Jump-Diffusion Affine Models¶

Pure diffusion models, however sophisticated, cannot capture the sudden large moves --- earnings surprises, flash crashes, central bank announcements --- that punctuate financial time series. The affine jump-diffusion framework extends the theory by adding compound Poisson jumps with state-dependent intensity to the continuous dynamics, while preserving the exponential-affine characteristic function. The jump contribution enters the Riccati system through an additional term involving the moment generating function of the jump size distribution, and the entire pricing machinery --- Fourier inversion, bond pricing, calibration --- carries over with minimal modification. This section derives the extended Riccati equations, presents the Bates model as the canonical example, and discusses the choice of jump size distributions.

Learning Objectives

By the end of this section, you will be able to:

Write down the SDE for an affine jump-diffusion and identify the jump compensator
Derive the extended Riccati system incorporating jump contributions
Explain why affine jump intensity and exponential moment generating functions preserve the affine structure
Formulate the Bates model (Heston + Merton jumps) and compute its characteristic function
Compare log-normal, double-exponential, and self-exciting jump specifications

Motivation¶

Why Jumps Matter¶

The Black-Scholes and Heston models produce continuous sample paths. However, empirical asset returns exhibit:

Excess kurtosis far beyond what stochastic volatility alone can generate at short maturities
Short-maturity implied volatility smiles that are too steep to be explained by diffusion models
Discontinuous price gaps at market open, around earnings announcements, and during crises

Jumps provide a natural mechanism for these features. At random Poisson times, the asset price experiences an instantaneous multiplicative shock, generating heavy tails in the return distribution and steep short-maturity smiles. The key question is: can we add jumps without destroying the analytical tractability of the affine framework?

The answer is yes, provided the jump intensity and the jump size distribution satisfy specific conditions that maintain the affine structure of the generator.

The Affine Jump-Diffusion SDE¶

General Specification¶

Definition: Affine Jump-Diffusion

A \(d\)-dimensional affine jump-diffusion \(X_t\) on \(D = \mathbb{R}_+^m \times \mathbb{R}^{d-m}\) satisfies

\[ dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t + dJ_t \]

where:

\(\mu(x) = b_0 + Bx\) is the affine drift (including jump compensation)
\(\sigma(x)\sigma(x)^\top = a_0 + \sum_i \alpha_i x^{(i)}\) is the affine diffusion
\(J_t\) is a jump process with affine compensator:

\[ \nu(X_t, dz, dt) = \left(\lambda_0\,\nu_0(dz) + \sum_{i=1}^d \lambda_i\,X_t^{(i)}\,\nu_i(dz)\right)dt \]

Here \(\lambda_0 \geq 0\) and \(\lambda_i \geq 0\) are intensity coefficients, and \(\nu_0, \nu_1, \ldots, \nu_d\) are probability distributions on \(\mathbb{R}^d \setminus \{0\}\) describing jump sizes.

The total jump intensity is \(\Lambda(x) = \lambda_0 + \sum_i \lambda_i x^{(i)}\), which is affine in the state. The jump size distribution can depend on which component triggers the jump (through the index \(i\)) but not on the state value \(x\).

Compensated Jump Martingale¶

The jump process is decomposed as

\[ dJ_t = \int_{\mathbb{R}^d \setminus \{0\}} z\,\tilde{N}(dt, dz) + \int_{\mathbb{R}^d \setminus \{0\}} z\,\nu(X_{t^-}, dz)\,dt \]

where \(\tilde{N}(dt, dz) = N(dt, dz) - \nu(X_{t^-}, dz)\,dt\) is the compensated Poisson random measure. The deterministic part (the compensator) is absorbed into the drift \(\mu(x)\).

