Quantitative model · Derivatives

Option valuation under Black–Scholes–Merton and Cox–Ross–Rubinstein

Research by Kyle Wisniewski · Quantitative Markets & Institutions Lab

The model compares closed-form and lattice values, sensitivities, convergence, and early-exercise effects under one set of inputs.

Decision summary

Use agreement to validate the implementation—not the market model

Interpretation

For European options, Black–Scholes and a sufficiently fine CRR tree should converge. For American exercise, the lattice also represents early-exercise value that the closed form omits.

Decision context

Cross-method agreement shows that a pricing implementation is numerically coherent. It does not show that constant volatility, continuous hedging, or Gaussian returns describe the market.

Intended analytical use

Derivatives practitioners, quantitative developers, model validators, and researchers use it to separate numerical error from model risk.

Principal limitation

This educational tool uses user-selected inputs rather than a live option chain. Its outputs are model values, not executable prices or investment advice.

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. The current strike, rate, volatility, and yield stay fixed. Each maturity has a distinct line pattern. The dashed gray line is intrinsic value.

View representative option-value data

Price against volatility; the slope is vega. An open circle marks the current volatility. Under these assumptions, option value generally rises with volatility.

View representative volatility data

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, with no arbitrage. Reasonable at short horizons; strained across regimes.
Continuing analytical utility: 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}

The lattice prices by backward induction from the terminal payoffs, discounting the expected value one step at a time. Two properties make it more than a numerical curiosity. First, the CRR price converges to Black-Scholes as \(N \to \infty\). The tree is the binomial construction of Brownian motion. Raising \(N\) makes that convergence visible. The oscillation 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. That is 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 numerical difference therefore carries two meanings. Read it by contract type.