Examples: Constructing Risk-Neutral Measures¶

This section provides detailed worked examples of constructing the risk-neutral measure in various market models, from the basic Black-Scholes setting to multi-asset and interest rate models.

Taxonomy: completeness and uniqueness

The relationship between the number of traded assets \(n\) and the number of independent Brownian motions \(d\) determines the market structure:

Complete market (\(n = d\)): the volatility matrix \(\Sigma\) is square and invertible, so \(\boldsymbol{\theta}\) is unique and \(\mathbb{Q}\) is the unique risk-neutral measure.
Incomplete market (\(n < d\)): \(\Sigma\) is underdetermined, leaving \(d - n\) free parameters in \(\boldsymbol{\theta}\) and a family of risk-neutral measures.
Overdetermined (\(n > d\)): the system is consistent only if no-arbitrage constraints are satisfied; otherwise arbitrage exists.

Example 1: Black-Scholes Model (Single Stock)¶

Physical Dynamics¶

Under \(\mathbb{P}\), the stock price follows:

\[ dS_t = \mu S_t\,dt + \sigma S_t\,dW_t^{\mathbb{P}} \]

with constant parameters \(\mu\) (expected return) and \(\sigma\) (volatility).

Step 1: Identify the Market Price of Risk¶

The market price of risk is:

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

Example: If \(\mu = 0.12\), \(r = 0.05\), \(\sigma = 0.20\):

\[ \theta = \frac{0.12 - 0.05}{0.20} = 0.35 \]

Step 2: Define the Radon-Nikodym Derivative¶

\[ Z_T = \exp\left(-\theta W_T^{\mathbb{P}} - \frac{1}{2}\theta^2 T\right) \]

\[ \frac{d\mathbb{Q}}{d\mathbb{P}} = Z_T \]

Step 3: Apply Girsanov¶

Under \(\mathbb{Q}\):

\[ W_t^{\mathbb{Q}} = W_t^{\mathbb{P}} + \theta t \]

is a Brownian motion.

Step 4: Risk-Neutral Dynamics¶

\[ dS_t = \mu S_t\,dt + \sigma S_t\,dW_t^{\mathbb{P}} \]

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

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

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

Verification¶

The discounted price \(\tilde{S}_t = e^{-rt}S_t\) satisfies:

\[ d\tilde{S}_t = \sigma\tilde{S}_t\,dW_t^{\mathbb{Q}} \]

No drift term—\(\tilde{S}_t\) is a \(\mathbb{Q}\)-martingale. ✓

Example 2: Stock with Dividends¶

Physical Dynamics¶

\[ dS_t = (\mu - q)S_t\,dt + \sigma S_t\,dW_t^{\mathbb{P}} \]

where \(q\) is the continuous dividend yield.

Market Price of Risk¶

\[ \theta = \frac{(\mu - q) - (r - q)}{\sigma} = \frac{\mu - r}{\sigma} \]

Same as before—dividends don't affect \(\theta\).

Risk-Neutral Dynamics¶

\[ dS_t = (r - q)S_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}} \]

Discounted Process¶

The appropriate discounted process is \(e^{-rt}S_t e^{qt} = e^{-(r-q)t}S_t\):

\[ d(e^{-(r-q)t}S_t) = \sigma e^{-(r-q)t}S_t\,dW_t^{\mathbb{Q}} \]

This is a martingale. ✓

Example 3: Two Correlated Stocks¶

Physical Dynamics¶

\[ \begin{cases} dS_t^1 = \mu_1 S_t^1\,dt + \sigma_1 S_t^1\,dW_t^{1,\mathbb{P}} \\ dS_t^2 = \mu_2 S_t^2\,dt + \sigma_2 S_t^2\,(\rho\,dW_t^{1,\mathbb{P}} + \sqrt{1-\rho^2}\,dW_t^{2,\mathbb{P}}) \end{cases} \]

where \(W^1, W^2\) are independent and \(\rho\) is the correlation.

