Measure Change and Financial Interpretation¶

The previous sections established how to construct the risk-neutral measure \(\mathbb{Q}\) using Girsanov’s theorem and the Radon--Nikodym derivative. This section answers the next natural question:

What do we do with \(\mathbb{Q}\)?

The answer is pricing. Under \(\mathbb{Q}\), the value of any contingent claim is a discounted expectation, and the complex problem of derivative valuation reduces to computing an integral. The remaining pages of this section organize the conceptual, mechanical, practical, and theoretical implications.

The Central Formula¶

The risk-neutral valuation principle states that the time-\(0\) price of a contingent claim with payoff \(X_T\) at maturity \(T\) is

\[ V_0 = \mathbb{E}^{\mathbb{Q}}\!\left[e^{-\int_0^T r_s\,ds}\,X_T\right] \]

where the expectation is taken under the risk-neutral measure \(\mathbb{Q}\) and \(r_s\) is the instantaneous risk-free rate. When \(r\) is constant, this simplifies to

\[ V_0 = e^{-rT}\,\mathbb{E}^{\mathbb{Q}}[X_T] \]

Pricing = expectation under the measure that makes discounted prices martingales.

Core Topics¶

The essential content focuses on three questions:

What is the difference between \(\mathbb{P}\) and \(\mathbb{Q}\)? The physical vs risk-neutral world clarifies the roles of the two measures and why they must not be confused.
Where does the risk premium go? The risk premium decomposition shows how Girsanov’s theorem absorbs the excess return \(\mu - r\) into the measure change, leaving only the risk-free drift under \(\mathbb{Q}\).
What is the difference between pricing and hedging? Pricing vs hedging explains that pricing is an expectation under \(\mathbb{Q}\), while hedging is a pathwise replication that does not depend on the choice of measure.

Deeper Perspectives¶

The remaining topics extend the pricing framework in different directions:

The stochastic discount factor provides the economic foundation --- it explains why the risk-neutral measure exists by connecting measure change to investor preferences and marginal utility.
From SDF to CAPM shows that equilibrium asset pricing models (CAPM, factor models) are special cases of the same structure.
The practitioner perspective examines how the theoretical framework is actually used: calibration, model risk, discrete hedging, and the implied volatility surface.
When measure change fails catalogs the failure modes --- Novikov violation, strict local martingales (bubbles), incomplete markets --- that mark the boundaries of the framework.

Guiding Principle¶

The distinction between physical and risk-neutral measures is developed in § Physical vs Risk-Neutral World; the mechanism by which \(\mu\) is replaced by \(r\) is established in § Risk Premium Decomposition. All other pages in this section reference these canonical treatments rather than re-deriving them.

Exercises¶

Exercise 1. A stock follows geometric Brownian motion under \(\mathbb{P}\) with drift \(\mu = 0.12\), volatility \(\sigma = 0.20\), and risk-free rate \(r = 0.03\). Using the risk-neutral valuation formula, compute the price of a derivative that pays \(S_T^2\) at time \(T = 1\) (take \(S_0 = 1\)).

Solution to Exercise 1

Under \(\mathbb{Q}\), the stock dynamics are \(dS_t = rS_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}}\), so

\[ S_T = S_0 \exp\!\left[\left(r - \tfrac{1}{2}\sigma^2\right)T + \sigma W_T^{\mathbb{Q}}\right] \]

Therefore

\[ S_T^2 = S_0^2 \exp\!\left[2\left(r - \tfrac{1}{2}\sigma^2\right)T + 2\sigma W_T^{\mathbb{Q}}\right] \]

Since \(W_T^{\mathbb{Q}} \sim N(0, T)\) under \(\mathbb{Q}\), the moment generating function gives

\[ \mathbb{E}^{\mathbb{Q}}[S_T^2] = S_0^2 \exp\!\left[2\left(r - \tfrac{1}{2}\sigma^2\right)T + \tfrac{1}{2}(2\sigma)^2 T\right] = S_0^2 \exp\!\left[(2r + \sigma^2)T\right] \]

The price is

\[ V_0 = e^{-rT}\,\mathbb{E}^{\mathbb{Q}}[S_T^2] = S_0^2 \exp\!\left[(r + \sigma^2)T\right] \]

Substituting \(S_0 = 1\), \(r = 0.03\), \(\sigma = 0.20\), \(T = 1\):

\[ V_0 = \exp(0.03 + 0.04) = e^{0.07} \approx 1.0725 \]

The physical drift \(\mu = 0.12\) does not appear --- only \(r\) and \(\sigma\) determine the price.

Exercise 2. A colleague argues that a stock with higher expected return \(\mu\) should produce more expensive call options. Using the risk-neutral valuation formula, identify the error in this reasoning and explain why derivative prices are independent of \(\mu\).

Solution to Exercise 2

The risk-neutral valuation formula computes an expectation under \(\mathbb{Q}\), where the stock dynamics are \(dS_t = rS_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}}\). Girsanov’s theorem absorbs the physical drift \(\mu\) into the change of measure: the Radon--Nikodym derivative depends on \(\theta = (\mu - r)/\sigma\), but once we compute under \(\mathbb{Q}\), only \(r\) and \(\sigma\) remain.

The colleague’s error is confusing the physical measure \(\mathbb{P}\) with the pricing measure \(\mathbb{Q}\). Under \(\mathbb{P}\), a higher \(\mu\) means the stock is more likely to reach higher values. But pricing uses \(\mathbb{Q}\), where every stock has drift \(r\) regardless of its physical drift. A higher \(\mu\) changes the Radon--Nikodym derivative (reweighting paths more aggressively) but does not change the risk-neutral distribution of \(S_T\). Therefore option prices are independent of \(\mu\).