Extended Riccati System with Jumps¶

The Key Result¶

Theorem: Riccati System for Affine Jump-Diffusions

The conditional characteristic function of an affine jump-diffusion has the exponential-affine form

\[ \mathbb{E}\!\left[e^{u^\top X_T} \mid X_t = x\right] = \exp\!\bigl(\phi(\tau, u) + \psi(\tau, u)^\top x\bigr) \]

where \(\phi\) and \(\psi\) satisfy the extended Riccati system:

\[ \frac{d\psi_i}{d\tau} = R_i(\psi) = (B^\top\psi)_i + \frac{1}{2}\psi^\top\alpha_i\psi + \lambda_i\!\left(\int e^{\psi^\top z}\,\nu_i(dz) - 1\right) \]

\[ \frac{d\phi}{d\tau} = F(\psi) = b_0^\top\psi + \frac{1}{2}\psi^\top a_0\psi + \lambda_0\!\left(\int e^{\psi^\top z}\,\nu_0(dz) - 1\right) \]

with \(\psi(0, u) = u\) and \(\phi(0, u) = 0\).

The jump contribution to each equation is \(\lambda_k(\hat{\nu}_k(\psi) - 1)\), where

\[ \hat{\nu}_k(u) = \int e^{u^\top z}\,\nu_k(dz) = \mathbb{E}_{\nu_k}\!\left[e^{u^\top Z}\right] \]

is the moment generating function (MGF) of the jump size distribution \(\nu_k\).

Why the Structure Is Preserved¶

The affine structure of the Riccati system is maintained because:

The jump intensity \(\Lambda(x) = \lambda_0 + \sum_i \lambda_i x^{(i)}\) is affine in \(x\)
The jump size MGF \(\hat{\nu}_k(\psi)\) depends on \(\psi\) but not on \(x\)
The product \(\Lambda(x) \cdot (\hat{\nu}(\psi) - 1)\) is therefore affine in \(x\): the constant part \(\lambda_0(\hat{\nu}_0 - 1)\) enters \(F(\psi)\), and the state-dependent parts \(\lambda_i(\hat{\nu}_i - 1)\) enter \(R_i(\psi)\)

If either the jump intensity or the jump size distribution depended nonlinearly on \(x\), the affine structure would break.

Standard Jump Size Distributions¶

Log-Normal Jumps (Merton)¶

The jump multiplier \(Y = e^Z\) has \(Z \sim N(\mu_J, \sigma_J^2)\). The MGF is

\[ \hat{\nu}(u) = \mathbb{E}[e^{uZ}] = \exp\!\left(\mu_J u + \frac{1}{2}\sigma_J^2 u^2\right) \]

The jump contribution to the Riccati equation becomes

\[ \lambda\!\left(\exp\!\left(\mu_J\psi + \frac{1}{2}\sigma_J^2\psi^2\right) - 1\right) \]

which is a smooth function of \(\psi\) that can be evaluated in closed form.

Double-Exponential Jumps (Kou)¶

The jump size has an asymmetric double-exponential distribution:

\[ \nu(dz) = \bigl(p\,\eta_1 e^{-\eta_1 z}\mathbf{1}_{\{z > 0\}} + (1-p)\,\eta_2 e^{\eta_2 z}\mathbf{1}_{\{z < 0\}}\bigr)dz \]

with \(\eta_1 > 1\) and \(\eta_2 > 0\). The MGF is

\[ \hat{\nu}(u) = p\,\frac{\eta_1}{\eta_1 - u} + (1-p)\,\frac{\eta_2}{\eta_2 + u} \]

defined for \(-\eta_2 < \operatorname{Re}(u) < \eta_1\). The rational form of \(\hat{\nu}\) leads to simpler Riccati equations than the log-normal case and admits analytical first-passage time distributions.

Self-Exciting Jumps¶

In self-exciting (Hawkes-type) jump specifications, each jump increases the intensity of future jumps:

\[ d\Lambda_t = \kappa(\bar{\lambda} - \Lambda_t)\,dt + \delta\,dN_t \]

where \(N_t\) is the point process and \(\delta > 0\) is the jump in intensity caused by each event. This specification is affine in the extended state \((X_t, \Lambda_t)\), placing it within the affine jump-diffusion framework at the cost of an additional state variable.