Market Price of Risk Vector¶

We need \(\boldsymbol{\theta} = (\theta_1, \theta_2)\) such that:

\[ \begin{pmatrix} \mu_1 - r \\ \mu_2 - r \end{pmatrix} = \begin{pmatrix} \sigma_1 & 0 \\ \sigma_2\rho & \sigma_2\sqrt{1-\rho^2} \end{pmatrix} \begin{pmatrix} \theta_1 \\ \theta_2 \end{pmatrix} \]

Solution¶

\[ \theta_1 = \frac{\mu_1 - r}{\sigma_1} \]

\[ \theta_2 = \frac{(\mu_2 - r) - \sigma_2\rho\theta_1}{\sigma_2\sqrt{1-\rho^2}} \]

Uniqueness¶

If the volatility matrix has full rank (2 assets, 2 Brownian motions), \(\boldsymbol{\theta}\) is unique and the market is complete.

Example 4: Incomplete Market (Stochastic Volatility)¶

Heston Model Dynamics¶

\[ \begin{cases} dS_t = \mu S_t\,dt + \sqrt{V_t}S_t\,dW_t^{1,\mathbb{P}} \\ dV_t = \kappa(\bar{V} - V_t)\,dt + \xi\sqrt{V_t}\,dW_t^{2,\mathbb{P}} \end{cases} \]

with \(\text{Corr}(dW^1, dW^2) = \rho\).

The Problem¶

Two sources of randomness (\(W^1, W^2\)) but only one traded asset (\(S\)).

The market price of risk for \(W^1\) is determined:

\[ \theta_1 = \frac{\mu - r}{\sqrt{V_t}} \]

But \(\theta_2\) (the volatility risk premium) is not determined by no-arbitrage alone.

Multiple Risk-Neutral Measures¶

Any choice of \(\theta_2(t, V_t)\) satisfying integrability conditions gives a valid risk-neutral measure.

Common choices: - \(\theta_2 = 0\) (no volatility risk premium) - \(\theta_2 = \lambda\sqrt{V_t}\) (proportional to volatility)

Implication¶

Different choices of \(\theta_2\) give different option prices. The market is incomplete.

Example 5: Foreign Exchange¶

Domestic Perspective¶

Under \(\mathbb{P}\):

\[ dX_t = (\mu_X)X_t\,dt + \sigma_X X_t\,dW_t^{\mathbb{P}} \]

where \(X_t\) is the exchange rate (domestic per foreign).

No-Arbitrage Condition¶

To prevent FX arbitrage:

\[ \mu_X = r_d - r_f \]

where \(r_d\) is domestic rate, \(r_f\) is foreign rate.

Market Price of Risk¶

\[ \theta = \frac{\mu_X - (r_d - r_f)}{\sigma_X} = 0 \]

The no-arbitrage condition implies that the Girsanov kernel is \(\theta = 0\), so the domestic risk-neutral measure already produces the correct drift.

Risk-Neutral Dynamics¶

\[ dX_t = (r_d - r_f)X_t\,dt + \sigma_X X_t\,dW_t^{\mathbb{Q}} \]

Example 6: Vasicek Interest Rate Model¶

Physical Dynamics¶

\[ dr_t = \kappa(\bar{r} - r_t)\,dt + \sigma\,dW_t^{\mathbb{P}} \]

Market Price of Risk¶

For interest rate models, \(\theta\) is typically specified exogenously:

\[ \theta_t = \lambda \quad \text{(constant)} \]

or

\[ \theta_t = \frac{\lambda}{\sigma}r_t \quad \text{(proportional to rate)} \]

Risk-Neutral Dynamics¶

\[ dr_t = \kappa(\bar{r} - r_t)\,dt + \sigma(dW_t^{\mathbb{Q}} - \theta\,dt) \]

\[ = [\kappa(\bar{r} - r_t) - \sigma\theta]\,dt + \sigma\,dW_t^{\mathbb{Q}} \]

\[ = \kappa^*(\bar{r}^* - r_t)\,dt + \sigma\,dW_t^{\mathbb{Q}} \]

where \(\kappa^* = \kappa\) and \(\bar{r}^* = \bar{r} - \sigma\theta/\kappa\).

