Risk-Neutral Dynamics for Stochastic-Volatility Models

Two Sources of Risk, Two Drifts to Transform

In a stochastic-volatility model, the asset price and its variance are both random, and their shocks may be correlated. Pricing changes the probability measure used to evaluate discounted payoffs. Under that measure, the asset’s drift must match the risk-free return after accounting for dividends. The variance drift also changes: its adjustment depends on both market prices of risk and on the correlation structure. Keeping those two drift changes separate is essential.

Work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})$ carrying two independent standard Brownian motions $B^1_t,B^2_t$. Fix a finite horizon $T$, and assume the filtration satisfies the usual conditions. For a progressively measurable process $H$, write $H\in\mathcal{L}^2$ on $[0,T]$ when $\mathbb{E}\int_0^T H_s^2\,ds<\infty$; Ito integrals below have integrands in this class. Define a second Brownian driver by $B^V_t=\rho B^1_t+\sqrt{1-\rho^2}B^2_t$, where $-1<\rho<1$. Thus the asset driver $B^1$ and the variance driver $B^V$ have instantaneous correlation $\rho$.

Definition: A physical-measure two-factor specification for a positive asset $S_t$ and a nonnegative variance state $V_t$ is $$ \frac{dS_t}{S_t}=\mu_t\,dt+\sqrt{V_t}\,dB^1_t,\qquad dV_t=a_t\,dt+\eta_t\,dB^V_t. $$ Here $\mu_t$ and $a_t$ are drift coefficients, while $\sqrt{V_t}$ and $\eta_t$ are diffusion coefficients. The coefficients must be adapted and locally integrable so the equations are defined; to use the ordinary $L^2$ Ito integral on $[0,T]$, require the corresponding diffusion integrands to be in $\mathcal{L}^2$. Nonnegativity of $V_t$ is a model requirement, not a consequence of this notation alone.

Let $r_t$ be the short rate and $q_t$ a continuous dividend yield. A risk-neutral measure $\mathbb{Q}$, equivalent to $\mathbb{P}$ on $\mathcal{F}_T$, is a pricing measure under which the asset’s drift is $r_t-q_t$. The market prices of risk are two progressively measurable processes $\theta^1_t,\theta^2_t$. Assume they are in $\mathcal{L}^2$, and impose Novikov’s condition $\mathbb{E}\exp\{\tfrac12\int_0^T[(\theta^1_s)^2+(\theta^2_s)^2]\,ds\}<\infty$.

Standard result (Girsanov’s theorem, with Novikov’s condition): The process $$ Z_t=\mathcal{E}\left(-\int_0^\cdot\theta^1_s\,dB^1_s-\int_0^\cdot\theta^2_s\,dB^2_s\right)_t =\exp\left(-\int_0^t\theta^1_s\,dB^1_s-\int_0^t\theta^2_s\,dB^2_s-\frac12\int_0^t\big((\theta^1_s)^2+(\theta^2_s)^2\big)\,ds\right) $$ is a true martingale on $[0,T]$ with expectation one. Defining $d\mathbb{Q}/d\mathbb{P}|_{\mathcal{F}_T}=Z_T$, the processes $B^{i,\mathbb{Q}}_t=B^i_t+\int_0^t\theta^i_s\,ds$, $i=1,2$, are independent Brownian motions under $\mathbb{Q}$. This is the standard Girsanov result; its proof is not given here.

The theorem’s martingale assertion matters: it makes $Z_T$ a probability density, rather than merely a positive local martingale with expectation possibly below one. In examples, bounded market prices of risk satisfy Novikov’s condition on every finite horizon. Under $\mathbb{Q}$, define $B^{V,\mathbb{Q}}_t=\rho B^{1,\mathbb{Q}}_t+\sqrt{1-\rho^2}B^{2,\mathbb{Q}}_t$. Substitution into the original Brownian drivers gives the drift transformations.

The Drift Transformation

Proposition: Under the measure change above, the two-factor specification becomes $$ \frac{dS_t}{S_t}=(\mu_t-\sqrt{V_t}\theta^1_t)\,dt+\sqrt{V_t}\,dB^{1,\mathbb{Q}}_t, \qquad dV_t=\big(a_t-\eta_t\beta_t\big)\,dt+\eta_t\,dB^{V,\mathbb{Q}}_t, $$ where $\beta_t=\rho\theta^1_t+\sqrt{1-\rho^2}\theta^2_t$. To impose the risk-neutral asset drift, require $\mu_t-\sqrt{V_t}\theta^1_t=r_t-q_t$. The variance drift under $\mathbb{Q}$ is $a^{\mathbb{Q}}_t=a_t-\eta_t\beta_t$, or, inversely, $a_t=a^{\mathbb{Q}}_t+\eta_t\beta_t$.

