Examples: Constructing Risk-Neutral Measures¶

This page is a catalog of how risk-neutral measures look in concrete models. Each example follows the same template from § Construction of the Risk-Neutral Measure:

1. identify \(\mathbb{P}\)-dynamics,
2. solve \(\boldsymbol{\mu} - r\mathbf{1} = \Sigma\boldsymbol{\theta}\) for the market price of risk,
3. apply Girsanov,
4. read off the risk-neutral dynamics.

Completeness vs incompleteness is determined by the rank of \(\Sigma\) — see § Market Price of Risk, multi-asset case for the taxonomy.

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

Under \(\mathbb{P}\): \(dS_t = \mu S_t\,dt + \sigma S_t\,dW_t^{\mathbb{P}}\) with constant \(\mu\), \(\sigma\), \(r\).

Market price of risk: \(\theta = (\mu - r)/\sigma\).

Under \(\mathbb{Q}\):

Example 2: Stock with Dividends¶

Under \(\mathbb{P}\): \(dS_t = (\mu - q)S_t\,dt + \sigma S_t\,dW_t^{\mathbb{P}}\) with continuous dividend yield \(q\).

Market price of risk: \(\theta = (\mu - r)/\sigma\) (dividends cancel).

Under \(\mathbb{Q}\):

Example 3: Two Correlated Stocks¶

Two stocks driven by Brownian motions \(W^1, W^2\) with correlation \(\rho\):

Recall (see § Market Price of Risk) the vector equation \(\boldsymbol{\mu} - r\mathbf{1} = \Sigma\boldsymbol{\theta}\). Here \(\Sigma\) is the \(2 \times 2\) lower-triangular matrix with rows \((\sigma_1, 0)\) and \((\sigma_2\rho,\,\sigma_2\sqrt{1-\rho^2})\), giving \(\theta_1 = (\mu_1 - r)/\sigma_1\) and \(\theta_2 = [(\mu_2 - r) - \sigma_2\rho\theta_1]/(\sigma_2\sqrt{1-\rho^2})\).

With \(n = d = 2\) and \(\Sigma\) invertible, \(\boldsymbol{\theta}\) is unique and the market is complete.

Example 4: Incomplete Market (Stochastic Volatility)¶

The Heston model has two sources of randomness but only one traded asset:

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

No-arbitrage determines \(\theta_1 = (\mu - r)/\sqrt{V_t}\), but \(\theta_2\) (the volatility risk premium) remains free — any choice satisfying integrability gives a valid \(\mathbb{Q}\). Common specifications include \(\theta_2 = 0\) and \(\theta_2 = \lambda\sqrt{V_t}\). Different choices yield different option prices: the market is incomplete.

Example 5: Foreign Exchange¶

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 (covered interest parity): \(\mu_X = r_d - r_f\), so \(\theta = 0\).

Under \(\mathbb{Q}\):

Example 6: Vasicek Interest Rate Model¶

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

Market price of risk: specified exogenously (e.g., \(\theta_t = \lambda\)) because the short rate is not a traded asset.

Under \(\mathbb{Q}\) (with \(\theta = \lambda\)):

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

Summary Table¶

The unifying structural insight, visible in every row of the table above:

See § Market Price of Risk, Exercise 2 for the Girsanov derivation.

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 \(\mathbb{Q}\)?

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.

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 \(\Sigma\boldsymbol{\theta}\).

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 \(\mathbb{Q}\)?

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 \(\mu_X \neq r_d - r_f\)?

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 \(r_0 = 0.03\).