Equivalent Local Martingale Measures in Diffusion Markets

From a Density Process to a Drift Change

In a diffusion market, an equivalent change of probability measure changes the drift of Brownian motion while preserving its diffusion scale. This makes it possible to ask whether a measure can be chosen so that every discounted asset price has zero drift. The resulting condition is a system of equations: the original discounted drifts must be attainable by applying the volatility matrix to a market-price-of-risk process.

Work on a finite horizon \([0,T]\) on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\), with the usual conditions and trivial \(\mathcal{F}_0\). Let \(\mathbf{B}_t\) be a \(d\)-dimensional Brownian motion, and take \((\mathcal{F}_t)\) to be its completed, right-continuous natural filtration. There are \(m\) discounted asset prices, collected in a vector \(\widetilde{\mathbf{S}}_t\), with dynamics under \(\mathbb{P}\)

Here \(\mathbf{b}_t\in\mathbb{R}^m\) is the drift vector and \(\boldsymbol{\sigma}_t\in\mathbb{R}^{m\times d}\) is the volatility matrix. Assume the coefficients are progressively measurable and, almost surely, \(\int_0^T|\mathbf{b}_t|\,dt<\infty\) and \(\int_0^T\|\boldsymbol{\sigma}_t\|^2\,dt<\infty\). A stochastic integral against \(\mathbf{B}\) is componentwise Ito integration: its progressively measurable integrands are in \(\mathcal{L}^2\) when \(\mathbb{E}\int_0^T\|H_t\|^2dt<\infty\); locally square-integrable integrands are integrated by localization.

To state the drift condition, use the following standard form of Girsanov’s theorem. If \(\boldsymbol{\theta}\) is progressively measurable, locally square-integrable, and \(Z_t=\mathcal{E}(-\int_0^\cdot\boldsymbol{\theta}_s\cdot d\mathbf{B}_s)_t\) is a true martingale with \(Z_T>0\) almost surely, then \(d\mathbb{Q}=Z_Td\mathbb{P}\) defines an equivalent measure and \(\mathbf{B}^{\mathbb{Q}}_t=\mathbf{B}_t+\int_0^t\boldsymbol{\theta}_sds\) is Brownian under \(\mathbb{Q}\). This is the standard Girsanov theorem; its proof is not given here. A useful sufficient condition for \(Z\) to be a true martingale is Novikov’s condition, \(\mathbb{E}[\exp(\tfrac12\int_0^T|\boldsymbol{\theta}_t|^2dt)]<\infty\). Novikov’s condition is sufficient, not necessary.

Since \(d\mathbf{B}_t=d\mathbf{B}^{\mathbb{Q}}_t-\boldsymbol{\theta}_t dt\), substitution into the asset dynamics changes the drift from \(\mathbf{b}_t\) to \(\mathbf{b}_t-\boldsymbol{\sigma}_t\boldsymbol{\theta}_t\). In particular, \(\boldsymbol{\theta}\) is measured in Brownian-risk coordinates; it is not itself the asset drift vector.

The Market-Price-of-Risk Characterization

Proof. First suppose such a process \(\boldsymbol{\theta}\) exists. By condition 2, \(Z_T\) has expectation one and is strictly positive almost surely, so \(d\mathbb{Q}=Z_Td\mathbb{P}\) defines a probability measure equivalent to \(\mathbb{P}\). Girsanov’s theorem gives the Brownian motion \(\mathbf{B}^{\mathbb{Q}}\) under \(\mathbb{Q}\). Substituting \(d\mathbf{B}_t=d\mathbf{B}^{\mathbb{Q}}_t-\boldsymbol{\theta}_t dt\) gives

The drift vanishes by condition 1, and the remaining stochastic integral is a continuous local martingale under \(\mathbb{Q}\). It need not be a true martingale: that requires additional integrability, such as suitable bounds on its second moments.

Conversely, suppose \(\mathbb{Q}\) is an equivalent local martingale measure and let \(Z_t=\mathbb{E}[d\mathbb{Q}/d\mathbb{P}\mid\mathcal{F}_t]\). The Brownian martingale representation theorem, a standard result for the completed natural Brownian filtration, gives \(dZ_t=\boldsymbol{\eta}_t\cdot d\mathbf{B}_t\) for a progressively measurable, locally square-integrable \(\boldsymbol{\eta}\). This representation theorem is invoked here as a standard result. Equivalence makes \(Z_T>0\) almost surely, and the conditional expectation of this strictly positive variable is positive at every time; continuity of \(Z\) then gives a version positive throughout \([0,T]\). Set \(\boldsymbol{\theta}_t=-\boldsymbol{\eta}_t/Z_t\), defined by localization. The stochastic differential for \(Z\), or equivalently the Ito formula for \(\log Z\), yields \(Z=\mathcal{E}(-\int_0^\cdot\boldsymbol{\theta}_s\cdot d\mathbf{B}_s)\). The density process is a true martingale by its conditional-expectation definition.

Girsanov’s theorem now gives \(d\mathbf{B}=d\mathbf{B}^{\mathbb{Q}}-\boldsymbol{\theta}dt\), so the \(\mathbb{Q}\)-dynamics of the assets have finite-variation part \((\mathbf{b}-\boldsymbol{\sigma}\boldsymbol{\theta})dt\). A continuous finite-variation process that is also a local martingale is constant. To see why, stop it when its absolute value and total variation are bounded; Ito’s formula for its square shows that the expected square of the stopped process, starting from zero, is zero, since its quadratic variation is zero. Removing the stops gives the assertion. Uniqueness of the continuous local-martingale plus finite-variation decomposition therefore forces \((\mathbf{b}-\boldsymbol{\sigma}\boldsymbol{\theta})dt=0\), which is condition 1. Since \(\mathbb{P}\) and \(\mathbb{Q}\) are equivalent, the condition holds \(dt\times\mathbb{P}\)-almost everywhere as stated. Both directions are proved. \(\square\)

Worked Applications

What the Condition Does—and Does Not—Say

The equation \(\boldsymbol{\sigma}_t\boldsymbol{\theta}_t=\mathbf{b}_t\) is a pointwise linear-algebra requirement, almost everywhere in time and probability. If the drift vector lies outside the column space of the volatility matrix, no change of Brownian drift can remove it. If solutions exist but the volatility matrix has a nontrivial null space, adding a suitable null-space component may produce other candidate processes. Each candidate must still generate a true, strictly positive density process; solving the linear equations alone does not establish equivalence.

The drift is the coefficient of \(dt\), while the diffusion coefficient multiplies Brownian increments. A change of measure of this form adjusts the former through the latter; it does not set the diffusion coefficient to zero. Also, the resulting discounted prices are guaranteed to be local martingales. A local martingale need not have the integrability needed for the conditional-expectation identity of a true martingale, so one must not infer that all discounted prices have constant expectations without checking further conditions.

Common pitfall—using the wrong drift or sign. In the first example, using \(\mu\) rather than the discounted drift \(\mu-r\) gives the incorrect value \(\theta=\mu/\sigma=(3/10)/(2/5)=3/4\). With the convention \(B^{\mathbb{Q}}=B+\theta t\), the discounted drift would then be \(1/5-(2/5)(3/4)=1/5-3/10=-1/10\), not zero. The correct value is \((\mu-r)/\sigma=(1/5)/(2/5)=1/2\), for which \(1/5-(2/5)(1/2)=0\). Keeping both the discounting and the sign convention explicit prevents this error.