The Bates Model¶

Specification¶

The Bates (1996) model combines the Heston stochastic volatility model with Merton-type log-normal jumps in the log-price:

\[ d\log S_t = \left(r - \tfrac{1}{2}V_t - \lambda\bar{k}\right)dt + \sqrt{V_t}\left(\rho\,dW_t^{(1)} + \sqrt{1-\rho^2}\,dW_t^{(2)}\right) + Z_t\,dN_t \]

\[ dV_t = \kappa(\theta - V_t)\,dt + \xi\sqrt{V_t}\,dW_t^{(1)} \]

where \(N_t\) is a Poisson process with constant intensity \(\lambda\), \(Z_t \sim N(\mu_J, \sigma_J^2)\) i.i.d., and \(\bar{k} = e^{\mu_J + \sigma_J^2/2} - 1\) is the expected jump size.

Affine Structure¶

With state \(X_t = (V_t, \log S_t)^\top\), the Bates model is an \(A_1(2)\) affine jump-diffusion:

Drift: \(b_0 = (\kappa\theta, r - \lambda\bar{k})^\top\), \(B = \begin{pmatrix} -\kappa & 0 \\ -1/2 & 0 \end{pmatrix}\)
Diffusion: \(a_0 = 0\), \(\alpha_1 = \begin{pmatrix} \xi^2 & \rho\xi \\ \rho\xi & 1 \end{pmatrix}\)
Jumps: \(\lambda_0 = \lambda\) (constant intensity), \(\nu_0(dz) = N(\mu_J, \sigma_J^2)\) on the log-price component only

Characteristic Function¶

The Riccati system for \(\psi = (\psi_1, \psi_2)^\top\):

\[ \psi_2' = 0 \implies \psi_2(\tau) = u_2 \]

\[ \psi_1' = -\kappa\psi_1 - \frac{1}{2}u_2 + \frac{1}{2}\bigl(\xi^2\psi_1^2 + 2\rho\xi\psi_1 u_2 + u_2^2\bigr) \]

\[ \phi' = \kappa\theta\psi_1 + (r - \lambda\bar{k})u_2 + \lambda\!\left(e^{\mu_J u_2 + \sigma_J^2 u_2^2/2} - 1\right) \]

The equation for \(\psi_1\) is exactly the Heston Riccati equation (independent of jumps). The jump term \(\lambda(e^{\mu_J u_2 + \sigma_J^2 u_2^2/2} - 1)\) adds to the \(\phi\) equation only, because the jump intensity is state-independent (\(\lambda_0 = \lambda\), \(\lambda_1 = 0\)).

Separation of Diffusion and Jump Effects

In the Bates model, jumps affect only the \(\phi\) equation, not the \(\psi\) equation. This means the factor loading \(\psi_1(\tau)\) (sensitivity to variance) is identical in Heston and Bates. The jump contribution modifies only the intercept \(\phi(\tau)\). This separation makes calibration convenient: one can first fit the Heston parameters from the variance smile, then adjust the jump parameters to match short-maturity behavior.

Impact on Pricing¶

Short-Maturity Behavior¶

The most visible effect of jumps is at short maturities. For \(\tau \to 0\), the diffusion contribution to option prices vanishes as \(O(\sqrt{\tau})\), while the jump contribution is \(O(\tau)\). This means:

The short-maturity implied volatility smile width is controlled by jump parameters (\(\lambda\), \(\sigma_J\))
The short-maturity skew is controlled by the mean jump size (\(\mu_J\))
Pure diffusion models (Heston without jumps) produce flat implied volatility at very short maturities

Bond Pricing with Jumps¶

For the affine short rate \(r_t = \rho_0 + \rho_1^\top X_t\) with jumps in the state, the bond price retains the exponential-affine form:

\[ P(t,T) = \exp\!\bigl(A(\tau) + B(\tau)^\top X_t\bigr) \]

The Riccati system for \((A, B)\) incorporates both the discounting terms (\(-\rho_0, -\rho_1\)) and the jump terms (\(\lambda_k(\hat{\nu}_k(B) - 1)\)). In interest rate models with jump risk (e.g., jump-diffusion short-rate models), the jump term can generate yield curve features that pure diffusion models cannot, including sudden kinks at maturities corresponding to anticipated policy events.