Proof. From the definition of $B^{i,\mathbb{Q}}$, $dB^i_t=dB^{i,\mathbb{Q}}_t-\theta^i_t\,dt$. Therefore $dB^V_t=dB^{V,\mathbb{Q}}_t-[\rho\theta^1_t+\sqrt{1-\rho^2}\theta^2_t]dt =dB^{V,\mathbb{Q}}_t-\beta_tdt$. Substituting $dB^1_t$ into the asset equation subtracts $\sqrt{V_t}\theta^1_t$ from its drift. Substituting $dB^V_t$ into the variance equation subtracts $\eta_t\beta_t$ from its drift. Equating the asset drift to $r_t-q_t$ gives the stated pricing restriction. Rearranging $a^{\mathbb{Q}}_t=a_t-\eta_t\beta_t$ gives $a_t=a^{\mathbb{Q}}_t+\eta_t\beta_t$, with a plus sign in the inverse relation. $\square$

The asset’s risk-neutral drift depends only on $\theta^1_t$, because the asset is driven by $B^1_t$. The variance drift depends on a projection of both risk prices onto its Brownian driver. This is why correlation affects the variance adjustment even though the asset drift formula itself contains no $\rho$.

Worked Example: The Joint Instantaneous Covariance

At a time when $V_t=4$, take $\eta_t=1$ and $\rho=\tfrac35$. Compute the covariance matrix of the asset’s relative return and the variance increment, per unit time.

The random parts of the increments are $2\,dB^1_t$ and $dB^V_t$. Since $dB^1_t\,dB^1_t=dt$, $dB^2_t\,dB^2_t=dt$, and $dB^1_t\,dB^2_t=0$, their covariation is $2\,dB^1_t\,dB^V_t=2\rho\,dt=\tfrac65dt$. The variances per unit time are $2^2=4$ and $1^2=1$, respectively. Thus the covariance matrix is $$ \begin{pmatrix} 4 & \tfrac65\\[2pt] \tfrac65 & 1 \end{pmatrix}. $$

Check: its determinant is $4\cdot1-(\tfrac65)^2=4-\tfrac{36}{25}=\tfrac{64}{25}>0$, consistent with two non-degenerate Brownian directions. The correlation is $(\tfrac65)/(\sqrt{4}\sqrt{1})=\tfrac35$, as specified. More generally, the covariance per unit time between $dS_t/S_t$ and $dV_t$ is $\rho\sqrt{V_t}\eta_t$. A correlation coefficient for the two instantaneous increments is defined only when both diffusion coefficients are nonzero.

Worked Drift Changes

Worked Example: Setting the Asset’s Pricing Drift

At a time with $V_t=4$, suppose $r_t=\tfrac1{20}$, $q_t=\tfrac1{100}$, and $\theta^1_t=\tfrac15$. Find the physical asset drift required for the risk-neutral drift to be $r_t-q_t$.

The pricing drift is $\tfrac1{20}-\tfrac1{100}=\tfrac5{100}-\tfrac1{100}=\tfrac4{100}=\tfrac1{25}$. The physical drift must satisfy $\mu_t-\sqrt{V_t}\theta^1_t=\tfrac1{25}$. Since $\sqrt{V_t}=2$, the risk adjustment is $2\cdot\tfrac15=\tfrac25$, and hence $\mu_t=\tfrac1{25}+\tfrac25=\tfrac1{25}+\tfrac{10}{25}=\tfrac{11}{25}$.

Check: substituting back gives $\tfrac{11}{25}-2\cdot\tfrac15=\tfrac{11}{25}-\tfrac{10}{25}=\tfrac1{25}=r_t-q_t$. If $V_t=0$, the diffusion is zero and this formula cannot adjust the asset drift there; the pricing restriction then requires $\mu_t=r_t-q_t$ at that state.

Worked Example: Transforming a Nonnegative Variance Drift

Take $\rho=\tfrac35$, $\eta_t=2V_t$, and physical variance drift $a_t=1-V_t$. Let $\theta^1_t=\tfrac14$ and $\theta^2_t=-\tfrac12$. Find the risk-neutral variance drift and verify that this example need not leave the variance state space.

First, $\sqrt{1-\rho^2}=\sqrt{1-\tfrac9{25}}=\tfrac45$, so $\beta_t=\tfrac35\cdot\tfrac14+\tfrac45\cdot(-\tfrac12)=\tfrac3{20}-\tfrac25=\tfrac3{20}-\tfrac8{20}=-\tfrac14$. The drift under $\mathbb{Q}$ is therefore $$ a^{\mathbb{Q}}_t=(1-V_t)-(2V_t)(-\tfrac14)=1-V_t+\tfrac12V_t=1-\tfrac12V_t. $$

Both variance equations have a constant positive immigration term and a diffusion proportional to $V_t$. To check nonnegativity rather than assume it, consider either equation $dV_t=(1-cV_t)dt+2V_t\,dW_t$, where $W$ is the relevant Brownian motion and $c=1$ under $\mathbb{P}$, or $c=\tfrac12$ under $\mathbb{Q}$. Set $E_t=\exp((-c-2)t+2W_t)$. Ito’s formula with $f(t,w)=\exp((-c-2)t+2w)$, $f_t=(-c-2)f$, $f_w=2f$, and $f_{ww}=4f$ gives $dE_t=[(-c-2)f+\tfrac12(4f)]dt+2f\,dW_t=-cE_tdt+2E_t\,dW_t$. The candidate solution is $$ V_t=E_t\left(V_0+\int_0^t E_s^{-1}\,ds\right). $$

