3.2.6 Advanced Extensions

Basic Prerequisites: 04 · Fundamental Theorems and 05 · Failure Modes.

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1. Intuition & Practical Objective

The binomial model is not just the toy that BSM approximates. Two of its features survive the limit and are harder in continuous time, so the tree is where they should be learned:

1. Early exercise. An American claim is a free-boundary / optimal-stopping problem. In a tree the boundary is discovered node by node; in continuous time it is a smooth-pasting condition. Everything a desk does with American puts, Bermudan swaptions, or callable structures starts here.
2. The passage to continuous time. Scaling the random walk $u=e^{\sigma \sqrt{\Delta t}}$, $d=e^{-\sigma \sqrt{\Delta t}}$ as $\Delta t\to 0$ produces Brownian motion and the lognormal - which is why the tree price converges to the BSM price, and why “binomial = discrete BSM” is a theorem and not an analogy.

This page closes the folder by making both directions concrete.

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2. Mathematical Ground Truth & Derivations

2.1 American pricing as optimal stopping (Shreve Ch 5–6)

$$v_n(x)=g(x),\boxed{v_k(x)=\max \left\{\frac{1}{1+r}\right[\tilde{p}{v}_{k+1}(ux)+\tilde{q}{v}_{k+1}(dx)],g(x)\}}$$

Characterizing properties (Shreve Def 6.1):

• (a) Value = optimal stopping value: $V_k=(1+r{)}^k{\max }_{\tau \ge k}\tilde{\mathbb{E}}[(1+r{)}^{-\tau }{G}_{\tau }\mid F_k]$ over stopping times $\tau$ ($\{\tau =k\}\in F_k$ - no look-ahead).
• (b) Smallest supermartingale: $\{(1+r{)}^{-k}{V}_k\}$ is the smallest $\tilde{\mathbb{P}}$-supermartingale dominating $\{G_k\}$.
• (c) Optimal exercise time: ${\tau }^{\ast }=\min \{k:V_k=G_k\}$ attains the maximum.
• (d) Hedge: same difference quotient as the European case, but with a consumption stream $C_k=V_k-\frac{1}{1+r}\tilde{\mathbb{E}}[V_{k+1}\mid F_k]\ge 0$ (Lemma 2.21).

The perpetual (infinite-horizon) version has a clean closed form. With $u=2,d=\frac{1}{2},r=\frac{1}{4},K=5$ and $\tilde{p}=\tilde{q}=\frac{1}{2}$, the candidate value is piecewise (Shreve Ch 8.8)

$$v(x)=\{\begin{array}{ll}\frac{6}{x},& x\ge 3,\\ 5-x,& 0<x\le 3,\end{array}\text{exercise the first time }S\le 2.$$

Verified the only way an American value can be: (i) $v(x)\ge (5-x{)}^{+}$ everywhere; (ii) $\{(4/5{)}^{k}v(S_k)\}$ is a supermartingale; (iii) $v$ is the smallest such process.

2.2 The random walk underneath (Shreve Ch 8)

Under $\tilde{\mathbb{P}}$, ${\log }_2{S}_k={\log }_2{S}_0+M_k$ (equivalently $S_k=S_0{2}^{M_k}$) with $M$ a symmetric $\pm 1$ walk. First-passage moments come from the exponential martingale: for $0<\alpha <1$,

$$\mathbb{E}[{\alpha }^{{\tau }_1}]=\frac{1-\sqrt{1-{\alpha }^2}}{\alpha }⟹\mathbb{E}[{\alpha }^{{\tau }_m}]={\left(\frac{1-\sqrt{1-{\alpha }^2}}{\alpha }\right)}^m.$$

With $\alpha =\frac{4}{5}$ this is $\frac{1}{2}$, which prices the perpetual put by hand: stopping at $S=2$ gives $3\cdot \frac{1}{2}=\frac{3}{2}$; stopping at $S=1$ gives $4\cdot (\frac{1}{2}{)}^2=1$; so $\frac{3}{2}$ (stop at 2) is optimal. (The walk bridges to Brownian motion through Donsker’s theorem - the same scaling this page verifies numerically.)