Bond Pricing¶

Under \(\mathbb{Q}\):

\[ P(t,T) = \mathbb{E}^{\mathbb{Q}}\left[e^{-\int_t^T r_s\,ds} \;\middle|\; r_t\right] \]

For Vasicek, this has closed form:

\[ P(t,T) = A(t,T)e^{-B(t,T)r_t} \]

Summary Table¶

Model	# Assets	# BMs	Complete?	\(\theta\) Unique?
Black-Scholes	1	1	Yes	Yes
Multi-stock	\(n\)	\(n\)	Yes	Yes
Stoch. vol.	1	2	No	No
FX	1	1	Yes	Yes
Interest rate	Many (bonds)	1	Often incomplete (state not directly spanned)	No

Key Takeaways¶

Across all examples, the same structural pattern emerges: the number of traded assets relative to the number of independent Brownian motions determines whether the risk-neutral measure is unique.

Complete markets (\(n = d\)): \(\boldsymbol{\theta}\) is uniquely determined, one risk-neutral measure.
Incomplete markets (\(n < d\)): multiple valid \(\boldsymbol{\theta}\), multiple risk-neutral measures; the extra degrees of freedom correspond to unhedgeable risk factors.
Calibration: In practice, \(\mathbb{Q}\) is inferred from market prices, not derived from \(\mathbb{P}\).
Volatility unchanged: Only drift changes under measure transformation.

The construction of \(\mathbb{Q}\) is the mathematical foundation of derivative pricing.

Exercises¶

Exercise 1. In the Black-Scholes model with \(\mu = 0.15\), \(r = 0.03\), and \(\sigma = 0.30\), compute the market price of risk \(\theta\) and the Radon-Nikodym derivative \(Z_1\) for a specific path where \(W_1^{\mathbb{P}} = -0.5\). Is this path upweighted or downweighted under \(\mathbb{Q}\)?

Solution to Exercise 1

With \(\mu = 0.15\), \(r = 0.03\), and \(\sigma = 0.30\):

\[ \theta = \frac{\mu - r}{\sigma} = \frac{0.15 - 0.03}{0.30} = 0.40 \]

For the specific path with \(W_1^{\mathbb{P}} = -0.5\) and \(T = 1\):

\[ Z_1 = \exp\!\left(-\theta W_1^{\mathbb{P}} - \frac{1}{2}\theta^2 \cdot 1\right) = \exp\!\left(-0.40 \cdot (-0.5) - \frac{1}{2}(0.16)\right) \]

\[ = \exp(0.20 - 0.08) = \exp(0.12) \approx 1.1275 \]

Since \(Z_1 > 1\), this path is upweighted under \(\mathbb{Q}\). Intuitively, \(W_1^{\mathbb{P}} = -0.5\) corresponds to a below-average stock return. The risk-neutral measure overweights adverse outcomes (and underweights favorable ones), reflecting the risk adjustment embedded in \(\mathbb{Q}\).

Exercise 2. A stock pays a continuous dividend yield \(q = 0.02\) with \(\mu = 0.08\), \(\sigma = 0.20\), and \(r = 0.05\). Verify that the market price of risk is the same as in the no-dividend case. Write the risk-neutral dynamics and check that the discounted reinvested-dividend process is a \(\mathbb{Q}\)-martingale.

Solution to Exercise 2

With dividends, the stock dynamics are \(dS_t = (\mu - q)S_t\,dt + \sigma S_t\,dW_t^{\mathbb{P}}\). The discounted price for a dividend-paying stock must account for the reinvested dividends. Consider \(\tilde{S}_t = e^{-rt}e^{qt}S_t \cdot e^{-qt} = e^{-rt}S_t\). More carefully, the total return from holding the stock includes dividends, so the relevant discounted process is \(e^{-(r-q)t}S_t\) (or equivalently \(e^{-rt}\) times the dividend-reinvested portfolio).

