Risk Premium Decomposition¶

Risk-neutral valuation replaces the physical drift \(\mu\) with the risk-free rate \(r\)---but what exactly is removed? The answer is the risk premium \(\sigma\theta\), where \(\theta = (\mu - r)/\sigma\) is the market price of risk. The decomposition \(\mu = r + \sigma\theta\) splits the physical drift into a time-value component and a risk-compensation component, revealing the financial content of Girsanov's theorem.

Core identity

The risk premium \(\sigma\theta = \mu - r\) is the wedge between how assets evolve under \(\mathbb{P}\) and how they are priced under \(\mathbb{Q}\). Girsanov's theorem removes exactly this wedge.

The Decomposition¶

Under \(\mathbb{P}\), a risky asset follows \(dS_t = \mu S_t\,dt + \sigma S_t\,dW_t^{\mathbb{P}}\). Recall (see § Market Price of Risk): \(\theta := (\mu - r)/\sigma\). Rearranging gives the risk premium decomposition:

\[ \mu = r + \sigma\theta \]

Component	Symbol	Interpretation
Risk-free rate	\(r\)	Compensation for the passage of time
Volatility	\(\sigma\)	Magnitude of randomness per unit time
Market price of risk	\(\theta\)	Compensation per unit of volatility
Risk premium	\(\sigma\theta\)	Total excess return for bearing risk

The risk premium \(\sigma\theta = \mu - r\) is exactly what Girsanov's theorem subtracts from the physical drift to produce the pricing measure.

Interpretation

The risk premium is not "extra drift"---it is the cost of changing measure. Removing it turns \(\mathbb{P}\) into \(\mathbb{Q}\).

How Girsanov Removes the Premium¶

Recall (see § Drift Adjustment): the Girsanov shift \(W_t^{\mathbb{Q}} = W_t^{\mathbb{P}} + \int_0^t \theta_s\,ds\) converts \(\mathbb{P}\)-dynamics \(dS_t = \mu S_t\,dt + \sigma S_t\,dW_t^{\mathbb{P}}\) into

\[ dS_t = rS_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}} \]

precisely because \(\sigma\theta = \mu - r\) cancels the excess drift. Volatility \(\sigma\) is unchanged: it equals quadratic variation \(\langle S\rangle_t = \sigma^2 S_t^2\,dt\), a pathwise quantity invariant under equivalent measure changes.

Drift changes, volatility does not

This is why option prices depend on \(\sigma\) but not on \(\mu\).

The Radon--Nikodym Derivative¶

Recall (see § The Exponential Martingale): for constant \(\theta\),

\[ \frac{d\mathbb{Q}}{d\mathbb{P}}\bigg|_{\mathcal{F}_T} = \exp\!\left(-\theta W_T^{\mathbb{P}} - \frac{1}{2}\theta^2 T\right) \]

A larger \(|\theta|\) means more aggressive reweighting: the further \(\mu\) is from \(r\), the more the measure must be tilted.

Different Measures, Different Drifts¶

The same process \(S_t\) has different drifts under different measures:

Measure	Drift of \(S_t\)	Interpretation
\(\mathbb{P}\) (physical)	\(\mu = r + \sigma\theta\)	Reflects risk preferences
\(\mathbb{Q}\) (risk-neutral)	\(r\)	Pricing measure
\(\mathbb{Q}^T\) (forward)	\(r - \sigma_P(t,T)\)	Bond-normalized pricing

Under every equivalent measure the volatility \(\sigma\) is preserved; only the drift changes. This is the content of Girsanov's theorem.

Why derivative prices do not depend on the physical drift

Since \(\mathbb{Q}\) replaces \(\mu\) with \(r\), the valuation formula \(V_0 = \mathbb{E}^{\mathbb{Q}}[e^{-rT}\Phi(S_T)]\) involves only \(r\) and \(\sigma\), not \(\mu\). See Physical vs Risk-Neutral World for further discussion.

Economic Interpretations¶

Sharpe Ratio Connection¶

For a single asset, the market price of risk equals the Sharpe ratio:

\[ \theta = \frac{\mu - r}{\sigma} = \text{Sharpe ratio} \]

The Sharpe ratio measures risk-adjusted excess return---a natural quantity to govern the measure change. In a multi-asset setting the analogous object is \(\|\boldsymbol{\theta}\|\), the maximum achievable Sharpe ratio across all portfolios (the slope of the capital market line).