Summary¶

Affine jump-diffusions extend the continuous affine framework by adding compound Poisson jumps with state-dependent intensity \(\Lambda(x) = \lambda_0 + \sum_i \lambda_i x^{(i)}\). The exponential-affine characteristic function is preserved because the jump contribution \(\lambda_k(\hat{\nu}_k(\psi) - 1)\) enters the Riccati system additively, maintaining the separation between state-independent terms (\(F\)) and state-dependent terms (\(R_i\)). The Bates model combines Heston stochastic volatility with Merton log-normal jumps and illustrates the clean separation: jumps modify only the scalar function \(\phi\), leaving the variance factor loading \(\psi_1\) unchanged. Log-normal and double-exponential jump distributions are the standard choices, each with distinct analytical properties. Self-exciting jump specifications extend the framework at the cost of additional state variables. The practical value of affine jump-diffusions lies in their ability to match short-maturity implied volatility smiles that pure diffusion models cannot capture.

Exercises¶

Exercise 1. For the Merton jump-diffusion model \(d\log S_t = (r - \frac{1}{2}\sigma^2 - \lambda\bar{k})\,dt + \sigma\,dW_t + J_t\,dN_t\) where \(J_t \sim N(\mu_J, \sigma_J^2)\) and \(N_t\) is Poisson with intensity \(\lambda\), identify the affine parameters and compute \(F(u)\) and \(R(u)\). Verify that \(R(u) = 0\) (no state-dependent jumps).

Solution to Exercise 1

The Merton jump-diffusion has a single state variable \(X_t = \log S_t\) on \(\mathbb{R}\). The dynamics under the risk-neutral measure are

\[ d(\log S_t) = \left(r - \tfrac{1}{2}\sigma^2 - \lambda\bar{k}\right)dt + \sigma\,dW_t + J_t\,dN_t \]

where \(J_t \sim N(\mu_J, \sigma_J^2)\), \(N_t\) is Poisson with constant intensity \(\lambda\), and \(\bar{k} = e^{\mu_J + \sigma_J^2/2} - 1\).

Affine parameters. The state is one-dimensional (\(d = 1\), \(m = 0\), so \(X_t \in \mathbb{R}\)):

Drift: \(b_0 = r - \frac{1}{2}\sigma^2 - \lambda\bar{k}\), \(B = 0\) (no state-dependent drift)
Diffusion: \(a_0 = \sigma^2\), \(\alpha_1 = 0\) (constant diffusion, no CIR-type component)
Jumps: \(\lambda_0 = \lambda\) (constant intensity), \(\lambda_1 = 0\) (no state-dependent intensity), \(\nu_0(dz) = N(\mu_J, \sigma_J^2)\)

Computing \(F(u)\) and \(R(u)\). From the Riccati system:

\[ R(u) = B^\top u + \frac{1}{2}\alpha_1 u^2 + \lambda_1\!\left(\hat{\nu}_1(u) - 1\right) = 0 + 0 + 0 = 0 \]

This confirms \(R(u) = 0\) --- there is no state-dependent component.

\[ F(u) = b_0 u + \frac{1}{2}a_0 u^2 + \lambda_0\!\left(\hat{\nu}_0(u) - 1\right) \]

\[ = \left(r - \tfrac{1}{2}\sigma^2 - \lambda\bar{k}\right)u + \frac{1}{2}\sigma^2 u^2 + \lambda\!\left(e^{\mu_J u + \sigma_J^2 u^2/2} - 1\right) \]

Since \(R(u) = 0\), we have \(\psi(\tau, u) = u\) for all \(\tau\), and \(\phi(\tau, u) = F(u)\cdot\tau\). The characteristic function is therefore

\[ \mathbb{E}\!\left[e^{u\log S_T}\right] = \exp\!\left(F(u)\,\tau + u\,\log S_t\right) \]

which is the well-known Merton formula.

Exercise 2. For the Bates model (Heston + Merton jumps in the log-price), write the extended Riccati system. Show that the jump contribution adds \(\lambda(\exp(\mu_J u + \frac{1}{2}\sigma_J^2 u^2) - 1)\) to the \(\phi\)-equation (or the appropriate component). How does this modify the characteristic function compared to pure Heston?