The market price of risk is computed from the excess return of the total return process. Since the stock pays dividends at rate \(q\), the total instantaneous return is \(\mu\,dt + \sigma\,dW_t^{\mathbb{P}}\). The excess over \(r\) divided by \(\sigma\) gives:

\[ \theta = \frac{\mu - r}{\sigma} = \frac{0.08 - 0.05}{0.20} = 0.15 \]

This is the same as without dividends (where \(\mu\) would be the total return). Under \(\mathbb{Q}\):

\[ dS_t = (r - q)S_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}} \]

The discounted reinvested-dividend process \(\tilde{S}_t = e^{-(r-q)t}S_t\) satisfies:

\[ d\tilde{S}_t = \sigma\tilde{S}_t\,dW_t^{\mathbb{Q}} \]

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

Exercise 3. For the two correlated stocks model, take \(\mu_1 = 0.12\), \(\mu_2 = 0.08\), \(r = 0.03\), \(\sigma_1 = 0.25\), \(\sigma_2 = 0.30\), and \(\rho = 0.4\). Solve for \(\theta_1\) and \(\theta_2\). Verify your answer by checking that both excess returns \(\mu_i - r\) are reproduced by \(\Sigma\boldsymbol{\theta}\).

Solution to Exercise 3

Given \(\mu_1 = 0.12\), \(\mu_2 = 0.08\), \(r = 0.03\), \(\sigma_1 = 0.25\), \(\sigma_2 = 0.30\), \(\rho = 0.4\):

From \(\theta_1 = (\mu_1 - r)/\sigma_1\):

\[ \theta_1 = \frac{0.12 - 0.03}{0.25} = 0.36 \]

From \(\theta_2 = [(\mu_2 - r) - \sigma_2\rho\theta_1]/(\sigma_2\sqrt{1 - \rho^2})\):

\[ \theta_2 = \frac{(0.08 - 0.03) - 0.30 \cdot 0.4 \cdot 0.36}{0.30\sqrt{1 - 0.16}} = \frac{0.05 - 0.0432}{0.30 \cdot \sqrt{0.84}} \]

\[ = \frac{0.0068}{0.30 \cdot 0.9165} = \frac{0.0068}{0.2750} \approx 0.0247 \]

Verification: The volatility matrix is \(\Sigma = \begin{pmatrix} 0.25 & 0 \\ 0.30 \cdot 0.4 & 0.30\sqrt{0.84} \end{pmatrix} = \begin{pmatrix} 0.25 & 0 \\ 0.12 & 0.2750 \end{pmatrix}\).

\[ \Sigma\boldsymbol{\theta} = \begin{pmatrix} 0.25 \cdot 0.36 + 0 \cdot 0.0247 \\ 0.12 \cdot 0.36 + 0.2750 \cdot 0.0247 \end{pmatrix} = \begin{pmatrix} 0.09 \\ 0.0432 + 0.0068 \end{pmatrix} = \begin{pmatrix} 0.09 \\ 0.05 \end{pmatrix} \]

This matches \(\boldsymbol{\mu} - r\mathbf{1} = (0.09, 0.05)^{\top}\).

Exercise 4. In the Heston stochastic volatility model, explain why the volatility risk premium \(\theta_2\) cannot be determined by no-arbitrage. If one practitioner sets \(\theta_2 = 0\) and another sets \(\theta_2 = -0.5\sqrt{V_t}\), write the risk-neutral variance dynamics under each choice. Which choice produces a lower long-run mean of variance under \(\mathbb{Q}\)?

Solution to Exercise 4

In the Heston model, \(dS_t = \mu S_t\,dt + \sqrt{V_t}S_t\,dW_t^{1,\mathbb{P}}\) and \(dV_t = \kappa(\bar{V} - V_t)\,dt + \xi\sqrt{V_t}\,dW_t^{2,\mathbb{P}}\). There are two Brownian motions but only one traded risky asset \(S\). The no-arbitrage condition for \(S\) determines \(\theta_1 = (\mu - r)/\sqrt{V_t}\), but \(\theta_2\) (associated with \(W^2\)) is unconstrained because no traded asset's return depends solely on \(W^2\) in a way that pins down \(\theta_2\). This is the hallmark of an incomplete market.

Under \(\mathbb{Q}\) with a general \(\theta_2\), the variance dynamics become:

\[ dV_t = \kappa(\bar{V} - V_t)\,dt + \xi\sqrt{V_t}(dW_t^{2,\mathbb{Q}} - \theta_2\,dt) \]

Choice 1: \(\theta_2 = 0\):

\[ dV_t = \kappa(\bar{V} - V_t)\,dt + \xi\sqrt{V_t}\,dW_t^{2,\mathbb{Q}} \]

Long-run mean: \(\bar{V}\).

