Risk-Neutral Pricing

Risk-neutral pricing is the central trick of derivatives theory: to price a claim, discount its expected payoff, but take the expectation under a different, artificial probability measure in which every asset earns the risk-free rate. The startling part is that the real-world drift $\mu$, and investors' risk preferences, drop out entirely. Two identical derivatives are priced the same whether the underlying is expected to soar or crash. This section explains why that is not a swindle but a theorem.

Why real-world expectation fails

The naive guess, price a claim as its discounted real-world expected payoff, $e^{-rT}\mathbb{E}^{\mathbb{P}}[\text{payoff}]$, is wrong, because it ignores risk. A risky payoff is worth less than its expectation discounted at $r$; how much less depends on a risk premium that varies by investor. Risk-neutral pricing sidesteps this by absorbing the risk premium into a change of probability measure, leaving a clean discounted expectation.

Numeraire and discounting

Fix the money-market account $B_t = e^{\int_0^t r_s\,ds}$ (with constant $r$, $B_t = e^{rt}$) as the numeraire, the unit in which we measure all other prices. Define discounted prices $\tilde S_t = S_t / B_t$. The core idea: prices should be arbitrage-free, and arbitrage-freeness is a statement about these discounted, numeraire-relative prices.

The Fundamental Theorem of Asset Pricing

First FTAP. A market is arbitrage-free if and only if there exists a probability measure $\mathbb{Q}$, equivalent to the real-world measure $\mathbb{P}$ (they agree on which events have zero probability), under which every discounted asset price $\tilde S_t = S_t/B_t$ is a martingale. $\mathbb{Q}$ is the equivalent martingale measure (or risk-neutral measure).

Second FTAP. The market is complete (every claim can be replicated by trading) if and only if that measure $\mathbb{Q}$ is unique.

Given $\mathbb{Q}$, the price at time $t$ of a claim paying $V_T$ at $T$ is the discounted conditional expectation

$\boxed{\,V_t = B_t\,\mathbb{E}^{\mathbb{Q}}\!\Big[\frac{V_T}{B_T}\;\Big|\;\mathcal{F}_t\Big] = e^{-r(T-t)}\,\mathbb{E}^{\mathbb{Q}}[V_T \mid \mathcal{F}_t].\,}$

The logic is a two-line arbitrage argument: if $\tilde V_t = V_t/B_t$ is a $\mathbb{Q}$-martingale, then $\tilde V_t = \mathbb{E}^{\mathbb{Q}}[\tilde V_T \mid \mathcal{F}_t]$ by the martingale property; multiply through by $B_t$. Replication (completeness) is what guarantees this price is the unique no-arbitrage price, a hedger can manufacture the payoff for exactly $V_t$.

Why the drift becomes $r$

This is the concrete payoff of the abstract theorem. Under $\mathbb{P}$ the stock follows geometric Brownian motion

$dS_t = \mu S_t\,dt + \sigma S_t\,dW_t^{\mathbb{P}}.$

For $\tilde S_t = e^{-rt}S_t$ to be a $\mathbb{Q}$-martingale it must be driftless. Compute its dynamics with Itô:

$d\tilde S_t = e^{-rt}\big(dS_t - r S_t\,dt\big) = e^{-rt}S_t\big((\mu - r)\,dt + \sigma\,dW_t^{\mathbb{P}}\big).$

The drift $(\mu - r)$ must be removed. Define the market price of risk $\lambda = (\mu - r)/\sigma$ and, by Girsanov's theorem, a new Brownian motion $W_t^{\mathbb{Q}} = W_t^{\mathbb{P}} + \lambda t$. Substituting $dW^{\mathbb{P}} = dW^{\mathbb{Q}} - \lambda\,dt$,

$d\tilde S_t = e^{-rt}S_t\big((\mu - r - \sigma\lambda)\,dt + \sigma\,dW_t^{\mathbb{Q}}\big) = e^{-rt}S_t\,\sigma\,dW_t^{\mathbb{Q}},$

because $\mu - r - \sigma\lambda = 0$ by construction. The discounted price is now driftless, a $\mathbb{Q}$-martingale, and under $\mathbb{Q}$ the undiscounted stock follows

$dS_t = r S_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}}.$

The real drift $\mu$ has been replaced by $r$. Crucially, the volatility $\sigma$ is unchanged, Girsanov shifts drift but not diffusion. This is why implied volatility is a $\mathbb{Q}$-object we can extract from prices, while $\mu$ is not.

Why preferences vanish

The change of measure has silently priced risk. Under $\mathbb{Q}$, all assets drift at $r$, so a risk-neutral agent (indifferent to risk) would be content, hence the name. But we did not assume anyone is risk-neutral; we constructed a measure in which pricing looks as if they were. The real-world premium $\mu - r$ was fully encoded in $\lambda$ and the Radon-Nikodym derivative $d\mathbb{Q}/d\mathbb{P}$. Because the replicating hedge removes all risk, the hedger's preferences never enter, only $r$, $\sigma$, and the payoff.

Worked example: a forward

A forward contract to buy one share at $K$ and time $T$ pays $S_T - K$. Its arbitrage-free value is

$V_0 = e^{-rT}\mathbb{E}^{\mathbb{Q}}[S_T - K] = e^{-rT}\big(\mathbb{E}^{\mathbb{Q}}[S_T] - K\big) = e^{-rT}\big(S_0 e^{rT} - K\big) = S_0 - Ke^{-rT},$

using $\mathbb{E}^{\mathbb{Q}}[S_T] = S_0 e^{rT}$ (the martingale property of $\tilde S$). Setting $V_0 = 0$ gives the fair forward price $F_0 = S_0 e^{rT}$, the classic cost-of-carry result, derived here with zero reference to $\mu$. Notice the expected spot under $\mathbb{Q}$ is the forward, not the real-world expected price.

What breaks in practice

• Incompleteness. Stochastic volatility, jumps, and transaction costs make markets incomplete: $\mathbb{Q}$ is no longer unique, so there is a range of arbitrage-free prices and the market must pick one (a variance risk premium, a jump premium). See Heston.
• The measure is not the real world. Risk-neutral probabilities are pricing weights, not forecasts. The $\mathbb{Q}$-density of a crash exceeds the $\mathbb{P}$-density because of risk premia, reading option-implied "probabilities" as real-world odds is a classic error.
• Discounting subtleties. Post-2008, collateralized derivatives discount at OIS, not Libor; the numeraire choice (multi-curve, funding, FVA) is where a lot of modern desk complexity lives.

In interviews

Expect "why can we price using the risk-free rate when the stock is risky?", the answer is the FTAP plus replication: a delta-hedged position is riskless, so it must earn $r$, and equivalently we price under $\mathbb{Q}$ where discounted assets are martingales. Be able to show that $e^{-rt}S_t$ being a $\mathbb{Q}$-martingale forces the drift to be $r$, and that $\sigma$ is invariant under the measure change. A sharp follow-up: "is the risk-neutral probability of an up-move the real probability?", no, and confusing the two is the deepest conceptual error in the field. This machinery is what makes Black-Scholes and Feynman-Kac work.