Solution to Exercise 2

The Bates model has state \(X_t = (V_t, \log S_t)^\top\) with \(\psi = (\psi_1, \psi_2)^\top\) and initial conditions \(\psi_1(0) = u_1\), \(\psi_2(0) = u_2\). For option pricing, typically \(u_1 = 0\) and \(u_2 = iu\) for \(u \in \mathbb{R}\).

The extended Riccati system. Since \(\log S_t\) is a Gaussian-type component (\(m = 1\), only \(V_t\) is CIR-type):

\[ \frac{d\psi_2}{d\tau} = R_2(\psi) = 0 \implies \psi_2(\tau) = u_2 \]

\[ \frac{d\psi_1}{d\tau} = R_1(\psi) = -\kappa\psi_1 - \frac{1}{2}u_2 + \frac{1}{2}\!\left(\xi^2\psi_1^2 + 2\rho\xi u_2\psi_1 + u_2^2\right) \]

\[ \frac{d\phi}{d\tau} = F(\psi) = \kappa\theta\psi_1 + (r - \lambda\bar{k})u_2 + \lambda\!\left(e^{\mu_J u_2 + \sigma_J^2 u_2^2/2} - 1\right) \]

Comparison with pure Heston. The equation for \(\psi_1\) is identical to the Heston Riccati equation --- it contains no jump terms because \(\lambda_1 = 0\) (the jump intensity is constant, not state-dependent). The jump contribution

\[ \lambda\!\left(\exp\!\left(\mu_J u_2 + \tfrac{1}{2}\sigma_J^2 u_2^2\right) - 1\right) \]

appears only in the \(\phi\)-equation. This means:

The factor loading \(\psi_1(\tau, u)\) on the variance is unchanged from Heston.
The scalar intercept \(\phi(\tau, u)\) receives an additive correction linear in \(\tau\) from the jump term.
The characteristic function of the Bates model is the Heston characteristic function multiplied by \(\exp(\lambda\tau(e^{\mu_J u_2 + \sigma_J^2 u_2^2/2} - 1 - \bar{k}u_2))\), which is precisely the Merton jump correction factor.

Exercise 3. Consider a jump-diffusion on \(\mathbb{R}_+\) with exponentially distributed jumps: \(m_1(dz) = \lambda\eta e^{-\eta z}\,dz\) for \(z > 0\). Compute the jump contribution to \(R(u)\) as \(\lambda(\frac{\eta}{\eta - u} - 1)\) and determine the domain of \(u\) for which this expression is finite. What happens at \(u = \eta\)?

Solution to Exercise 3

The jump size distribution is exponential on \((0, \infty)\) with density \(\nu_1(dz) = \eta e^{-\eta z}\,dz\) for \(z > 0\). The jump contribution to \(R(u)\) involves the MGF:

\[ \hat{\nu}_1(u) = \int_0^\infty e^{uz}\,\eta e^{-\eta z}\,dz = \eta\int_0^\infty e^{-(\ \eta - u)z}\,dz = \frac{\eta}{\eta - u} \]

This integral converges if and only if \(\operatorname{Re}(\eta - u) > 0\), i.e., \(\operatorname{Re}(u) < \eta\). The jump contribution to \(R(u)\) is therefore

\[ \lambda\!\left(\frac{\eta}{\eta - u} - 1\right) = \lambda\,\frac{u}{\eta - u} \]

Domain. The expression is well-defined for \(\operatorname{Re}(u) < \eta\). At \(u = \eta\), the MGF \(\hat{\nu}_1(u) = \eta/(\eta - u) \to \infty\), which means the exponential moment \(\mathbb{E}[e^{\eta Z}]\) is infinite. This corresponds to the fact that the exponential distribution has a moment generating function that diverges at its rate parameter.

For the Riccati ODE to have a well-defined solution, the trajectory \(\psi(\tau)\) must remain in the domain \(\operatorname{Re}(\psi) < \eta\). If the ODE drives \(\psi\) toward \(\eta\), the solution blows up in finite time --- this is the moment explosion phenomenon, which places constraints on the maturity \(T\) and the transform variable \(u\) for which the characteristic function is finite.

