Exponential Martingales¶
Exponential martingales built from affine processes serve two fundamental purposes in mathematical finance. First, they provide the mechanism for computing characteristic functions: the conditional expectation \(\mathbb{E}[e^{u^\top X_T} \mid \mathcal{F}_t]\) equals the martingale evaluated at time \(t\). Second, they serve as Radon-Nikodym derivatives for measure changes, enabling the passage from the physical measure \(\mathbb{P}\) to the risk-neutral measure \(\mathbb{Q}\) while preserving the affine structure. This section constructs exponential martingales from the Riccati solutions, proves the martingale property using the infinitesimal generator, and establishes the integrability conditions that separate true martingales from mere local martingales.
Learning Objectives
By the end of this section, you will be able to:
- Construct the exponential martingale \(M_t = \exp(\phi(T-t, u) + \psi(T-t, u)^\top X_t)\) from the Riccati solutions
- Prove that \(M_t\) is a local martingale by applying Ito's lemma and showing the drift vanishes
- State sufficient conditions for the local martingale to be a true martingale
- Explain the role of exponential martingales in characteristic function computation and measure change
- Compute explicit exponential martingales for the Vasicek and CIR models
Motivation¶
From Semigroups to Martingales¶
Recall from the generator section that the infinitesimal generator of an affine process acts on exponential functions as \(\mathcal{A}e_u(x) = [F(u) + R(u)^\top x]\,e_u(x)\). This computation suggests that if we can find functions \(\phi(\tau, u)\) and \(\psi(\tau, u)\) that "absorb" the generator's action, then the process
should have zero drift --- that is, it should be a martingale. The Riccati system provides exactly the \(\phi\) and \(\psi\) that achieve this cancellation.
This martingale perspective is more than a technical convenience. Every conditional expectation of the form \(\mathbb{E}[g(X_T) \mid \mathcal{F}_t]\) can be expressed as a martingale evaluated at time \(t\). For exponential-affine functions \(g(x) = e^{u^\top x}\), the martingale \(M_t\) does the job. And when \(M_t\) is used as a Radon-Nikodym derivative, it changes the probability measure while preserving the affine class.
Construction of the Exponential Martingale¶
The Riccati Solutions¶
Let \(X_t\) be a \(d\)-dimensional affine process with generator characterized by the functions \(F(u)\) and \(R(u)\) defined in the previous section. Define \(\phi : \mathbb{R}_+ \times \mathcal{U} \to \mathbb{C}\) and \(\psi : \mathbb{R}_+ \times \mathcal{U} \to \mathbb{C}^d\) as the solutions to the Riccati system
where \(\mathcal{U} \subseteq \mathbb{C}^d\) is the domain on which the Riccati system has a solution up to time \(T - t\).
The Martingale Process¶
Definition: Exponential Martingale of an Affine Process
For \(u \in \mathcal{U}\) and \(0 \leq t \leq T\), define
This is the exponential martingale associated with the affine process \(X\) and the parameter \(u\).
At the terminal time \(t = T\): \(M_T^{(u,T)} = \exp(\phi(0, u) + \psi(0, u)^\top X_T) = e^{u^\top X_T}\) since \(\phi(0,u) = 0\) and \(\psi(0,u) = u\).
At the initial time \(t = 0\): \(M_0^{(u,T)} = \exp(\phi(T, u) + \psi(T, u)^\top X_0)\).
The Martingale Property¶
Statement¶
Theorem: Exponential Martingale Property
If \(\phi\) and \(\psi\) solve the Riccati system and the integrability condition \(\mathbb{E}[|M_t^{(u,T)}|] < \infty\) holds for all \(t \in [0, T]\), then \(M_t^{(u,T)}\) is a martingale with respect to the filtration \((\mathcal{F}_t)_{t \geq 0}\) generated by \(X\).
Proof via Ito's Lemma¶
Define \(g(t, x) = \phi(T-t, u) + \psi(T-t, u)^\top x\) so that \(M_t = e^{g(t, X_t)}\).
Step 1: Compute \(dg\). Since \(\tau = T - t\), the time derivatives give
The spatial derivatives are \(\nabla_x g = \psi\) and \(\nabla_x^2 g = 0\) (since \(g\) is linear in \(x\)).