Equilibrium Connection¶

The market price of risk \(\theta\) also appears in equilibrium asset pricing. The CAPM relation \(\mu_i - r = \beta_i(\mu_M - r)\) is equivalent to the risk premium decomposition when \(\theta\) is driven by a single market factor. For the full development---including multi-factor models and the stochastic discount factor---see From SDF to CAPM.

Time-Varying Risk Premia¶

In general, \(\theta_t = (\mu_t - r_t)/\sigma_t\) is stochastic. Empirical evidence shows it varies with business cycle conditions, market volatility, and interest rate levels. Time-varying risk premia are central to term structure models and to explaining observed patterns in asset returns.

The market price of risk is not directly observable

In practice, \(\theta_t\) must be inferred jointly with model assumptions---either estimated from historical returns under \(\mathbb{P}\) (confounded with estimation error) or extracted from option prices under \(\mathbb{Q}\) (absorbed into calibrated dynamics). This unobservability is a fundamental source of model risk.

Worked Example: Black-Scholes Model¶

Consider a stock with \(\mu = 0.12\), \(\sigma = 0.20\), and \(r = 0.03\). The market price of risk is

\[ \theta = \frac{0.12 - 0.03}{0.20} = 0.45 \]

The decomposition reads

\[ \underbrace{0.12}_{\mu} = \underbrace{0.03}_{r} + \underbrace{0.20 \times 0.45}_{\sigma\theta = 0.09} \]

Under \(\mathbb{P}\), the stock earns 12% per year. Under \(\mathbb{Q}\), it earns only 3%. The 9% difference is the risk premium.

The Radon--Nikodym derivative for a one-year horizon is

\[ \frac{d\mathbb{Q}}{d\mathbb{P}}\bigg|_{\mathcal{F}_1} = \exp\!\left(-0.45\,W_1^{\mathbb{P}} - \tfrac{1}{2}(0.45)^2\right) \]

Paths with positive \(W_1^{\mathbb{P}}\) (above-average returns) receive weight below 1 under \(\mathbb{Q}\); paths with negative \(W_1^{\mathbb{P}}\) receive weight above 1. The measure change systematically penalizes high-return paths to remove the drift.

Summary¶

The risk premium decomposition \(\mu = r + \sigma\theta\) is the financial content of Girsanov's theorem:

\(r\) compensates for time.
\(\sigma\theta\) compensates for bearing volatility.
\(\theta\) determines the measure change.

Under \(\mathbb{P}\) the drift reflects risk preferences; under \(\mathbb{Q}\) it is \(r\). Volatility is invariant across equivalent measures. In complete markets \(\theta\) is unique; in incomplete markets the decomposition admits multiple solutions, each corresponding to a different pricing measure.

For the implications when the decomposition breaks down, see When Measure Change Fails. For calibration in practice, see Practitioner Perspective.

Exercises¶

Exercise 1. A stock has physical drift \(\mu = 0.10\), volatility \(\sigma = 0.30\), and risk-free rate \(r = 0.02\). Compute the market price of risk \(\theta\), the risk premium \(\sigma\theta\), and write the Radon--Nikodym derivative \(d\mathbb{Q}/d\mathbb{P}|_{\mathcal{F}_T}\) for \(T = 1\).

Solution to Exercise 1

The market price of risk is

\[ \theta = \frac{\mu - r}{\sigma} = \frac{0.10 - 0.02}{0.30} = \frac{4}{15} \approx 0.2667 \]

The risk premium is

\[ \sigma\theta = 0.30 \times \frac{4}{15} = 0.08 \]

This confirms \(\mu - r = 0.10 - 0.02 = 0.08\).

The Radon--Nikodym derivative for \(T = 1\) is

\[ \frac{d\mathbb{Q}}{d\mathbb{P}}\bigg|_{\mathcal{F}_1} = \exp\!\left(-\frac{4}{15}W_1^{\mathbb{P}} - \frac{1}{2}\cdot\frac{16}{225}\right) = \exp\!\left(-\frac{4}{15}W_1^{\mathbb{P}} - \frac{8}{225}\right) \]

Paths with positive \(W_1^{\mathbb{P}}\) are downweighted and paths with negative \(W_1^{\mathbb{P}}\) are upweighted, removing the risk premium from the drift.

