3.2 No-Arbitrage & the Binomial Model

Basic Prerequisites: Probability & Measure Theory and high-school algebra. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

Two prices for the same payoff cannot coexist, because the difference is free money. That single sentence - the law of one price - is the whole engine of derivative pricing, and it is worth nothing unless you can build the alternative: a portfolio of traded assets that reproduces the payoff exactly. The binomial model is the smallest market in which that construction is possible, and it is the model where every concept of continuous-time finance (risk-neutral measure, martingale, state price, completeness, dynamic hedging) appears in a form you can compute by hand.

This folder is the hub of that topic. It gives (a) the fast formula lookup below - job #1 of this pillar - and (b) six sub-pages that walk from one-period replication through multiperiod trees, the Fundamental Theorems, the failure modes, and the road to continuous time.

The one-sentence essence. “A derivative’s price is the cost of the cheapest self-financing portfolio that replicates it; because that cost is independent of the real-world drift, the price can always be written as a discounted expectation under an artificial measure $P$ - and under that measure the discounted stock is a martingale.”

2. Mathematical Ground Truth & Derivations

Quick-Reference Lookup (job #1). Everything below is transcribed from the verified corpus (Shreve Vol I Ch 1–3, 5–8; Björk Ch 2–3; Hull Ch 13; Haug §4.1–4.2) and every number in the check column was re-executed from the scripts in §3 and the sub-pages.

Notation: $S_{k}$ stock at step $k$ , $u > 1$ up-factor, $d < 1$ down-factor, $r$ one-period risk-free rate, $T$ maturity, $n$ number of steps, $Δ t = T / n$ , $f_{u}, f_{d}$ the claim’s up/down payoffs, $b$ cost-of-carry, $σ$ volatility, $ζ (ω)$ the state price of state $ω$ .

Quantity	Formula	Verified check
One-period no-arbitrage bracket (Shreve 1.2; Björk Prop 2.3)	$d < 1 + r < u$	$S_{0} = 4, u = 2, d = \frac{1}{2}, r = \frac{1}{4}$ : $0.5 < 1.25 < 2$ ✓
Risk-neutral probabilities (Shreve 1.8)	$\tilde{p} = \frac{1 + r - d}{u - d}, \tilde{q} = \frac{u - 1 - r}{u - d}$	$\tilde{p} = \tilde{q} = 0.5$ , $\tilde{p} + \tilde{q} = 1$
Replicating delta (Shreve 1.6)	$Δ_{0} = \frac{f _{u} - f _{d}}{S _{0} ( u - d )}$	put $K = 5$ : $Δ_{0} = - 0.5000$ , bond $= 3.2000$
One-period price (Shreve 1.9)	$V_{0} = \frac{1}{1 + r} [\tilde{p} f_{u} + \tilde{q} f_{d}]$	$V_{0} = 1.2000$ ; replication both states $= 0.0000/3.0000$ ✓
Multiperiod RN valuation (Shreve §3.4)	$V_{k} = (1 + r)^{k} E [\frac{V _{m}}{( 1 + r ) ^{m}} F_{k}]$	discounted stock: $E [S_{1} / (1 + r)] = 4.000000 = S_{0}$
State price (Arrow–Debreu)	$ζ (ω) = \frac{P ( ω )}{1 + r}$	$ζ (H) = ζ (T) = 0.4$ , $\sum ζ = 0.8 = \frac{1}{1 + r}$
State-price valuation	$V_{0} = \sum_{ω} ζ (ω) V_{1} (ω) = E^{P} [ζ V_{1}]$	$1.2000$ for any physical $p$ (checked $p = 0.5, 0.6, 0.9$ )
CRR up/down (CRR 1979; Hull 13.15/13.16)	$u = e^{σ Δ t}, d = \frac{1}{u} = e^{- σ Δ t}$	$n = 100$ , $T = 0.5$ : $u = 1.021440$ , $d = 0.979010$
CRR risk-neutral prob (Hull 13.17)	$p = \frac{e ^{b Δ t} - d}{u - d}$ ( $b = r$ stock, $b = r - q$ index)	same: $p = 0.504126$ , $d < e^{r Δ t} < u$ ✓
CRR European put (Haug 7.1–7.6)	$e^{- r T} i = 0 \sum n (i n) p^{i} (1 - p)^{n - i} (X - S u^{i} d^{n - i})^{+}$	$n = 1000 \Rightarrow 4.4496$ vs BSM $4.4494$ (Haug-verified)
CRR American put (Haug 7.9–7.11)	$P_{j, i} = max (X - S u^{i} d^{j - i}, e^{- r Δ t} (p P_{j + 1, i + 1} + (1 - p) P_{j + 1, i}))$	$n = 1000 \Rightarrow 4.6921$ (early-exercise premium $0.2427$ )
First Fundamental Theorem (Shreve II 5.4.7; Björk Ch 3)	arbitrage-free $⟺$ an equivalent martingale measure exists	$1$ -stock $3$ -state market: EMM set is $1$ -dimensional
Second Fundamental Theorem (Shreve II 5.4.9; Björk Prop 3.15)	complete $⟺$ the EMM is unique	same market: claim price range $[0.5952, 0.6667]$ ; after adding a $3$ rd asset: $0.628571$
Perpetual American put (Shreve Ch 8.8)	$v (x) = \frac{6}{x}$ for $x \geq 3$ ; $v (x) = 5 - x$ for $0 < x \leq 3$	$v (4) = 1.5000$ ; $v (8) = 0.7500$