Step 2: Apply Ito's formula to \(M_t = e^{g(t, X_t)}\). For the diffusion case (no jumps):
Substituting \(dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\):
Step 3: The drift vanishes. Using the affine structure \(\mu(x) = b_0 + Bx\) and \(a(x) = a_0 + \sum_i \alpha_i x^{(i)}\):
Therefore the drift coefficient equals \(-F(\psi) - R(\psi)^\top x + F(\psi) + R(\psi)^\top x = 0\).
The SDE reduces to
which has zero drift, making \(M_t\) a local martingale. Under the integrability condition, it is a true martingale. \(\square\)
The Cancellation Mechanism
The proof reveals the precise mechanism: the time derivatives of \(\phi\) and \(\psi\) (given by the Riccati system) exactly cancel the drift generated by the state dynamics through the generator. This cancellation is the raison d'etre of the Riccati equations.
From Local Martingale to True Martingale¶
The Distinction¶
A local martingale need not be a true martingale. The stochastic integral \(\int_0^t M_s\,\psi^\top\sigma(X_s)\,dW_s\) is a local martingale by construction, but it is a true martingale only if it satisfies additional integrability conditions.
Sufficient Conditions¶
Proposition: Sufficient Conditions for True Martingale
The process \(M_t^{(u,T)}\) is a true martingale if any of the following hold:
- Novikov condition: \(\mathbb{E}\!\left[\exp\!\left(\frac{1}{2}\int_0^T |\psi(T-s, u)^\top\sigma(X_s)|^2\,ds\right)\right] < \infty\)
- Kazamaki condition: \(\exp\!\left(\int_0^t \psi(T-s,u)^\top\sigma(X_s)\,dW_s - \frac{1}{2}\int_0^t |\psi(T-s,u)^\top\sigma(X_s)|^2\,ds\right)\) is a uniformly integrable martingale
- Boundedness: \(u\) is purely imaginary, \(u = iv\) with \(v \in \mathbb{R}^d\), so that \(|M_t| = |\exp(\phi + i\psi^\top X_t)|\) is bounded by the exponential of a deterministic function
For the characteristic function computation (\(u = iv\)), condition (3) often applies: when \(u\) is purely imaginary and the Riccati solution \(\psi\) remains purely imaginary, the modulus \(|M_t|\) is bounded.
When the Martingale Property Fails
For real-valued \(u > 0\) (relevant for Laplace transforms and moment generating functions), the Novikov or Kazamaki conditions may fail. For example, in the CIR model with \(u > 0\) large enough, the Riccati solution \(\psi(\tau, u)\) can explode in finite time, causing \(M_t\) to be only a strict local martingale. This is related to the existence of moment explosions studied by Andersen and Piterbarg (2007).
Characteristic Function as Martingale Expectation¶
The martingale property immediately gives the characteristic function. Since \(M_t\) is a martingale:
Therefore
This is the log-affine expectation property that defines the affine class. The martingale construction provides a probabilistic proof of this property, complementing the PDE-based proof via the Feynman-Kac theorem.
Example: Vasicek Exponential Martingale¶
For the Vasicek model \(dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t\) with \(F(u) = \kappa\theta u + \frac{1}{2}\sigma^2 u^2\) and \(R(u) = -\kappa u\), the Riccati solutions are
The exponential martingale is
Verification for Vasicek
The volatility of \(M_t\) is \(M_t \cdot \psi(T-t, u) \cdot \sigma = M_t \cdot u\,e^{-\kappa(T-t)} \cdot \sigma\), which is deterministic in the sense that the random factor is only through \(M_t\) itself. The Novikov condition is satisfied for all \(u \in \mathbb{C}\) since the integrand \(|u\,e^{-\kappa(T-t)}\sigma|^2\) is deterministic and bounded.
