arbitrage-free price of a European call on an asset paying a continuous dividend yield \(q\) is

C(S,t) = S e^{-q\tau}\,N(d_1) - K e^{-r\tau}\,N(d_2)$$ $$d_1 = \frac{\ln(S/K) + \left(r - q + \tfrac{1}{2}\sigma^2\right)\tau}{\sigma\sqrt{\tau}}, \qquad d_2 = d_1 - \sigma\sqrt{\tau}, \qquad \tau = T - t

with \(N(\cdot)\) the standard normal CDF, evaluated here with an erf approximation (Abramowitz & Stegun 7.1.26, accurate to about \(1.5\times10^{-7}\)). The put follows from put-call parity. The formula's power comes from what it assumes away:

Geometric Brownian motion: The underlying follows \(dS = \mu S\,dt + \sigma S\,dW\) with constant volatility — log-returns are i.i.d. Gaussian. Real returns have fat tails and volatility that clusters.
Frictionless, continuous hedging: The derivation replicates the option by trading the stock continuously with no costs or gaps. Discrete hedging and transaction costs leave residual risk the price ignores.
Constant rate, no arbitrage: A single risk-free rate for borrowing and lending, and no free lunches. Reasonable at short horizons; strained across regimes.
Why it survives anyway: Because the market uses it in reverse: quote the price, invert for implied volatility. The formula became a coordinate system rather than a forecast — a theme the case study takes up in full.

The lattice

Cox-Ross-Rubinstein

The binomial model discretizes time into \(N\) steps of length \(\Delta t = \tau/N\). Each step the price moves up by \(u = e^{\sigma\sqrt{\Delta t}}\) or down by \(d = 1/u\), with risk-neutral probability

p = \frac{e^{(r-q)\Delta t} - d}{u - d}

Prices are computed by backward induction from the terminal payoffs, discounting the expected value one step at a time. Two properties make the lattice more than a numerical curiosity. First, as \(N \to \infty\) the CRR price converges to Black-Scholes — the tree is the binomial construction of Brownian motion, and watching the two prices agree as you raise \(N\) above is convergence made visible (the oscillation you see at low \(N\) is the tree's discrete strikes straddling the true value). Second, the lattice handles what the closed form cannot: early exercise. At each node an American option is worth the greater of continuation value and immediate exercise, a free-boundary problem with no closed-form solution. Toggle the American put above and note the premium over its European twin — that gap is the value of the right to exercise early, largest deep in the money and when rates are high.

Reading the difference column For European contracts the BS-CRR gap is pure discretization error, shrinking like \(O(1/N)\). For American contracts the gap is economically real: early-exercise value the European formula structurally omits. The same number on screen, two entirely different meanings — a small lesson in why model output needs interpretation before it needs decimals.