38. Change Probability Measure
本章给出测度变换 (change of probability measure) 的离散时间构造。给定标准高斯冲击 \(W_{t+1}\sim\mathcal N(0,1)\)，用一个随机变量 \(H_{t+1}\)（即 Radon-Nikodym 导数）定义新测度下的条件期望 \(\hat{\mathbb E}[Y_{t+1}\mid\mathcal F_t]\equiv\mathbb E[H_{t+1}Y_{t+1}\mid\mathcal F_t]\)。\(H_{t+1}\) 是合法测度变换的两个条件：(1) \(H_{t+1}\ge0\)（密度非负）；(2) \(\mathbb E[H_{t+1}\mid\mathcal F_t]=1\)（密度积分为 1）。当 \(\ln H_{t+1}\) 关于 \(W_{t+1}\) 线性 (38.1) 时，新测度仅平移冲击均值：\(\hat{\mathbb E}[W_{t+1}\mid\mathcal F_t]=b(X_t)\) (38.4)；当含 \(W_{t+1}^2\) 的二次项 (38.2) 时，新测度同时改变均值与方差：\(\hat\mu=\frac{b(X_t)}{1-2\gamma(X_t)}\)、\(\hat\sigma^2=\frac1{1-2\gamma(X_t)}\)。这是资产定价中从物理测度到风险中性测度变换的基础。
This chapter gives the discrete-time construction of a change of probability measure. Given a standard Gaussian shock \(W_{t+1}\sim\mathcal N(0,1)\), a random variable \(H_{t+1}\) (the Radon-Nikodym derivative) defines the conditional expectation under the new measure \(\hat{\mathbb E}[Y_{t+1}\mid\mathcal F_t]\equiv\mathbb E[H_{t+1}Y_{t+1}\mid\mathcal F_t]\). The two conditions for \(H_{t+1}\) to be a legitimate change of measure: (1) \(H_{t+1}\ge0\) (density nonnegative); (2) \(\mathbb E[H_{t+1}\mid\mathcal F_t]=1\) (density integrates to 1). When \(\ln H_{t+1}\) is linear in \(W_{t+1}\) (38.1), the new measure merely shifts the shock's mean: \(\hat{\mathbb E}[W_{t+1}\mid\mathcal F_t]=b(X_t)\) (38.4); when there is a quadratic term in \(W_{t+1}^2\) (38.2), the new measure changes both mean and variance: \(\hat\mu=\frac{b(X_t)}{1-2\gamma(X_t)}\), \(\hat\sigma^2=\frac1{1-2\gamma(X_t)}\). This is the foundation of the physical-to-risk-neutral measure change in asset pricing.
38.1 Setup
设随机冲击为标量 \(W_{t+1}\sim\mathcal N(0,1)\)。设随机变量 \(H_{t+1}\) 由某过程定义，例如 (38.1) 或 (38.2)（\(X_t\) 为状态变量向量）：
Let the random shock be a scalar \(W_{t+1}\sim\mathcal N(0,1)\). Suppose a random variable \(H_{t+1}\) is defined by some process, such as (38.1) or (38.2) (\(X_t\) a vector of state variables):
$$\ln(H_{t+1})=\alpha(X_t)+b(X_t)\,W_{t+1}\tag{38.1}$$
$$\ln(H_{t+1})=\alpha(X_t)+b(X_t)\,W_{t+1}+\gamma(X_t)\,W_{t+1}^2\tag{38.2}$$
38.2 Conditions for Change of Measure
称 \(H_{t+1}\) 为 \(t\) 时刻适当定义的测度变换，若：
- \(H_{t+1}\ge0\)：保证新测度下密度函数非负；
- \(\mathbb E[H_{t+1}\mid\mathcal F_t]=1\)：保证新测度下密度积分为 1。
38.3 Characterize the New Probability Measure
用 \(\hat{\mathbb E}[\,\cdot\mid\mathcal F_t]\) 记新测度下的条件期望，使得对任意 \(Y_{t+1}\in\mathcal F_{t+1}\)，
$$\hat{\mathbb E}[Y_{t+1}\mid\mathcal F_t]=\mathbb E[H_{t+1}Y_{t+1}\mid\mathcal F_t].$$
条件 (2) \(\mathbb E[H_{t+1}\mid\mathcal F_t]=1\) 蕴含 (38.3)：
The random variable \(H_{t+1}\) is said to be an appropriately defined change of probability measure at time \(t\) if:
- \(H_{t+1}\ge0\): this guarantees the density function under the new measure is nonnegative;
- \(\mathbb E[H_{t+1}\mid\mathcal F_t]=1\): this guarantees the density integrates to 1 under the new measure.