Example: CIR Exponential Martingale¶
For the CIR model \(dX_t = \kappa(\theta - X_t)\,dt + \xi\sqrt{X_t}\,dW_t\) with \(F(u) = \kappa\theta u\) and \(R(u) = -\kappa u + \frac{1}{2}\xi^2 u^2\), the Riccati solutions involve
where \(D(\tau, u)\) depends on \(u\) and \(\tau\) through the discriminant \(\gamma = \sqrt{\kappa^2 - 2\xi^2 u}\). The exponential martingale is
The state-dependent volatility \(M_t \cdot \psi(T-t, u) \cdot \xi\sqrt{X_t}\) means the Novikov condition is not automatically satisfied. For purely imaginary \(u = iv\), the modulus \(|e^{iv X_T}| = 1\), and the martingale property holds. For real \(u > 0\), the Riccati solution exists only up to the explosion time \(\tau^* = \tau^*(u)\), limiting the time horizon over which \(M_t\) is a martingale.
Role in Measure Change¶
Exponential martingales serve as Radon-Nikodym derivatives for changing the probability measure. If \(M_t^{(u,T)}\) is a strictly positive true martingale with \(\mathbb{E}[M_T] = 1\) (which requires \(\phi(T, u) + \psi(T, u)^\top x = 0\) at \(x = X_0\) with appropriate normalization), then
defines a new probability measure \(\mathbb{Q}\). Under \(\mathbb{Q}\), the process \(X_t\) remains affine but with modified parameters. This closure property --- that the affine class is preserved under exponential-affine measure changes --- is developed in detail in the next section.
Summary¶
The exponential martingale \(M_t = \exp(\phi(T-t,u) + \psi(T-t,u)^\top X_t)\) is the probabilistic object that connects the Riccati ODE system to the characteristic function of an affine process. The martingale property holds because the Riccati equations are precisely the conditions for the drift of \(M_t\) to vanish under Ito's lemma. For purely imaginary \(u\), the martingale property is automatic; for real \(u\), additional integrability conditions (Novikov or Kazamaki) are needed to upgrade from local to true martingale. Beyond characteristic function computation, exponential martingales serve as Radon-Nikodym derivatives for measure changes that preserve the affine structure.
Further Reading¶
- Duffie, D., Filipovic, D., and Schachermayer, W. (2003). "Affine Processes and Applications in Finance." Annals of Applied Probability, 13(3), 984--1053.
- Andersen, L. and Piterbarg, V. (2007). "Moment Explosions in Stochastic Volatility Models." Finance and Stochastics, 11(1), 29--50.
- Filipovic, D. (2009). Term-Structure Models: A Graduate Course. Springer.
Exercises¶
Exercise 1. For the Vasicek model with \(\psi(\tau, u) = ue^{-\kappa\tau}\), write out \(M_t = \exp(\phi(T-t, u) + ue^{-\kappa(T-t)}X_t)\) explicitly. Compute \(dM_t\) using Ito's lemma and verify that the drift term vanishes, leaving only the stochastic integral \(dM_t = M_t \cdot ue^{-\kappa(T-t)}\sigma\,dW_t\).
Solution to Exercise 1
The Vasicek exponential martingale with \(\psi(\tau, u) = ue^{-\kappa\tau}\) and \(\phi(\tau, u) = \kappa\theta u\frac{1 - e^{-\kappa\tau}}{\kappa} + \frac{\sigma^2 u^2}{4\kappa}(1 - e^{-2\kappa\tau})\) is
Write \(g(t, x) = \phi(T-t, u) + ue^{-\kappa(T-t)}x\). Then \(M_t = e^{g(t, X_t)}\) and by Ito's formula:
Since \(g\) is linear in \(x\), \(\frac{\partial^2 g}{\partial x^2} = 0\). The spatial derivative is \(\frac{\partial g}{\partial x} = ue^{-\kappa(T-t)}\). For the time derivative, using \(\tau = T - t\):
where \(F(\psi) = \kappa\theta\psi + \frac{1}{2}\sigma^2\psi^2\) and \(R(\psi) = -\kappa\psi\). Substituting \(dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t\):
Since \(M_t = e^{g}\), the Ito formula for the exponential gives \(dM_t = M_t\,dg + \frac{1}{2}M_t\,(dg)^2\). With \(dg = [\text{drift}]\,dt + \psi\sigma\,dW_t\), the quadratic variation contributes \((dg)^2 = \psi^2\sigma^2\,dt\). The full drift of \(M_t\) is therefore \(M_t[-F(\psi) - R(\psi)X_t + \psi\kappa(\theta - X_t) + \frac{1}{2}\psi^2\sigma^2]\). Expanding: \(\psi\kappa\theta + \frac{1}{2}\sigma^2\psi^2 = F(\psi)\) and \(-\kappa\psi = R(\psi)\), so the drift is \(M_t[-F(\psi) - R(\psi)X_t + F(\psi) + R(\psi)X_t] = 0\). The remaining stochastic part is
confirming that \(M_t\) is a local martingale with zero drift.