Choice 2: \(\theta_2 = -0.5\sqrt{V_t}\):

\[ dV_t = [\kappa(\bar{V} - V_t) + 0.5\xi V_t]\,dt + \xi\sqrt{V_t}\,dW_t^{2,\mathbb{Q}} \]

\[ = [(\kappa\bar{V}) - (\kappa - 0.5\xi)V_t]\,dt + \xi\sqrt{V_t}\,dW_t^{2,\mathbb{Q}} \]

This has the form \(\kappa^*(\bar{V}^* - V_t)\,dt\) with \(\kappa^* = \kappa - 0.5\xi\) and \(\bar{V}^* = \kappa\bar{V}/(\kappa - 0.5\xi)\).

Assuming \(\kappa > 0.5\xi\) (so mean reversion is maintained), \(\bar{V}^* = \kappa\bar{V}/(\kappa - 0.5\xi) > \bar{V}\). Therefore, Choice 2 produces a higher long-run mean of variance under \(\mathbb{Q}\), and Choice 1 produces the lower long-run mean \(\bar{V}\).

Exercise 5. In the FX example, the no-arbitrage condition implies \(\mu_X = r_d - r_f\). Derive this condition by requiring the domestic-currency value of a foreign money market investment to grow at rate \(r_d\) under \(\mathbb{Q}\). What happens to the market price of risk if \(\mu_X \neq r_d - r_f\)?

Solution to Exercise 5

The domestic-currency value of investing 1 unit of foreign currency in the foreign money market is \(V_t = X_t e^{r_f t}\). By Itô's formula:

\[ dV_t = e^{r_f t}dX_t + r_f X_t e^{r_f t}\,dt = V_t\!\left(\frac{dX_t}{X_t} + r_f\,dt\right) \]

Under \(\mathbb{Q}\), this discounted process \(\tilde{V}_t = e^{-r_d t}V_t\) must be a martingale:

\[ d\tilde{V}_t = \tilde{V}_t\!\left(\frac{dX_t}{X_t} + r_f\,dt - r_d\,dt\right) \]

For the drift of \(\tilde{V}_t\) to vanish, the drift of \(dX_t/X_t\) must equal \(r_d - r_f\):

\[ \mu_X = r_d - r_f \]

If \(\mu_X \neq r_d - r_f\), the market price of risk becomes

\[ \theta = \frac{\mu_X - (r_d - r_f)}{\sigma_X} \neq 0 \]

A Girsanov change of measure with this \(\theta\) is required to construct \(\mathbb{Q}\), and \(W_t^{\mathbb{Q}} = W_t^{\mathbb{P}} + \theta t\) absorbs the deviation. Under \(\mathbb{Q}\), the FX dynamics always become \(dX_t = (r_d - r_f)X_t\,dt + \sigma_X X_t\,dW_t^{\mathbb{Q}}\).

Exercise 6. In the Vasicek model with \(\kappa = 0.5\), \(\bar{r} = 0.04\), \(\sigma = 0.01\), and constant market price of risk \(\theta = 0.2\), compute \(\bar{r}^* = \bar{r} - \sigma\theta/\kappa\). Using the risk-neutral dynamics, compute the zero-coupon bond price \(P(0, T)\) for \(T = 5\) with \(r_0 = 0.03\).

Solution to Exercise 6

Given \(\kappa = 0.5\), \(\bar{r} = 0.04\), \(\sigma = 0.01\), \(\theta = 0.2\):

\[ \bar{r}^* = \bar{r} - \frac{\sigma\theta}{\kappa} = 0.04 - \frac{0.01 \cdot 0.2}{0.5} = 0.04 - 0.004 = 0.036 \]

Under \(\mathbb{Q}\), the dynamics are \(dr_t = \kappa(\bar{r}^* - r_t)\,dt + \sigma\,dW_t^{\mathbb{Q}}\) with \(\kappa^* = \kappa = 0.5\) and \(\bar{r}^* = 0.036\).