Exercise 2. Starting from \(dS_t = \mu S_t\,dt + \sigma S_t\,dW_t^{\mathbb{P}}\) and the Girsanov relation \(W_t^{\mathbb{Q}} = W_t^{\mathbb{P}} + \theta t\), derive the \(\mathbb{Q}\)-dynamics of \(S_t\) and verify that the drift becomes \(r\). Explain why \(\sigma\) is unchanged.

Solution to Exercise 2

From the Girsanov relation, \(dW_t^{\mathbb{P}} = dW_t^{\mathbb{Q}} - \theta\,dt\). Substituting:

\[ dS_t = \mu S_t\,dt + \sigma S_t\left(dW_t^{\mathbb{Q}} - \theta\,dt\right) = (\mu - \sigma\theta)S_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}} \]

Using \(\theta = (\mu - r)/\sigma\) gives \(\sigma\theta = \mu - r\), so \(\mu - \sigma\theta = r\) and

\[ dS_t = rS_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}} \]

The drift changes from \(\mu\) to \(r\), confirming the measure change removes the risk premium.

The volatility \(\sigma\) is unchanged because it equals the diffusion coefficient, determined by the quadratic variation \(\langle S \rangle_t = \sigma^2 S_t^2\,dt\). Quadratic variation is a path-by-path property depending on sample paths, not on probability weights. Since \(\mathbb{P}\) and \(\mathbb{Q}\) share the same paths, the volatility is identical under both measures.

Exercise 3. In the Vasicek model, the short rate follows \(dr_t = \kappa(\bar{r} - r_t)\,dt + \sigma_r\,dW_t^{\mathbb{P}}\) with \(\kappa = 0.5\), \(\bar{r} = 0.04\), \(\sigma_r = 0.01\), and market price of interest rate risk \(\theta = 0.3\). Compute the risk-neutral long-run mean \(\bar{r}^{\mathbb{Q}}\) and explain why \(\bar{r}^{\mathbb{Q}} < \bar{r}\) when \(\theta > 0\).

Solution to Exercise 3

Under \(\mathbb{Q}\), the Vasicek dynamics become

\[ dr_t = \left[\kappa(\bar{r} - r_t) - \sigma_r\theta\right]dt + \sigma_r\,dW_t^{\mathbb{Q}} = \kappa\!\left(\bar{r} - \frac{\sigma_r\theta}{\kappa} - r_t\right)dt + \sigma_r\,dW_t^{\mathbb{Q}} \]

The risk-neutral long-run mean is

\[ \bar{r}^{\mathbb{Q}} = \bar{r} - \frac{\sigma_r\theta}{\kappa} = 0.04 - \frac{0.01 \times 0.3}{0.5} = 0.04 - 0.006 = 0.034 \]

Since \(\theta > 0\), we have \(\bar{r}^{\mathbb{Q}} = 0.034 < 0.04 = \bar{r}\).

Economic intuition. A positive \(\theta\) means investors demand compensation for interest rate risk. Under \(\mathbb{P}\), the long-run mean \(\bar{r}\) includes this compensation. Under \(\mathbb{Q}\), the premium is removed, so the long-run mean is lower. Bond prices---computed as \(\mathbb{Q}\)-expectations of discounted payoffs---use this lower mean, embedding the term premium into the yield curve.

Exercise 4. Prove that the Sharpe ratio \((\mu_i - r)/\sigma_i\) is the same for all assets in a single-factor complete market.

Solution to Exercise 4

In a single-factor complete market (\(n = d = 1\)), each asset satisfies

\[ dS_t^i = \mu_i S_t^i\,dt + \sigma_i S_t^i\,dW_t^{\mathbb{P}} \]

The risk premium decomposition gives

\[ \mu_i - r = \sigma_i\theta \]

where \(\theta\) is the unique market price of risk. Dividing by \(\sigma_i\):

\[ \frac{\mu_i - r}{\sigma_i} = \theta \]

The left side is the Sharpe ratio of asset \(i\). Since \(\theta\) is a property of the market (not of individual assets) and is unique in a complete market, the Sharpe ratio is identical for every traded asset.