2.3 Toward continuous time

Each step is $\log (S_{k+1}/S_k)=\pm \sigma \sqrt{\Delta t}$. Summing $n$ steps gives

$$\log \frac{S_T}{S_0}=\sigma \sqrt{\Delta t}\sum _{k=1}^n{\epsilon }_k,{\epsilon }_k=\pm 1,\Delta t=\frac{T}{n},$$

whose variance is ${\sigma }^{2}T$ and whose standardized sum converges (CLT / Donsker) to $N((r-\frac{1}{2}{\sigma }^2)T,{\sigma }^{2}T)$ - the lognormal of geometric Brownian motion. Substituting that limiting distribution into the risk-neutral expectation $e^{-rT}\tilde{\mathbb{E}}[(S_T-K{)}^{+}]$ is the Black–Scholes–Merton derivation (BSM · 02) - which is exactly why the CRR price converges to the BSM price $O(1/n)$.

Beyond the limit, the two structural extensions are jump-diffusion (Merton 1976: a Poisson mixture of lognormals, closed form but an incomplete market - FT2 fails, 04) and stochastic/local volatility (Heston & SABR).

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3. Computational Implementation - three checks, stdlib only

(a) The step scaling becomes Gaussian - $\log u=-\log d$ at every $\Delta t$, so the walk is a symmetric random walk in the log.

(b) The perpetual American put - the piecewise closed form and the first-passage moments, against Shreve’s worked numbers.

$v(4)=1.5$ and $v(8)=0.75$ match the random-walk rule values exactly, and $v(x)=6/x$ reproduces $3\cdot (\frac{1}{2}{)}^{j-1}$ on $x=2^j$ - the piecewise formula and the stopping rule are the same object.

(c) American via CRR - the tree as the practical American engine.

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4. Failure Modes & First-Principles Breakdowns

1. Applying the European recursion to an American claim. The single max with intrinsic value is the American algorithm; omitting it silently returns the European price and a $5\%$-scale error.
2. Getting the measurability wrong. A stopping time must satisfy $\{\tau =k\}\in F_k$. “Exercise when the running minimum hits a level” is a valid stopping rule only if evaluated with information available at $k$; using $S_{k+1}$ is look-ahead and overstates value (Shreve §5.2’s explicit non-example).
3. Assuming an American hedge is the European hedge. Where exercise is optimal the delta hedge is undefined by the two terminal equations (Shreve Ex. 5.1 yields inconsistent ${\Delta }_1(T)$), which is exactly why the supermartingale-plus-consumption formulation (Lemma 2.21) is the correct object.
4. Reading the continuous limit too literally. Convergence to BSM is $O(1/n)$ with oscillation and assumes the lognormal; a tree calibrated to make the one-step distribution lognormal does not make the multi-step distribution lognormal at finite $n$.
5. Confusing “converges to BSM” with “BSM is right”. The tree converges to BSM because both assume a single lognormal driver. Real markets jump and smile; where the tree adds value is precisely where BSM has no closed form (American, discrete dividends, path-dependence), not where it claims to be exact.

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5. References

• Shreve, Stochastic Calculus for Finance I
• Hull, Options, Futures, and Other Derivatives
• Haug, The Complete Guide to Option Pricing Formulas
• Björk, Arbitrage Theory in Continuous Time

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6. Connected Graph Bridges

• Back: 05 · Failure Modes · Index Hub
• Forward: Black–Scholes–Merton · Implied Volatility Surfaces · Heston & SABR · Interest-Rate & Term-Structure Models · The Greeks & Dynamic Hedging
• Base: Stochastic Calculus & Itô · Probability & Measure Theory