Financial Mathematics · Lecture 10 of 12

The Binomial Model: Replication with Arithmetic

All of the previous lecture's measure theory, done with one step, two states and high-school algebra. Then stack the steps into a tree and watch it converge to Black–Scholes.

November 15, 2026By Diganta Bhattacharya5 min read

One step, two states

Today the stock is $S$. Tomorrow it is $uS$ or $dS$ (with $d \lt e^{r\Delta t} \lt u$, else there's an arbitrage). An option pays $V_u$ or $V_d$. Build a portfolio of $\Delta$ shares and $B$ in the bank that matches both outcomes:

$$\Delta uS+Be^{r\Delta t}=V_u,\qquad \Delta dS+Be^{r\Delta t}=V_d.$$

Subtract to get the hedge ratio, the discrete delta:

$$\Delta=\frac{V_u-V_d}{uS-dS}.$$

The portfolio costs $\Delta S+B$, so by no-arbitrage the option costs the same. Rearranging:

$$V=e^{-r\Delta t}\big(qV_u+(1-q)V_d\big),\qquad q=\frac{e^{r\Delta t}-d}{u-d}.$$

$q$ lies in $(0,1)$ and acts like a probability, but it was never assumed. It fell out of replication. The real probability of an up-move doesn't appear. $q$ is the risk-neutral probability, and $\mathbb E^q[S_{\Delta t}]=Se^{r\Delta t}$: the stock earns the risk-free rate on average under $q$, exactly like $\mathbb Q$ in Lecture 9.

Worked example$S=100$, $u=1.2$, $d=0.8$, $r=0$, call with $K=100$: $V_u=20$, $V_d=0$. Then $\Delta=20/40=0.5$ and $q=(1-0.8)/0.4=0.5$, so $V=10$. Check the hedge: buy ½ share for 50 and borrow 40, cost 10. Up: $60-40=20$ ✓. Down: $40-40=0$ ✓.

Many steps: the CRR tree

Split $[0,T]$ into $n$ steps of $\Delta t=T/n$ and choose (Cox–Ross–Rubinstein)

$$u=e^{\sigma\sqrt{\Delta t}},\qquad d=1/u,\qquad q=\frac{e^{r\Delta t}-d}{u-d}.$$

The $\sqrt{\Delta t}$ is the Brownian scaling from Lecture 6, so the log-price variance per step is $\sigma^2\Delta t$. Price by backward induction: payoffs at the leaves, then repeatedly apply the one-step formula back to the root:

def crr_call(S, K, T, r, sigma, n):
    dt = T / n; u = exp(sigma * sqrt(dt)); d = 1 / u
    q = (exp(r * dt) - d) / (u - d); disc = exp(-r * dt)
    V = [max(S * u**j * d**(n - j) - K, 0) for j in range(n + 1)]
    for i in range(n - 1, -1, -1):
        V = [disc * (q * V[j + 1] + (1 - q) * V[j]) for j in range(i + 1)]
    return V[0]

Why the limit is Black–Scholes

At expiry, the number of up-moves $j$ is binomial$(n,q)$ and $\ln S_T=\ln S+(2j-n)\sigma\sqrt{\Delta t}$. By the CLT (Lecture 1), as $n\to\infty$ this is normal with mean $(r-\tfrac12\sigma^2)T$ and variance $\sigma^2T$, which is the risk-neutral lognormal law. So the tree price converges to $e^{-rT}\mathbb E^{\mathbb Q}[(S_T-K)^+]$, the Black–Scholes price. The zig-zag in the figure (odd vs even $n$) comes from where the strike falls relative to the terminal nodes.

What trees can do that formulas can't

American options: at each node take $\max(\text{continuation},\text{exercise})$. No closed form exists, and trees handle it in one line (see the callable-bond and Bermudan-swaption posts in Finance in Banks).
Dividends, barriers, varying rates: adjust the tree node by node.
Intuition: delta at each node is the hedge ratio, and you can watch it move.

Key takeaways

One-step replication gives Δ = (V_u − V_d)/(uS − dS) and V = e^{−rΔt}(qV_u + (1−q)V_d).
q = (e^{rΔt} − d)/(u − d) is the risk-neutral probability. Real-world probabilities never enter.
With u = e^{σ√Δt}, the CRR tree converges to Black–Scholes by the CLT and handles American exercise.

Exercises

Price a one-step put with S = 50, u = 1.1, d = 0.9, r = 0, K = 50, and give the replicating portfolio.
Build a 3-step CRR tree for S = K = 100, σ = 20%, r = 5%, T = 1. Compare with Black–Scholes.
Modify the code for an American put. When is early exercise optimal?