This is the continuous-time analogue of CAPM: in equilibrium, all assets on the capital market line share the same Sharpe ratio, equal to the market price of risk. \(\square\)

Exercise 5. For a time-varying market price of risk \(\theta_t\) with \(\int_0^T \theta_t^2\,dt < \infty\) a.s., write the Radon--Nikodym derivative \(d\mathbb{Q}/d\mathbb{P}|_{\mathcal{F}_T}\) and explain why the Novikov condition

\[ \mathbb{E}^{\mathbb{P}}\!\left[\exp\!\left(\tfrac{1}{2}\int_0^T \theta_t^2\,dt\right)\right] < \infty \]

guarantees the stochastic exponential is a true martingale.

Solution to Exercise 5

The Radon--Nikodym derivative is

\[ \frac{d\mathbb{Q}}{d\mathbb{P}}\bigg|_{\mathcal{F}_T} = Z_T = \exp\!\left(-\int_0^T \theta_t\,dW_t^{\mathbb{P}} - \frac{1}{2}\int_0^T \theta_t^2\,dt\right) \]

This is \(Z_T = \mathcal{E}(M)_T\) where \(M_t = -\int_0^t \theta_s\,dW_s^{\mathbb{P}}\) is a continuous local martingale with quadratic variation \(\langle M \rangle_T = \int_0^T \theta_t^2\,dt\).

The process \(Z_t\) is a non-negative local martingale, hence a supermartingale by Fatou's lemma. For \(Z_T\) to define a valid probability density we need \(\mathbb{E}^{\mathbb{P}}[Z_T] = 1\), requiring \(Z_t\) to be a true martingale.

The Novikov condition controls the exponential moment of \(\langle M \rangle_T / 2\). When this moment is finite, the fluctuations of \(M_t\) are bounded enough to prevent probability mass from leaking to infinity. This ensures \(Z_t\) cannot drift systematically downward in expectation, guaranteeing \(\mathbb{E}[Z_T] = 1\) and making \(\mathbb{Q}\) a well-defined probability measure.

Exercise 6. Consider three assets driven by two independent Brownian motions with \(\boldsymbol{\mu} = (0.08, 0.12, 0.06)^{\top}\), \(r = 0.02\), and volatility matrix

\[ \Sigma = \begin{pmatrix} 0.20 & 0.10 \\ 0.15 & 0.25 \\ 0.10 & 0.05 \end{pmatrix} \]

Determine whether \(\boldsymbol{\mu} - r\mathbf{1} = \Sigma\boldsymbol{\theta}\) has a solution. What is the no-arbitrage implication?

Solution to Exercise 6

The system is

\[ \begin{pmatrix} 0.06 \\ 0.10 \\ 0.04 \end{pmatrix} = \begin{pmatrix} 0.20 & 0.10 \\ 0.15 & 0.25 \\ 0.10 & 0.05 \end{pmatrix} \begin{pmatrix} \theta_1 \\ \theta_2 \end{pmatrix} \]

This is overdetermined (3 equations, 2 unknowns). Using the first two equations: from \(0.20\theta_1 + 0.10\theta_2 = 0.06\) we get \(\theta_2 = 0.6 - 2\theta_1\). Substituting into the second equation:

\[ 0.15\theta_1 + 0.25(0.6 - 2\theta_1) = 0.10 \implies -0.35\theta_1 = -0.05 \implies \theta_1 = \tfrac{1}{7} \]

\[ \theta_2 = 0.6 - \tfrac{2}{7} = \tfrac{16}{35} \]

Checking the third equation: \(0.10 \times \frac{1}{7} + 0.05 \times \frac{16}{35} = \frac{0.50 + 0.80}{35} = \frac{1.30}{35} \approx 0.0371 \neq 0.04\).

The system has no solution. The three excess returns are inconsistent with any single market price of risk vector. This implies an arbitrage opportunity: one can construct a portfolio of the three assets with zero exposure to both Brownian motions but positive excess return, violating no-arbitrage.