How does the binomial option pricing model converge to the Black-Scholes formula as the number of steps increases?

The binomial model converges to Black-Scholes because, as the number of time steps increases, the binomial distribution of log-returns approaches a normal distribution by the Central Limit Theorem. BSM is literally the continuous-time limit of the discrete binomial tree.

The Connection:

In a binomial model with N steps over time T:

• Each step has length dt = T/N
• Up factor: u = exp(sigma x sqrt(dt))
• Down factor: d = exp(-sigma x sqrt(dt)) = 1/u
• Risk-neutral probability: p = (exp(r x dt) - d) / (u - d)

As N increases to infinity, dt approaches 0, and the sum of N binomial steps (each being a small up or down move) converges to a continuous geometric Brownian motion with drift r and volatility sigma. The binomial option price converges to the BSM analytical price.

Numerical Demonstration:

Pricing a European call on Halford Tech: S_0 = $50, K =$52, r = 5%, sigma = 25%, T = 1 year.

Steps (N)	Binomial Price	BSM Price	Difference
5	$5.02	$5.06	-$0.04
25	$5.05	$5.06	-$0.01
100	$5.06	$5.06	$0.00

The binomial price oscillates around the BSM value, converging smoothly. The oscillation occurs because odd and even numbers of steps produce slightly different lattice configurations. Using the average of N and N+1 step prices (the Richardson extrapolation) accelerates convergence.

Why This Matters:

1. Theoretical foundation: The convergence proof validates BSM by showing it emerges naturally from simple discrete no-arbitrage arguments, not just from stochastic calculus.

2. Practical application: The binomial model can price American options (with early exercise) and exotic options where BSM has no closed-form solution. Knowing it converges to BSM for European options gives confidence in the framework.

3. Intuition bridge: The binomial model is easier to understand intuitively (stock goes up or down each period). Its convergence to BSM shows that the elegant but abstract BSM formula is simply the "fast version" of an intuitive process.

Convergence Rate:

The binomial model converges to BSM at a rate of O(1/N), meaning doubling the steps roughly halves the error. For practical purposes, 200-500 steps give accuracy within a fraction of a cent for standard options.

Key Parameters in the Limit:

As N approaches infinity:

• u approaches 1 + sigma x sqrt(dt) (for small dt)
• p approaches 0.5 + (r - 0.5 x sigma^2) x sqrt(dt) / (2 x sigma)
• The binomial log-return distribution approaches N((r - 0.5 x sigma^2) x T, sigma^2 x T)
• This is exactly the BSM assumption of log-normal terminal prices

Practice binomial pricing in our CFA Derivatives question bank.