This is the fundamental insight of Black--Scholes: option prices depend on volatility and the risk-free rate, not on the expected return of the underlying.

Exercise 3. In the risk-neutral valuation formula with stochastic rates, the discount factor is \(e^{-\int_0^T r_s\,ds}\). Explain why discounting at the path-dependent rate \(r_s\) is necessary rather than at a fixed rate. What goes wrong if one discounts at \(\bar{r} = \mathbb{E}^{\mathbb{Q}}[\frac{1}{T}\int_0^T r_s\,ds]\) instead?

Solution to Exercise 3

The discount factor \(e^{-\int_0^T r_s\,ds}\) appears because the money market account \(B_t = e^{\int_0^t r_s\,ds}\) is the numéraire. The discounted price process \(V_t / B_t\) must be a \(\mathbb{Q}\)-martingale, which requires discounting at the actual realized path of short rates, not at their expected value.

Replacing the stochastic discount factor with a deterministic one introduces an error because

\[ e^{-\bar{r}T} \neq \mathbb{E}^{\mathbb{Q}}\!\left[e^{-\int_0^T r_s\,ds}\right] \]

by Jensen’s inequality (since the exponential is convex). More importantly, the correct formula \(V_0 = \mathbb{E}^{\mathbb{Q}}[e^{-\int_0^T r_s\,ds}\,X_T]\) captures the covariance between the discount factor and the payoff. When interest rates and the payoff are correlated (as they generally are when rates affect asset prices), replacing the stochastic discount with a constant ignores this covariance and misprices the claim.

Exercise 4. Consider a European call with payoff \((S_T - K)^+\) and a digital option with payoff \(\mathbf{1}_{S_T > K}\). Write the risk-neutral pricing formula for each. Then show that \(\partial C / \partial K = -e^{-rT}\,\mathbb{Q}(S_T > K)\), where \(C\) is the call price.

Solution to Exercise 4

The risk-neutral prices are

\[ C = e^{-rT}\,\mathbb{E}^{\mathbb{Q}}[(S_T - K)^+], \qquad D = e^{-rT}\,\mathbb{E}^{\mathbb{Q}}[\mathbf{1}_{S_T > K}] = e^{-rT}\,\mathbb{Q}(S_T > K) \]

Write the call price as an integral over the risk-neutral density \(q(s)\) of \(S_T\):

\[ C = e^{-rT}\int_K^{\infty}(s - K)\,q(s)\,ds \]

Differentiating under the integral with respect to \(K\):

\[ \frac{\partial C}{\partial K} = e^{-rT}\!\left[-(K - K)\,q(K) - \int_K^{\infty} q(s)\,ds\right] = -e^{-rT}\int_K^{\infty} q(s)\,ds \]

\[ = -e^{-rT}\,\mathbb{Q}(S_T > K) \]

This shows \(D = -\partial C / \partial K\): the digital call price equals the negative of the strike-derivative of the European call price. In practice, this relationship is used to hedge digital options with call spreads.

Exercise 5. Prove that if the discounted price process \(\tilde{S}_t = e^{-rt}S_t\) is a \(\mathbb{Q}\)-martingale, then \(\mathbb{E}^{\mathbb{Q}}[e^{-rT}S_T] = S_0\). Explain why this is an essential consistency check for any risk-neutral pricing model.

Solution to Exercise 5

If \(\tilde{S}_t = e^{-rt}S_t\) is a \(\mathbb{Q}\)-martingale, then by the martingale property:

\[ \mathbb{E}^{\mathbb{Q}}[\tilde{S}_T \mid \mathcal{F}_0] = \tilde{S}_0 \]

Since \(\tilde{S}_T = e^{-rT}S_T\) and \(\tilde{S}_0 = S_0\):

\[ \mathbb{E}^{\mathbb{Q}}[e^{-rT}S_T] = S_0 \]

This is an essential consistency check because the stock is a traded asset with known price \(S_0\). Applying the risk-neutral valuation formula to the stock (whose "payoff" at \(T\) is \(S_T\)) must recover its current price. Any model that fails this check is internally inconsistent: it would misprice the underlying, and all derivative prices built on it would be unreliable.

When this equality fails --- specifically when \(\mathbb{E}^{\mathbb{Q}}[e^{-rT}S_T] < S_0\) --- the discounted price is a strict local martingale rather than a true martingale, which is interpreted as an asset price bubble.

Exercise 6. List which subsection of this chapter is the canonical home for each of the following ideas, and explain why the assignment removes redundancy: (a) the meaning of \(\mathbb{P}\) vs \(\mathbb{Q}\); (b) the algebraic mechanism by which \(\mu\) becomes \(r\); (c) the contrast between expectation-based pricing and pathwise hedging; (d) the economic origin of risk-neutral reweighting.

Solution to Exercise 6

(a) § Physical vs Risk-Neutral World owns the conceptual boundary between \(\mathbb{P}\) (real-world dynamics) and \(\mathbb{Q}\) (pricing weights).

(b) § Risk Premium Decomposition owns the identity \(\mu = r + \sigma\theta\) and the Girsanov substitution that replaces \(\mu\) with \(r\).

(c) § Pricing vs Hedging owns the contrast between measure-dependent expectation and measure-invariant replication.

(d) § The Stochastic Discount Factor owns the marginal-utility explanation of why \(\mathbb{Q}\) differs from \(\mathbb{P}\).

Assigning each idea a single home eliminates redundancy: every other page references the canonical treatment instead of re-deriving it.