3.2.3 Binomial Trees & Convergence to Black–Scholes

Basic Prerequisites: 02 · No-Arbitrage & Risk-Neutral and Stochastic Calculus & Itô’s Lemma.

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1. Intuition & Practical Objective

A one-period binomial has two states; a real market has a continuum. The multi-period binomial tree bridges the gap: chop $[0,T]$ into $n$ steps, put a two-state bet at each, and let the coin be re-tossed at every node. The stock after $n$ steps is $S_0{u}^Y{d}^{n-Y}$ with $Y\sim \text{Binomial}(n,p)$ - a discrete approximation to the lognormal. Price each node by backward induction and, as $n\to \infty$, the price converges to the Black–Scholes–Merton value.

The practical objective of this page is the Cox–Ross–Rubinstein (CRR) recipe - how to choose $u,d,p$ so the tree has the right volatility - plus an honest picture of the error: CRR converges at $O(1/n)$, not monotonically.

Why it matters beyond pedagogy. A tree is the default engine for American options, discrete dividends, and any payoff with a kink or an exercise boundary - cases where BSM has no closed form. Knowing exactly how the discretization errs is what separates a reliable tree price from a plausible-looking wrong one.

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2. Mathematical Ground Truth & Derivations

2.1 CRR parameters (Cox–Ross–Rubinstein 1979; Hull 13.15–13.18; Haug 7.5–7.6)

With $\Delta t=T/n$, choose the up/down factors to match the volatility of geometric Brownian motion:

$$\boxed{u=e^{\sigma \sqrt{\Delta t}},d=\frac{1}{u}=e^{-\sigma \sqrt{\Delta t}}}$$

so the log-increment per step is $\pm \sigma \sqrt{\Delta t}$ and the per-step log-variance is $4{\sigma }^2\Delta tp(1-p)\approx {\sigma }^2\Delta t$ for $p\approx \frac{1}{2}$ - matching GBM’s ${\sigma }^2\Delta t$. The tree is recombining: an up-then-down equals a down-then-up, so after $n$ steps there are $n+1$ nodes (not $2^n$), with

$$S_{j,i}=S{u}^i{d}^{j-i}\text{at step }j\text{ after }i\text{ up-moves},\#\text{paths}=\left(\genfrac{}{}{0ex}{}{j}{i}\right).$$

The risk-neutral probability is the same bracket-derived expression as before, with $e^{b\Delta t}$ the per-step growth of the underlying’s carry ($b=r$ stock, $b=r-q$ index/continuous yield):

$$\boxed{p=\frac{e^{b\Delta t}-d}{u-d}}\text{(Hull 13.17)}$$

2.2 European closed form (CRR sum) and the efficient truncation

Because the terminal distribution is binomial, a European claim has a finite sum over terminal nodes (Haug eq. 7.1):

$$V_0=e^{-rT}\sum _{i=0}^n\left(\genfrac{}{}{0ex}{}{n}{i}\right)p^i(1-p{)}^{n-i}\max (S{u}^i{d}^{n-i}-X,0).$$

Only nodes with $S{u}^i{d}^{n-i}>X$ contribute, so one starts the sum at the smallest integer $i$ with $S{u}^i{d}^{n-i}>X$, i.e. $i>(\ln (X/S)-n\ln d)/\ln (u/d)$ - the standard speed-up from $O(n^2)$ backward induction to $O(n)$ for European claims (Haug eq. 7.3–7.4).

2.3 American backward induction (Haug 7.9–7.11; Hull §13.5)

For an American contract the holder compares continuation with intrinsic value at every node:

$$\boxed{P_{j,i}=\max (X-S{u}^i{d}^{j-i},e^{-r\Delta t}[p{P}_{j+1,i+1}+(1-p)P_{j+1,i}\left]\right)}$$

The tree is the natural American engine: no free boundary to solve, the exercise region is discovered node by node. (Full treatment on 06 · Advanced Extensions.)

2.4 Convergence

As $n\to \infty$ with $\Delta t=T/n\to 0$, the tree’s terminal distribution converges to the lognormal $S_T=S_0{e}^{(r-\frac{1}{2}{\sigma }^2)T+\sigma \sqrt{T}Z}$, and the CRR price converges to the BSM price at rate $O(1/n)$ - with a non-monotone, oscillating error. The convergence theorem (Hull §13.9 and its appendix) is exact in the limit; the oscillation is a finite-$n$ artifact.

Other parameterizations trade one property for another (Haug §4.3–4.5): Rendleman–Bartter ($p=\frac{1}{2}$, exponential $u/d$), Jarrow–Rudd (equal-probability lognormal matching), Leisen–Reimer (Peizer–Pratt inversion, $O(1/n^2)$-class accuracy at odd $n$), and the trinomial tree (Boyle 1986) with a third, middle state.

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3. Computational Implementation - convergence and its oscillation

Stdlib only. Two experiments: ($a$) the European put price as $n$ grows against the closed form; ($b$) the CRR error for an at-the-money American put, which shows the oscillation.

Read the table as two messages. First, CRR does converge: $n=1000$ gives $4.4496$ against BSM’s $4.4494$ (exactly Haug’s verified pair). Second, the error does not fall monotonically - it drops to $0.00026$ at $n=50$, rises to $0.00581$ at $n=200$, falls again. The ATM American series is worse: values jump by $\sim 0.5$ as $n$ changes parity, because the terminal payoff kink sits exactly on a tree node for even $n$ and between nodes for odd $n$. This is the well-documented CRR oscillation.

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4. Failure Modes & First-Principles Breakdowns

1. Trusting a single $n$. Because the error oscillates, doubling $n$ can make the price worse. Standard practice: average $n$ and $n+1$ (or $n\pm 1$) prices, or use a smoothing scheme (Leisen–Reimer, or averaging the payoff over $\pm$ half a node). Reporting one tree value without an $n$-study is not a price.
2. Negative “probabilities”. CRR needs $d<e^{r\Delta t}<u$, i.e. $r\sqrt{\Delta t}<\sigma$. With small $\sigma$ or a coarse step (long-dated, few nodes) $p$ leaves $[0,1]$ and the tree admits arbitrage - see 05 · Failure Modes.
3. Drift vs carry. Using $b=r$ for an index/FX/commodity with a yield or foreign rate silently drops an $e^{-q\Delta t}$ / $e^{-r_f\Delta t}$ factor from every node; the per-step growth factor is $e^{b\Delta t}$, not $e^{r\Delta t}$ (Hull §13.11).
4. Recombination assumed, not checked. The CRR tree recombines because $d=1/u$. Non-CRR variants (e.g. non-constant local volatility, “displaced” trees) may not; then node counts explode and the $O(n^2)$ cost becomes a real constraint.
5. Tree “Greeks” carry discretization noise. Delta from adjacent nodes, $\Delta =\frac{f_{1,1}-f_{1,0}}{Su-Sd}$ (Haug 7.9), is a finite difference on a coarse grid; its error is larger than the price error and inherited from the same oscillation.

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5. References

• Hull, Options, Futures, and Other Derivatives
• Haug, The Complete Guide to Option Pricing Formulas
• Shreve, Stochastic Calculus for Finance I
• Cox, Ross & Rubinstein (1979), Option pricing: a simplified approach, J. Financial Economics 7

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6. Connected Graph Bridges

• Back: 02 · No-Arbitrage & Risk-Neutral · Index Hub
• Forward: 04 · Fundamental Theorems · 06 · Advanced Extensions
• Siblings: Black–Scholes–Merton · The Greeks & Dynamic Hedging