38.3 Characterize the New Probability Measure
Denote the conditional expectation under the new measure by \(\hat{\mathbb E}[\,\cdot\mid\mathcal F_t]\), such that for any \(Y_{t+1}\in\mathcal F_{t+1}\),
$$\hat{\mathbb E}[Y_{t+1}\mid\mathcal F_t]=\mathbb E[H_{t+1}Y_{t+1}\mid\mathcal F_t].$$
Condition (2) \(\mathbb E[H_{t+1}\mid\mathcal F_t]=1\) implies (38.3):
$$1=\mathbb E\big[e^{\alpha(X_t)+b(X_t)W_{t+1}}\mid\mathcal F_t\big]=e^{\alpha(X_t)+\frac12(b(X_t))^2}\tag{38.3}$$
38.4 Case (38.1): Linear — Mean Shift
设 (38.1) 成立，即 \(\ln(H_{t+1})=\alpha(X_t)+b(X_t)W_{t+1}\)。则 \(W_{t+1}\) 在新测度下的均值为 (38.4)：
Suppose (38.1) is true, i.e. \(\ln(H_{t+1})=\alpha(X_t)+b(X_t)W_{t+1}\). Then the mean of \(W_{t+1}\) under the new measure is (38.4):
$$\hat{\mathbb E}[W_{t+1}\mid\mathcal F_t]=\mathbb E[H_{t+1}W_{t+1}\mid\mathcal F_t]=\int_{-\infty}^\infty w\,\frac1{\sqrt{2\pi}}e^{-\frac12(w-b(X_t))^2}\,dw=b(X_t)\tag{38.4}$$
最后一步成立，因为 \(\frac1{\sqrt{2\pi}}e^{-\frac12(w-b(X_t))^2}\) 恰是 \(\mathcal N(b(X_t),1)\) 的密度（推导中用 (38.3) 消去常数项）。故新测度下 \(W_{t+1}\) 的均值是 \(b(X_t)\)——测度变换只平移了冲击的均值，方差仍为 1。
38.5 Case (38.2): Quadratic — Mean and Variance Change
设 (38.2) 成立，即 \(\ln(H_{t+1})=\alpha(X_t)+b(X_t)W_{t+1}+\gamma(X_t)W_{t+1}^2\)。此时改用 \(\dfrac{H_{t+1}}{\mathbb E[H_{t+1}\mid\mathcal F_t]}\) 做测度变换（满足 \(\mathbb E\big[\frac{H_{t+1}}{\mathbb E[H_{t+1}\mid\mathcal F_t]}\mid\mathcal F_t\big]=1\)）。则 \(W_{t+1}\) 在新测度下的均值
The last line is true because \(\frac1{\sqrt{2\pi}}e^{-\frac12(w-b(X_t))^2}\) is exactly the density of \(\mathcal N(b(X_t),1)\) (the derivation uses (38.3) to cancel the constant term). So the mean of \(W_{t+1}\) under the new measure is \(b(X_t)\) — the change of measure only shifts the shock's mean; the variance is still 1.
38.5 Case (38.2): Quadratic — Mean and Variance Change
Suppose (38.2) is true, i.e. \(\ln(H_{t+1})=\alpha(X_t)+b(X_t)W_{t+1}+\gamma(X_t)W_{t+1}^2\). Now use \(\dfrac{H_{t+1}}{\mathbb E[H_{t+1}\mid\mathcal F_t]}\) to change the measure (satisfying \(\mathbb E\big[\frac{H_{t+1}}{\mathbb E[H_{t+1}\mid\mathcal F_t]}\mid\mathcal F_t\big]=1\)). Then the mean of \(W_{t+1}\) under the new measure
$$\hat{\mathbb E}[W_{t+1}\mid\mathcal F_t]=\int_{-\infty}^\infty w\,\frac1{\sqrt{2\pi}}e^{-\frac12\left(\frac{w-\mu}{\sigma}\right)^2}\,dw$$
通过配方匹配指数，得 \(-\dfrac1{2\sigma^2}=\gamma(X_t)-\dfrac12\) 与 \(\dfrac\mu{\sigma^2}=b(X_t)\)，解出：
Matching the exponent by completing the square gives \(-\dfrac1{2\sigma^2}=\gamma(X_t)-\dfrac12\) and \(\dfrac\mu{\sigma^2}=b(X_t)\), which solve to:
$$\sigma^2=\frac1{1-2\gamma(X_t)},\qquad \mu=\sigma^2 b(X_t)=\frac{b(X_t)}{1-2\gamma(X_t)}$$
故新测度下 \(W_{t+1}\) 的均值为 \(\dfrac{b(X_t)}{1-2\gamma(X_t)}\)、方差为 \(\dfrac1{1-2\gamma(X_t)}\)——二次项 \(\gamma\) 同时改变了均值与方差。
So under the new measure \(W_{t+1}\) has mean \(\dfrac{b(X_t)}{1-2\gamma(X_t)}\) and variance \(\dfrac1{1-2\gamma(X_t)}\) — the quadratic term \(\gamma\) changes both the mean and the variance.
References
- He, X. (2019d). Stochastic Calculus Notes by Xindi He.
- He, X. (2020–2024). Asset Pricing (lecture notes), Ch. 38.