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

Basic Prerequisites: 01 · From Zero and Probability & Measure Theory.

---

1. Intuition & Practical Objective

Page 01 built the price by replication. This page names the structure underneath it, because that structure - not the algebra - is what generalizes to continuous time.

Three objects are equivalent, and knowing they are the same object is most of the subject:

1. No arbitrage - no self-financing portfolio starts at zero and ends non-negative with a positive chance of being positive.
2. A risk-neutral measure $\tilde{\mathbb{P}}$ - a probability measure, equivalent to the real one, under which the discounted stock is a martingale.
3. State prices $\zeta (\omega )$ - Arrow–Debreu prices of a dollar in state $\omega$; every payoff is priced as ${\sum }_{\omega }\zeta (\omega )f(\omega )$, and they sum to $1/(1+r)$ because a dollar in every state is the bond.

The practical objective: be able to move freely between “expected payoff”, “martingale”, and “state price” language - because a desk quotes in the first, proves in the second, and calibrates in the third.

---

2. Mathematical Ground Truth & Derivations

2.1 The risk-neutral measure and the martingale property

With $\tilde{p},\tilde{q}$ from 01, the one-step recursion $V_k=\frac{1}{1+r}[\tilde{p}{V}_{k+1}(H)+\tilde{q}{V}_{k+1}(T)]$ rearranges to

$$V_k=(1+r{)}^k\tilde{\mathbb{E}}\left[\frac{V_m}{(1+r{)}^m}|F_k\right]\text{(risk-neutral valuation, Shreve §3.4)}$$

and in particular, applied to the stock itself,

$$\tilde{\mathbb{E}}\left[\frac{S_{k+1}}{(1+r{)}^{k+1}}|F_k\right]=\frac{S_k}{(1+r{)}^k}⟺\tilde{p}u+\tilde{q}d=1+r.$$

The discounted stock - not the stock - is the $\tilde{\mathbb{P}}$-martingale. (Under the physical measure, $S_k$ is a martingale only if $pu+qd=1$; here $pu+qd\ne 1+r$ in general, which is exactly why the real measure cannot price.) This single fact is the discrete statement of "$dS=rSdt+\sigma Sd\tilde{W}$" in BSM · 02.

2.2 Self-financing wealth

A portfolio process $\Delta =({\Delta }_0,\dots ,{\Delta }_{n-1})$ with each ${\Delta }_k$ $F_k$-measurable (no inside information) is self-financing if wealth evolves as (Shreve §3.3)

$$X_{k+1}={\Delta }_k{S}_{k+1}+(1+r)(X_k-{\Delta }_k{S}_k)=(1+r)X_k+{\Delta }_k(S_{k+1}-(1+r)S_k).$$

The bracketed term is a $\tilde{\mathbb{P}}$-martingale increment, so $X_k/(1+r{)}^k$ is a $\tilde{\mathbb{P}}$-martingale for every self-financing, adapted strategy. Discounted prices are martingales; that is the whole content of the measure.

2.3 State prices (Arrow–Debreu)

Define

$$\boxed{\zeta (\omega )=\frac{\tilde{\mathbb{P}}(\omega )}{1+r}}$$

Then ${\sum }_{\omega }\zeta (\omega )=\frac{1}{1+r}$ (the bond) and ${\sum }_{\omega }\zeta (\omega )S_1(\omega )=S_0$ (the stock) - the two traded assets are reproduced by the state-price system. Any claim is priced by

$$V_0=\sum _{\omega }\zeta (\omega )f(\omega ).$$

Equivalently, writing $Z(\omega )=d\tilde{\mathbb{P}}/d\mathbb{P}$ (the Radon–Nikodym derivative) and $\zeta =\frac{Z}{1+r}$, the price is a physical-measure expectation

$$V_0={\mathbb{E}}^{\mathbb{P}}\left[\zeta f\right].$$

$Z$ is the stochastic discount factor / state-price density (Björk Prop 3.18, $\Pi (0;X)={\mathbb{E}}^{\mathbb{P}}[\Lambda X]$ with $\Lambda =\frac{1}{1+R}\frac{dQ}{dP}$; Shreve Vol I Ch 9, ${\zeta }_k=(1+r{)}^{-k}{Z}_k$). Crucially $\zeta$ depends on the physical measure while $V_0$ does not - the same price is recovered for every $\mathbb{P}$ consistent with the same $\tilde{\mathbb{P}}$.

---

3. Computational Implementation - martingale, state prices, measure-invariance

All checks on Shreve’s $S_0=4,u=2,d=\frac{1}{2},r=\frac{1}{4}$, $K=5$ put. Stdlib only.

Two things are visible in that output. The state prices reproduce both traded assets, so they are the market’s pricing kernel; and the price $1.200000$ is recovered under physical probabilities $p=0.5$, $0.6$, and $0.9$ - the state-price density $\zeta =Z/(1+r)$ absorbs the change of measure exactly. The price is a property of the traded assets, not of anyone’s beliefs.

---

4. Failure Modes & First-Principles Breakdowns

1. Discounting under the wrong measure. ${\mathbb{E}}^{\mathbb{P}}[f]/(1+r)$ is not the price unless $\mathbb{P}=\tilde{\mathbb{P}}$. The state-price form ${\mathbb{E}}^{\mathbb{P}}[\zeta f]$ is correct for any $\mathbb{P}$ - provided $\zeta$ is right.
2. Martingaling the wrong object. $S_k$ itself is not the $\tilde{\mathbb{P}}$-martingale; $S_k/(1+r{)}^k$ is. Confusing the two produces a spurious drift and mispricing of every path-dependent claim.
3. Forgetting measurability. ${\Delta }_k$ must be $F_k$-measurable - chosen from information available before the step. A “hedge” that uses $S_{k+1}$ is look-ahead and its apparent profit is fictitious.
4. Reading $\zeta$ as a probability. State prices sum to $1/(1+r)<1$, not to $1$. They are prices, and their discount factor is baked in; treating them as probabilities double-discounts.
5. Assuming $\zeta$ is unique. In an incomplete market the state-price vector is a set, so a claim has a price interval, not a price (04 · Fundamental Theorems).

---

5. References

• Shreve, Stochastic Calculus for Finance I
• Björk, Arbitrage Theory in Continuous Time
• Shreve, Stochastic Calculus for Finance II

---

6. Connected Graph Bridges

• Back: 01 · From Zero · Index Hub
• Forward: 03 · Trees & Convergence · 04 · Fundamental Theorems
• Theory: Probability & Measure Theory · BSM · 02 PDE & Derivation