Two caveats to carry into every calculation. (1) The risk-neutral probabilities are derived from the replication equations - they are not the real coin-toss probabilities and carry no forecasting content (Shreve §1.1). (2) The bracket $d < 1 + r < u$ is a condition, not a convention: a tree that violates it has arbitrage and its " $p$ " leaves $[0, 1]$ (see 05 · Failure Modes).

3. Computational Implementation - the tree engine

Stdlib only (math), one function for European and American puts/calls. It reproduces the Haug-verified CRR numbers used throughout this folder.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

Violating the bracket. Any tree with $e^{r Δ t} \geq u$ (or $\leq d$ ) has arbitrage and a “probability” $p \in / [0, 1]$ ; the up/down factors must bracket the risk-free growth.
$p \neq = \tilde{p}$ confusion. Discounting the real-world expected payoff - $E^{P} [\cdot] / (1 + r)$ - gives the wrong number whenever $\tilde{p} \neq = p$ ; only the $P$ expectation prices.
Ignoring early exercise. A European formula applied to an American put understates value by the early-exercise premium ( $0.2427$ on the running example).
Treating convergence as monotone. CRR error is $O (1/ n)$ but oscillates, badly so for at-the-money American puts - “more steps” is not uniformly better.

5. References

Shreve, Steven E.: Stochastic Calculus for Finance I: The Binomial Asset Pricing Model
Shreve, Steven E.: Stochastic Calculus for Finance II
Björk, Tomas: Arbitrage Theory in Continuous Time
Hull, John C.: Options, Futures, and Other Derivatives (11th ed.)
Haug, Espen Gaarder: The Complete Guide to Option Pricing Formulas

6. Connected Graph Bridges

Foundational base: Probability & Measure Theory · Stochastic Calculus & Itô
Sibling topic: Black–Scholes–Merton (the continuous-time limit of exactly this model)
Related flat notes: No-Arbitrage Foundations & Binomial Trees · BSM & Feynman–Kac · The Greeks & Dynamic Hedging
Sub-pages (in-folder): 01 From Zero · 02 No-Arbitrage & Risk-Neutral · 03 Trees & Convergence · 04 Fundamental Theorems · 05 Failure Modes · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05

Kwant Atlas

Explorer

3.2 No-Arbitrage & the Binomial Model

1. Intuition & Practical Objective

2. Mathematical Ground Truth & Derivations

3. Computational Implementation - the tree engine

4. Failure Modes & First-Principles Breakdowns

5. References

6. Connected Graph Bridges

3.2.1 No-Arbitrage from Zero

3.2.2 No-Arbitrage, the Risk-Neutral Measure & State Prices

3.2.3 Binomial Trees & Convergence to Black–Scholes

3.2.4 The Fundamental Theorems of Asset Pricing

3.2.5 Failure Modes & Practice

3.2.6 Advanced Extensions