The integral is an ordinary time integral. Applying the product rule, the finite-variation factor has differential $E_t^{-1}dt$, and its covariation with $E_t$ is zero. Thus $dV_t=1\,dt-cV_tdt+2V_t\,dW_t$, with $V_0$ as prescribed. Since $E_t>0$, $V_0\ge0$, and $\int_0^t E_s^{-1}ds\ge0$, this solution is nonnegative. The constant market prices of risk satisfy Novikov’s condition on finite horizons, so the measure change used here is valid.

Check: for the pricing drift $c=\tfrac12$, the exponential factor is $E_t=\exp(-\tfrac52t+2B^{V,\mathbb{Q}}_t)$. Substitution into the verified equation gives drift $1-\tfrac12V_t$ and diffusion $2V_t$, matching the transformed coefficients. The inverse relation also checks: $a^{\mathbb{Q}}_t+\eta_t\beta_t=(1-\tfrac12V_t)+(2V_t)(-\tfrac14)=1-V_t=a_t$.

What Correlation Changes—and What It Does Not

Under $\mathbb{Q}$, the two independent Brownian motions $B^{1,\mathbb{Q}},B^{2,\mathbb{Q}}$ still generate the correlated pair $B^{1,\mathbb{Q}},B^{V,\mathbb{Q}}$. Their instantaneous covariance remains $\rho\,dt$. A change of measure alters drifts, not the quadratic covariation structure. In differential notation, $dB_t\,dB_t=dt$, $dB_t\,dt=0$, and $dt\,dt=0$; the two-factor cross rule here is $dB^{1,\mathbb{Q}}_t\,dB^{V,\mathbb{Q}}_t=\rho\,dt$.

Consequently, the joint diffusion term in a smooth function of $S_t,V_t$ contains a mixed second-derivative contribution proportional to $\rho$. The instantaneous covariance calculation above gives its coefficient before any pricing equation is formed. The correlation affects the joint evolution and the variance risk adjustment $\eta_t\beta_t$; it does not replace the asset’s own market price of risk $\theta^1_t$.

Important distinction: Under $\mathbb{P}$, the asset drift $\mu_t$ describes its physical expected return (when the relevant expectation exists). Under $\mathbb{Q}$, its pricing drift is $r_t-q_t$. These are different measures, not two competing descriptions under one probability law. The density process $Z_t$ is a true martingale under the stated Novikov condition. A discounted asset or a stochastic integral arising from it may instead be only a local martingale unless additional integrability conditions ensure a true martingale.

A strong solution is a process constructed on a specified probability space from specified Brownian motions and initial data. A weak solution specifies a probability space and Brownian motions on which the equation holds, without fixing them in advance. Pathwise uniqueness means two solutions driven by the same Brownian motions and initial data are equal almost surely; uniqueness in law means their distributions agree. Writing a risk-neutral coefficient transformation alone establishes none of these existence or uniqueness properties. They require separate conditions on the coefficients, such as suitable Lipschitz and linear-growth assumptions for standard existence-and-uniqueness results; square-root or boundary coefficients may require different arguments.

Common Pitfall: Reversing the Drift Shift

A frequent error is to solve $a^{\mathbb{Q}}=a-\eta\beta$ as $a=a^{\mathbb{Q}}-\eta\beta$. That keeps the wrong sign. For instance, in the variance example $\eta=2V$ and $\beta=-\tfrac14$. The incorrect inverse would give $(1-\tfrac12V)-(2V)(-\tfrac14)=1$, which is not the physical drift $1-V$. The correct inverse adds $\eta\beta$: $(1-\tfrac12V)+(2V)(-\tfrac14)=1-V$. A reliable check is to substitute the proposed physical drift into the forward formula $a^{\mathbb{Q}}=a-\eta\beta$ and recover the intended pricing drift.

Another error is to infer nonnegative variance merely from calling a state “variance.” Nonnegativity is a property of the chosen dynamics and initial condition. The example verified it directly; other coefficient specifications need their own boundary or solution argument. In particular, an additive constant diffusion can cross zero even if its drift points upward at zero.

Check Your Understanding

State assumptions when needed, and keep the physical and pricing measures distinct.

For $\rho=\tfrac35$, $\theta^1=\tfrac14$, and $\theta^2=-\tfrac12$, compute $\beta$ and identify the variance drift shift.
If $\mu=0.3$, $V=1$, and $\theta^1=0.1$, what is the risk-neutral asset drift before imposing a particular short rate or dividend yield?
For $\rho=0$, express the variance risk adjustment $\eta\beta$ in terms of $\eta,\theta^1,\theta^2$.
Why does the correlation between the asset and variance Brownian drivers remain unchanged under the stated measure change?
What additional condition is needed before a positive local-martingale density can be used as a probability density of expectation one?