Construction of the Risk-Neutral Measure¶
The risk-neutral measure is a probability measure under which discounted asset prices are martingales. Its existence is the mathematical expression of the absence of arbitrage.
The goal of this construction is to eliminate risk premia from asset dynamics so that pricing reduces to computing expectations: no-arbitrage prices are conditional expectations of discounted payoffs under the new measure.
Construction roadmap
- Discount: form the discounted price \(\tilde S_t = S_t / B_t\).
- Identify the obstruction: the drift \((\mu_t - r_t)\) prevents \(\tilde S_t\) from being a martingale.
- Define the Girsanov kernel: set \(\theta_t = (\mu_t - r_t)/\sigma_t\).
- Apply Girsanov’s theorem: the tilted measure \(\mathbb{Q}\) makes \(\tilde S_t\) a martingale.
Market Model¶
Consider a financial market consisting of: - a risk-free asset \(B_t\), - a risky asset \(S_t\).
Under the physical measure \(\mathbb{P}\), assume [ dS_t = \mu_t S_t\,dt + \sigma_t S_t\,dW_t^{\mathbb{P}}, \qquad dB_t = r_t B_t\,dt, ] where \(\mu_t\), \(\sigma_t\), and \(r_t\) are adapted processes.
Discounted Asset Price¶
Define the discounted price process [ \tilde S_t := \frac{S_t}{B_t}. ]
Applying Itô’s formula, [ d\tilde S_t = (\mu_t - r_t)\tilde S_t\,dt + \sigma_t \tilde S_t\,dW_t^{\mathbb{P}}. ]
The presence of the drift term \((\mu_t - r_t)\) prevents \(\tilde S_t\) from being a martingale under \(\mathbb{P}\).
Measure Change¶
Define the process [ \theta_t := \frac{\mu_t - r_t}{\sigma_t}, ] and the stochastic exponential [ Z_t = \exp!\left( - \int_0^t \theta_s\,dW_s^{\mathbb{P}} - \frac12 \int_0^t \theta_s^2\,ds \right). ]
For \(Z_t\) to be a true martingale (not merely a local martingale), the Novikov condition must hold: [ \mathbb{E}^{\mathbb{P}}!\left[\exp!\left(\tfrac12\int_0^T \theta_s^2\,ds\right)\right] < \infty. ]
When this condition is satisfied, \(Z_t\) defines a new probability measure \(\mathbb{Q}\) by [ \frac{d\mathbb{Q}}{d\mathbb{P}} \Big|_{\mathcal{F}_t} = Z_t. ]
Risk-Neutral Dynamics¶
Invariance principle
A change of measure does not change the sample paths of the process, nor does it change the volatility. It only changes the probability weights assigned to those paths. The same set of outcomes is possible under both \(\mathbb{P}\) and \(\mathbb{Q}\); only their likelihoods differ.
By Girsanov’s theorem, the process [ W_t^{\mathbb{Q}} := W_t^{\mathbb{P}} + \int_0^t \theta_s\,ds ] is a Brownian motion under \(\mathbb{Q}\).
The discounted asset price then satisfies [ d\tilde S_t = \sigma_t \tilde S_t\,dW_t^{\mathbb{Q}}, ] Since the drift term has vanished, the semimartingale decomposition of \(\tilde S_t\) under \(\mathbb{Q}\) contains no finite-variation (predictable) part beyond the initial value. A continuous process with this property is a local martingale. Since \(\tilde S_t\) is a stochastic exponential driven by a Brownian motion with bounded (or suitably integrable) coefficients, it satisfies standard integrability conditions ensuring that the local martingale is a true martingale: \(\mathbb{E}^{\mathbb{Q}}[\tilde S_t \mid \mathcal{F}_s] = \tilde S_s\) for \(s \le t\).
Definition¶
A risk-neutral measure is a probability measure \(\mathbb{Q}\) equivalent to \(\mathbb{P}\) under which all discounted traded asset prices are martingales.