For the Vasicek model, the bond price is \(P(0,T) = A(0,T)e^{-B(0,T)r_0}\) where

\[ B(0,T) = \frac{1 - e^{-\kappa T}}{\kappa} = \frac{1 - e^{-0.5 \cdot 5}}{0.5} = \frac{1 - e^{-2.5}}{0.5} = \frac{1 - 0.08209}{0.5} = \frac{0.91791}{0.5} = 1.83583 \]

\[ A(0,T) = \exp\!\left[\left(\bar{r}^* - \frac{\sigma^2}{2\kappa^2}\right)(B(0,T) - T) - \frac{\sigma^2}{4\kappa}B(0,T)^2\right] \]

Computing: \(\sigma^2/(2\kappa^2) = 0.0001/0.5 = 0.0002\) and \(\sigma^2/(4\kappa) = 0.0001/2 = 0.00005\).

\[ A(0,5) = \exp\!\left[(0.036 - 0.0002)(1.83583 - 5) - 0.00005 \cdot (1.83583)^2\right] \]

\[ = \exp\!\left[0.0358 \cdot (-3.16417) - 0.00005 \cdot 3.37028\right] \]

\[ = \exp(-0.11328 - 0.00017) = \exp(-0.11345) \approx 0.89280 \]

Therefore:

\[ P(0,5) = 0.89280 \cdot e^{-1.83583 \cdot 0.03} = 0.89280 \cdot e^{-0.05507} = 0.89280 \cdot 0.94645 \approx 0.84498 \]

Exercise 7. Consider a market with 3 stocks and 2 Brownian motions. Write the system \(\boldsymbol{\mu} - r\mathbf{1} = \Sigma\boldsymbol{\theta}\) where \(\Sigma\) is \(3 \times 2\). This is an overdetermined system. State the condition for a solution to exist and interpret it as a no-arbitrage condition. If the condition is violated, construct a specific arbitrage strategy.

Solution to Exercise 7

With 3 stocks and 2 Brownian motions, each stock has dynamics \(dS_t^i = \mu_i S_t^i\,dt + \sum_{j=1}^{2}\Sigma_{ij}S_t^i\,dW_t^{j,\mathbb{P}}\). The system is:

\[ \begin{pmatrix} \mu_1 - r \\ \mu_2 - r \\ \mu_3 - r \end{pmatrix} = \begin{pmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \\ \Sigma_{31} & \Sigma_{32} \end{pmatrix} \begin{pmatrix} \theta_1 \\ \theta_2 \end{pmatrix} \]

This is an overdetermined system: 3 equations, 2 unknowns. A solution exists if and only if the vector \(\boldsymbol{\mu} - r\mathbf{1}\) lies in the column space of \(\Sigma\). Equivalently, \(\boldsymbol{\mu} - r\mathbf{1}\) must be orthogonal to the left null space of \(\Sigma\). If \(\mathbf{n}\) spans the left null space (so \(\mathbf{n}^{\top}\Sigma = 0\)), the condition is:

\[ \mathbf{n}^{\top}(\boldsymbol{\mu} - r\mathbf{1}) = 0 \]

No-arbitrage interpretation: The vector \(\mathbf{n}\) represents portfolio weights. The condition \(\mathbf{n}^{\top}\Sigma = 0\) means this portfolio has zero exposure to both Brownian motions (zero volatility, hence risk-free). No-arbitrage then requires its expected return to equal \(r\), i.e., \(\mathbf{n}^{\top}\boldsymbol{\mu} = r(\mathbf{n}^{\top}\mathbf{1})\), which is \(\mathbf{n}^{\top}(\boldsymbol{\mu} - r\mathbf{1}) = 0\).

Constructing arbitrage when violated: If \(\mathbf{n}^{\top}(\boldsymbol{\mu} - r\mathbf{1}) > 0\), hold the portfolio \(\mathbf{n}\) (long/short the three stocks in proportions \(n_1, n_2, n_3\)) and finance it at rate \(r\). The portfolio has zero volatility (deterministic returns) but earns more than \(r\), producing a risk-free arbitrage profit.