Risk-Neutral Valuation as a Theorem

Motivation: why this matters in quant finance

where $B_t$ is the bank account (numéraire) and $\mathbb{Q}$ is the unique probability measure under which discounted asset prices are martingales. This is the bridge between two views of pricing: replication (you build a portfolio whose value matches $X$) and expectation (you compute an integral). The theorem is what justifies replacing one with the other.

This note states and proves the theorem rigorously, isolates the role of completeness vs incompleteness, and connects it to the Fundamental Theorems of Asset Pricing (FTAP).

The informal idea

Three ingredients:

1. Bank account / numéraire. Some traded asset whose price never goes negative — typically the cash account $B_t = e^{rt}$.
2. Risk-neutral measure $\mathbb{Q}$. A probability measure equivalent to the physical $\mathbb{P}$ such that all tradeable asset prices, when divided by $B_t$, are $\mathbb{Q}$-martingales. Existence is the First Fundamental Theorem of Asset Pricing (FTAP1: no arbitrage $\Leftrightarrow \mathbb{Q}$ exists). Uniqueness is the Second Fundamental Theorem (FTAP2: market completeness $\Leftrightarrow \mathbb{Q}$ unique).
3. Replication. A claim $X$ is attainable if there exists a self-financing trading strategy with terminal wealth $X$. In a complete market, every $X$ is attainable.

That's the theorem. The technical work is in the three pieces above.

Formal statement and proof

(i) The market admits no arbitrage.

(ii) There exists an equivalent probability measure $\mathbb{Q} \sim \mathbb{P}$ under which the discounted price processes $\tilde S^{(i)}_t := S^{(i)}_t / B_t$ are martingales.

(iii) The claim $X \in L^1(\mathbb{Q}, \mathcal{F}_T)$ is attainable, replicated by self-financing strategy $\phi$ with value $V_t$.

Then the price of $X$ at time $t$ is

Let $\tilde V_t = V_t / B_t$. Apply Itô's product rule (or in the elementary case, the discrete-time analogue) to $V_t \cdot B_t^{-1}$:

For the standard model where $B_t$ is locally riskless ($dB_t = r_t B_t \, dt$, no martingale part), the quadratic variation contributions vanish, and computation reduces to

Since each $\tilde S^{(i)}$ is a $\mathbb{Q}$-martingale and $\phi^{(i)}$ is predictable and bounded enough (technical: integrability conditions), $\tilde V_t$ is a $\mathbb{Q}$-local-martingale. Under integrability conditions (e.g., $V_t$ bounded below), it's a true $\mathbb{Q}$-martingale.

By the martingale property:

Multiplying both sides by $B_t$:

The Fundamental Theorems

Two companion results give the existence and uniqueness of $\mathbb{Q}$:

Key properties

• Independence of $\mathbb{P}$. The price formula uses only $\mathbb{Q}$. The physical drift of the stock disappears (cf. the Black-Scholes PDE, where $\mu$ disappears for the same reason).
• Numéraire invariance. Pricing is the same under any choice of numéraire, with the appropriate measure. Using $S$ as numéraire gives the stock-measure $\mathbb{Q}^S$ with $\mathbb{Q}^S$-martingale property of $B/S$.
• Linearity. Price is a linear functional on payoffs, which gives put-call parity for free.
• Time-consistency. The price at any future time satisfies $V_s = B_s \mathbb{E}^{\mathbb{Q}}[V_t/B_t \mid \mathcal{F}_s]$ for $s < t$ — the dynamics are consistent.
• Incomplete markets. If multiple $\mathbb{Q}$ exist, the formula gives a range of prices — the no-arbitrage interval. Picking a single $\mathbb{Q}$ requires extra information (calibration to liquid instruments).

Worked example: Black-Scholes call price

Standard Black-Scholes: $S_t = S_0 \exp((\mu - \sigma^2/2)t + \sigma W_t)$ under $\mathbb{P}$, $B_t = e^{rt}$.

By Girsanov, define $\mathbb{Q}$ via $d\mathbb{Q}/d\mathbb{P} = \exp(\theta W_T - \theta^2 T/2)$ where $\theta = (\mu - r)/\sigma$. Under $\mathbb{Q}$, $W^{\mathbb{Q}}_t = W_t - \theta t$ is a Brownian motion, and

so $\tilde S_t = S_t/B_t$ is a $\mathbb{Q}$-martingale.

European call: $X = (S_T - K)^+$. Apply the theorem:

Compute the expectation under the lognormal distribution of $S_T$ under $\mathbb{Q}$ (mean parameter $r - \sigma^2/2$, vol $\sigma$). The result is the Black-Scholes formula:

The expectation reduces to the Black-Scholes formula in closed form because lognormal expectations of European payoffs admit a closed form. Importantly, the $\mu$ from the original $\mathbb{P}$ dynamics never appears.

Common confusions and pitfalls

• $\mathbb{Q}$ is not "the real-world probability." It's a mathematical pricing measure. Probabilities under $\mathbb{Q}$ don't represent likelihoods of stock movements; they're shadow weights consistent with no-arbitrage.
• Risk-neutral does not mean investors are risk-neutral. It's a change of measure trick. Real investors are risk-averse; the Girsanov shift absorbs the risk premium $(\mu - r)/\sigma$ into the measure.
• Discounting matters. Always price discounted payoffs. Pricing $\mathbb{E}^{\mathbb{Q}}[X]$ without the factor $B_t/B_T$ is wrong unless $r = 0$.
• Choice of numéraire. For interest-rate products, the bank account is unsuitable (rates are stochastic). The $T$-forward measure (using zero-coupon bonds as numéraire) is more convenient.
• Incomplete markets. Stochastic volatility, jump models, real-world frictions — these break completeness, and there's no unique $\mathbb{Q}$. The theorem still holds for any specific $\mathbb{Q}$, but model selection becomes a calibration problem.
• Local vs true martingale. Technical issue in continuous time: discounted prices are local martingales under $\mathbb{Q}$. To turn local-martingale arguments into expectation identities, integrability conditions are needed (uniform integrability or admissibility constraints on strategies).

Where this goes next

• Monte Carlo pricing (basic) — the direct numerical implementation of the risk-neutral valuation formula.
• Change of measure — the mechanism (Girsanov) that constructs $\mathbb{Q}$.
• Risk-neutral measure — the foundational measure-theoretic setup.