Exercise 2. The Novikov condition requires \(\mathbb{E}[\exp(\frac{1}{2}\int_0^T |\psi(T-s,u)^\top\sigma(X_s)|^2\,ds)] < \infty\). For the Vasicek model, show that the integrand \(|\psi(T-s,u)|^2 \sigma^2 = |u|^2 e^{-2\kappa(T-s)}\sigma^2\) is deterministic and bounded, so the Novikov condition is satisfied for all \(u \in \mathbb{C}\).
Solution to Exercise 2
For the Vasicek model, \(\psi(T-s, u) = ue^{-\kappa(T-s)}\) and \(\sigma(X_s) = \sigma\) (constant). The integrand in the Novikov condition is
This is a deterministic function of \(s\) only --- it does not depend on the path of \(X_s\). Therefore
for any finite \(T\) and any \(u \in \mathbb{C}\). Since the exponent in the Novikov condition is a finite deterministic constant, the exponential of it is also finite, and the Novikov condition \(\mathbb{E}[\exp(\frac{1}{2}\int_0^T |\psi\sigma|^2\,ds)] < \infty\) is trivially satisfied. Hence \(M_t^{(u,T)}\) is a true martingale for all \(u \in \mathbb{C}\) and all \(T > 0\).
Exercise 3. For the CIR model with \(u = iv\) (purely imaginary), explain why \(|M_t^{(iv,T)}| = |\exp(\phi(T-t,iv) + \psi(T-t,iv)X_t)|\) is bounded by a deterministic constant times \(\exp(|\operatorname{Re}(\psi(T-t,iv))| \cdot X_t)\). Under what conditions on \(\psi\) does this guarantee boundedness of \(|M_t|\)?
Solution to Exercise 3
Write \(M_t = \exp(\phi(T-t, iv) + \psi(T-t, iv)X_t)\). Taking the modulus:
since \(|e^{z}| = e^{\operatorname{Re}(z)}\) for \(z \in \mathbb{C}\). Note that \(X_t \geq 0\) for the CIR process. Therefore
For \(|M_t|\) to be bounded, we need \(\operatorname{Re}(\psi(T-t, iv)) \leq 0\) for all \(t \in [0, T]\), so that the \(X_t\)-dependent term does not blow up as \(X_t\) grows. When \(u = iv\) is purely imaginary, the Riccati equation for \(\psi\) often yields \(\operatorname{Re}(\psi(\tau, iv)) \leq 0\) for all \(\tau \geq 0\) (this follows from the admissibility conditions in the CIR model). Under this condition, \(|M_t| \leq \exp(\sup_{\tau \in [0,T]} |\operatorname{Re}(\phi(\tau, iv))|)\), which is a finite deterministic constant. Boundedness of \(|M_t|\) immediately implies that \(M_t\) is a true (uniformly integrable) martingale.
Exercise 4. Prove that if \(M_t^{(u,T)}\) is a true martingale, then \(\mathbb{E}[M_T^{(u,T)}] = M_0^{(u,T)}\). Use this to derive the log-affine expectation formula \(\mathbb{E}[e^{u^\top X_T} \mid X_0 = x] = \exp(\phi(T,u) + \psi(T,u)^\top x)\) directly from the martingale property.
Solution to Exercise 4
Since \(M_t^{(u,T)}\) is a true martingale, \(\mathbb{E}[M_t^{(u,T)} \mid \mathcal{F}_s] = M_s^{(u,T)}\) for \(s \leq t\). Setting \(s = 0\) and \(t = T\):
Now \(M_T^{(u,T)} = \exp(\phi(0, u) + \psi(0, u)^\top X_T) = e^{u^\top X_T}\) since \(\phi(0,u) = 0\) and \(\psi(0,u) = u\). Also, \(M_0^{(u,T)} = \exp(\phi(T, u) + \psi(T, u)^\top X_0)\). Therefore
More generally, for the conditional expectation, the martingale property gives \(M_t = \mathbb{E}[M_T \mid \mathcal{F}_t]\), so
Since \(X_t\) is Markov, conditioning on \(\mathcal{F}_t\) is the same as conditioning on \(X_t = x\):
This is the log-affine expectation formula, derived purely from the martingale property without solving the Kolmogorov backward PDE.
