Construction of the Risk-Neutral Measure¶

Given \(\mu = 0.10\), \(\sigma = 0.25\), and \(r = 0.04\), the market price of risk is

The Radon–Nikodym derivative at time \(T\) is

Under \(\mathbb{Q}\), the process \(W_t^{\mathbb{Q}} = W_t^{\mathbb{P}} + \theta t = W_t^{\mathbb{P}} + 0.24\,t\) is a Brownian motion. Substituting \(dW_t^{\mathbb{P}} = dW_t^{\mathbb{Q}} - 0.24\,dt\) into the stock dynamics:

The discounted price \(\tilde{S}_t = e^{-0.04\,t}S_t\) satisfies

Since the drift vanishes, \(\tilde{S}_t\) is a \(\mathbb{Q}\)-martingale.

Starting from the discounted price dynamics

we define \(\theta_t = (\mu_t - r_t)/\sigma_t\) and \(W_t^{\mathbb{Q}} = W_t^{\mathbb{P}} + \int_0^t \theta_s\,ds\). Then \(dW_t^{\mathbb{P}} = dW_t^{\mathbb{Q}} - \theta_t\,dt\), and substituting:

Since \(\sigma_t\theta_t = \mu_t - r_t\), the drift terms cancel exactly:

The drift removal is equivalent to the martingale property because a continuous local martingale of the form \(dX_t = \sigma_t X_t\,dW_t\) has zero drift, meaning \(\mathbb{E}^{\mathbb{Q}}[\tilde{S}_t \mid \mathcal{F}_s] = \tilde{S}_s\) for \(s \le t\) (assuming appropriate integrability). A process is a martingale if and only if it has zero drift in its semimartingale decomposition, so eliminating the drift is precisely what makes \(\tilde{S}_t\) a \(\mathbb{Q}\)-martingale.

The risk-neutral measure \(\mathbb{Q}\) must be equivalent to \(\mathbb{P}\), meaning they agree on which events have probability zero. This equivalence is essential for two reasons:

If \(\mathbb{Q}(A) > 0\) for some event \(A\) with \(\mathbb{P}(A) = 0\): Then \(\mathbb{Q}\) assigns positive probability to a physically impossible event. This means derivative prices under \(\mathbb{Q}\) could depend on payoffs in scenarios that can never occur, leading to economically nonsensical prices. One could construct "arbitrage" by selling claims that pay in impossible states, collecting premium for risk that can never materialize.

If \(\mathbb{Q}(A) = 0\) for some event \(A\) with \(\mathbb{P}(A) > 0\): Then there exists a physically possible event whose payoff is completely ignored in pricing. An agent could buy a claim that pays in state \(A\) for zero cost (since \(\mathbb{Q}\) gives it zero weight) but receives a positive payoff with positive physical probability, creating an arbitrage opportunity.

Equivalence ensures a one-to-one correspondence between possible and priced events, which is exactly the no-arbitrage condition.

The Novikov condition states that \(Z_t\) is a true martingale (and hence \(\mathbb{Q}\) is well-defined) if

This is a sufficient condition ensuring that the stochastic exponential \(Z_t = \exp\!\left(-\int_0^t \theta_s\,dW_s^{\mathbb{P}} - \frac{1}{2}\int_0^t \theta_s^2\,ds\right)\) is a uniformly integrable martingale, not merely a local martingale.

Example where Novikov fails: Consider \(\sigma_t = \sigma\) constant and \(\mu_t - r_t = \sigma / \sqrt{T - t}\) for \(t < T\). Then \(\theta_t = 1/\sqrt{T-t}\) and

The integral diverges, so the Novikov condition fails. In this case, \(Z_t\) may fail to be a true martingale, and a well-defined equivalent measure \(\mathbb{Q}\) may not exist.

With two risky assets \(S^1, S^2\) and one Brownian motion \(W_t\), the dynamics are

For both discounted prices to be \(\mathbb{Q}\)-martingales, we need a single \(\theta\) satisfying

From the first equation, \(\theta = (\mu_1 - r)/\sigma_1\). Substituting into the second:

This simplifies to the no-arbitrage consistency condition:

Both assets must have the same Sharpe ratio. If this condition is violated, say \((\mu_1 - r)/\sigma_1 > (\mu_2 - r)/\sigma_2\), then no risk-neutral measure exists, and an arbitrage strategy can be constructed: go long asset 1 and short asset 2 in proportions that eliminate the Brownian motion exposure while retaining a positive drift.

The risk-neutral measure \(\mathbb{Q}\) is not the probability measure investors use to form expectations. Under the physical measure \(\mathbb{P}\), investors form beliefs about actual probabilities of future events and demand compensation (risk premia) for bearing risk.

The correct interpretation: \(\mathbb{Q}\) is an artificial probability measure constructed so that discounted asset prices are martingales. Under \(\mathbb{Q}\), all assets earn the risk-free rate in expectation — risk premia have been absorbed into the probability weighting. The measure \(\mathbb{Q}\) encodes both the physical probabilities and the market prices of risk into a single object.

Pricing under \(\mathbb{Q}\) is a mathematical convenience, not a statement about investor beliefs. The formula \(V_t = \mathbb{E}^{\mathbb{Q}}[e^{-\int_t^T r_s\,ds}\Phi_T \mid \mathcal{F}_t]\) gives the no-arbitrage price, which reflects both the probability of outcomes (from \(\mathbb{P}\)) and the risk adjustment (from \(\theta\)), combined into the tilted measure \(\mathbb{Q}\). In markets with risk-averse investors, \(\mathbb{Q}\) typically overweights adverse outcomes relative to \(\mathbb{P}\).

With \(dr_t = \alpha(r_t)\,dt + \beta(r_t)\,dW_t^{\mathbb{P}}\) and \(dS_t = \mu_t S_t\,dt + \sigma_t S_t\,dW_t^{\mathbb{P}}\) driven by the same Brownian motion, the discounted price \(\tilde{S}_t = S_t / B_t\) where \(B_t = \exp(\int_0^t r_s\,ds)\) satisfies (by Itô's formula):

The market price of risk is

Note that \(\theta_t\) is now stochastic since both \(\mu_t\) and \(r_t\) may depend on the state. Under the measure change with \(W_t^{\mathbb{Q}} = W_t^{\mathbb{P}} + \int_0^t \theta_s\,ds\), the risk-neutral dynamics become \(d\tilde{S}_t = \sigma_t\tilde{S}_t\,dW_t^{\mathbb{Q}}\) and

The market is complete because there is exactly one source of randomness (one Brownian motion) and one traded risky asset (the stock). The single equation \(\mu_t - r_t = \sigma_t\theta_t\) uniquely determines \(\theta_t\), leaving no free parameters. This means there is a unique risk-neutral measure, and every contingent claim can be replicated by a dynamic portfolio of the stock and the bond.