Its existence ensures arbitrage-free pricing. This is the constructive direction of the First Fundamental Theorem of Asset Pricing: the existence of an equivalent martingale measure is equivalent to the absence of arbitrage, and its uniqueness is equivalent to market completeness. This result tells us that pricing is not a separate theory: it is a direct consequence of no-arbitrage.
Economic Interpretation¶
Economically, the measure change from \(\mathbb{P}\) to \(\mathbb{Q}\) absorbs risk premia into the probability weighting. Under \(\mathbb{P}\), risky assets earn \(\mu > r\) to compensate investors for bearing uncertainty. Under \(\mathbb{Q}\), every asset earns the risk-free rate in expectation, because the excess return has been removed by tilting probabilities: adverse outcomes receive higher weight, favorable outcomes receive lower weight. The resulting pricing rule [ V_t = \mathbb{E}^{\mathbb{Q}}!\left[e^{-\int_t^T r_s\,ds}\,\Phi_T \;\middle|\; \mathcal{F}_t\right] ] encodes both the physical likelihood of outcomes and the market's risk adjustment in a single expectation.
Exercises¶
Exercise 1. A stock has physical dynamics \(dS_t = 0.10\,S_t\,dt + 0.25\,S_t\,dW_t^{\mathbb{P}}\) with risk-free rate \(r = 0.04\). Compute the market price of risk \(\theta\), write the Radon-Nikodym derivative \(Z_T\), and derive the discounted price dynamics under \(\mathbb{Q}\). Verify that the discounted price is a \(\mathbb{Q}\)-martingale.
Solution to Exercise 1
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.
Exercise 2. Starting from the discounted price \(\tilde{S}_t = S_t / B_t\) and its dynamics \(d\tilde{S}_t = (\mu_t - r_t)\tilde{S}_t\,dt + \sigma_t\tilde{S}_t\,dW_t^{\mathbb{P}}\), show that choosing \(\theta_t = (\mu_t - r_t)/\sigma_t\) and defining \(W_t^{\mathbb{Q}} = W_t^{\mathbb{P}} + \int_0^t \theta_s\,ds\) eliminates the drift term in the \(\tilde{S}_t\) dynamics. Why is this drift removal equivalent to the martingale property?
Solution to Exercise 2
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.
Exercise 3. Explain why the risk-neutral measure \(\mathbb{Q}\) must be equivalent to \(\mathbb{P}\) (i.e., both measures agree on which events have probability zero). What would go wrong financially if \(\mathbb{Q}\) assigned positive probability to an event that is impossible under \(\mathbb{P}\)?
Solution to Exercise 3
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.
Exercise 4. For time-varying coefficients \(\mu_t\), \(\sigma_t\), and \(r_t\), the market price of risk \(\theta_t = (\mu_t - r_t)/\sigma_t\) is a stochastic process. State the Novikov condition that ensures \(Z_t\) is a true martingale and \(\mathbb{Q}\) is well-defined. Give an example where the condition fails.
Solution to Exercise 4
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.
Exercise 5. Consider a market with two risky assets and one Brownian motion. Write the system of equations that \(\theta\) must satisfy for both discounted prices to be \(\mathbb{Q}\)-martingales. Under what condition on \(\mu_1, \mu_2, \sigma_1, \sigma_2, r\) is the system consistent (i.e., no arbitrage)?
Solution to Exercise 5
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.
Exercise 6. A student claims: "The risk-neutral measure is the probability measure that investors actually use to form expectations." Explain why this is incorrect and describe the correct interpretation of \(\mathbb{Q}\).
Solution to Exercise 6
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 \(\mathbb{P}\).
Exercise 7. Suppose the risk-free rate is stochastic: \(dr_t = \alpha(r_t)\,dt + \beta(r_t)\,dW_t^{\mathbb{P}}\), and the same Brownian motion drives the stock. Write the discounted stock price dynamics and determine \(\theta_t\). Explain why this market is complete (one Brownian motion, one traded asset besides the bond).
Solution to Exercise 7
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.