Exercise 5. Consider the exponential martingale \(M_t = \exp(\phi(T-t,u) + \psi(T-t,u)X_t)\) for a CIR process with \(u = 5\) and parameters \(\kappa = 2\), \(\xi = 1\). The critical threshold is \(u^* = 4 < u = 5\), so the Riccati solution \(\psi(\tau, 5)\) explodes at some finite time \(T^*\). Explain why \(M_t\) is only a strict local martingale (not a true martingale) for \(T > T^*\), and what this implies for the moment generating function \(\mathbb{E}[e^{5X_T}]\).
Solution to Exercise 5
For the CIR process with parameters \(\kappa = 2\), \(\xi = 1\), the Riccati equation for \(\psi\) is \(\psi' = R(\psi) = -2\psi + \frac{1}{2}\psi^2\). The discriminant is \(\gamma = \sqrt{\kappa^2 - 2\xi^2 u} = \sqrt{4 - 2u}\). For \(u = 5\), \(\gamma = \sqrt{4 - 10} = \sqrt{-6} = i\sqrt{6}\), which is purely imaginary, causing the Riccati solution to involve tangent functions that blow up at a finite time \(T^*\).
At the explosion time \(T^*\), \(\psi(\tau, 5) \to \infty\), meaning \(\exp(\psi(\tau, 5)\,X_t)\) is not integrable. For \(T > T^*\), the Riccati solution does not exist on \([0, T]\), and \(M_t\) cannot be a true martingale because \(\mathbb{E}[M_T] = \mathbb{E}[e^{5X_T}]\) is infinite.
Concretely, \(M_t\) remains a local martingale (the Ito drift still cancels formally), but it fails the integrability condition: \(\mathbb{E}[|M_t|]\) becomes infinite for \(t\) close to \(T\) when \(T > T^*\). This is a strict local martingale --- a local martingale that is not a true martingale. The consequence for the moment generating function is that \(\mathbb{E}[e^{5X_T}] = +\infty\) for \(T > T^*\): the fifth exponential moment of the CIR process explodes in finite time, a phenomenon known as moment explosion.
Exercise 6. Suppose you want to use \(M_t^{(u,T)}\) as a Radon-Nikodym derivative to define a new measure \(\mathbb{Q}\). What normalization condition must hold so that \(\mathbb{E}^{\mathbb{P}}[M_T^{(u,T)}] = 1\)? Express this condition in terms of \(\phi(T, u)\), \(\psi(T, u)\), and \(X_0\), and explain why it constrains the choice of \(u\).
Solution to Exercise 6
For \(M_T^{(u,T)}\) to serve as a valid Radon-Nikodym derivative, we need \(\mathbb{E}^{\mathbb{P}}[M_T^{(u,T)}] = 1\). Since \(M_T^{(u,T)} = e^{u^\top X_T}\) and \(M_0^{(u,T)} = \exp(\phi(T, u) + \psi(T, u)^\top X_0)\), the martingale property gives
The normalization condition is therefore
This is a constraint that links \(u\), \(T\), and the initial state \(X_0\). Given the Riccati solutions \(\phi(T, u)\) and \(\psi(T, u)\) (which are determined by the process parameters), this equation implicitly defines the admissible \(u\) for each \((T, X_0)\).
In practice, one often normalizes differently: define the Radon-Nikodym derivative as \(\frac{d\mathbb{Q}}{d\mathbb{P}}\big|_{\mathcal{F}_t} = \frac{M_t^{(u,T)}}{M_0^{(u,T)}}\), which automatically satisfies \(\mathbb{E}[\frac{d\mathbb{Q}}{d\mathbb{P}}\big|_{\mathcal{F}_T}] = 1\). The constraint on \(u\) then reduces to requiring that \(M_t^{(u,T)}\) be a strictly positive true martingale, which restricts \(u\) to the set where the Riccati solution exists on \([0, T]\) and the integrability conditions hold.