Exercise 4. Explain why log-normal jump sizes (Merton model) produce heavier tails than double-exponential jump sizes (Kou model) for extreme returns. Which distribution generates a steeper short-maturity implied volatility smile? Justify your answer in terms of the moment generating functions of the two distributions.

Solution to Exercise 4

Log-normal jumps (Merton). The jump size \(Z \sim N(\mu_J, \sigma_J^2)\) has a Gaussian distribution with tails decaying as \(\exp(-z^2/(2\sigma_J^2))\). However, the return impact is \(e^Z - 1\), so extremely large returns correspond to large \(|Z|\). The log-normal distribution of \(e^Z\) has all polynomial moments finite but super-exponential tails in the return space: \(\mathbb{P}(e^Z > x) \sim \exp(-(\ln x)^2/(2\sigma_J^2))\), which decays slowly (sub-exponentially) for large \(x\).

Double-exponential jumps (Kou). The jump size has density \(p\eta_1 e^{-\eta_1 z}\mathbf{1}_{z > 0} + (1-p)\eta_2 e^{\eta_2 z}\mathbf{1}_{z < 0}\), so the tails decay as \(e^{-\eta_1 z}\) for positive jumps and \(e^{\eta_2 z}\) for negative jumps. These are exponential tails, which decay faster than the log-normal tails for extreme values.

Since log-normal jump sizes produce heavier tails than double-exponential jump sizes, the Merton model generates more probability mass in the extreme tails of the return distribution. For short-maturity options, this means:

Merton (log-normal) produces a steeper short-maturity implied volatility smile, especially in the deep out-of-the-money wings, because the heavier tails assign more probability to extreme moves.
Kou (double-exponential) produces a more moderate smile with wings that flatten more quickly.

This can also be seen from the MGFs. The log-normal MGF \(\hat{\nu}(u) = e^{\mu_J u + \sigma_J^2 u^2/2}\) grows as \(e^{u^2}\) for large \(|u|\) (superexponential in \(u\)), while the double-exponential MGF \(\hat{\nu}(u) = p\eta_1/(\eta_1 - u) + (1-p)\eta_2/(\eta_2 + u)\) is rational in \(u\) (polynomial growth). The faster growth of the log-normal MGF reflects heavier tails in the jump size distribution.

Exercise 5. For a self-exciting (Hawkes-type) jump process where the jump intensity \(\lambda(X_t) = \lambda_0 + \lambda_1 X_t\) depends affinely on the state, verify that the jump contribution to \(R_j(u)\) is \(\lambda_1 \int(e^{uz} - 1)\nu(dz)\). Why does affine intensity preserve the affine structure, while a nonlinear intensity (e.g., \(\lambda(x) = \lambda_0 + \lambda_2 x^2\)) would break it?

Solution to Exercise 5

The affine jump-diffusion framework requires that the jump compensator be affine in the state:

\[ \nu(X_t, dz, dt) = \left(\lambda_0\,\nu_0(dz) + \sum_i \lambda_i X_t^{(i)}\,\nu_i(dz)\right)dt \]

With \(\lambda(X_t) = \lambda_0 + \lambda_1 X_t\) (affine intensity), the jump contribution to the generator acting on \(e^{u^\top x}\) is

\[ \int \left(e^{u^\top(x+z)} - e^{u^\top x}\right)\lambda(x)\,\nu(dz) = e^{u^\top x}\,\lambda(x)\!\left(\int e^{u^\top z}\nu(dz) - 1\right) \]

\[ = e^{u^\top x}\!\left(\lambda_0 + \lambda_1 x\right)\!\left(\hat{\nu}(u) - 1\right) \]

This is \(e^{u^\top x}\) times a function that is affine in \(x\): the constant part \(\lambda_0(\hat{\nu}(u) - 1)\) enters \(F(u)\), and the linear part \(\lambda_1(\hat{\nu}(u) - 1)\) enters \(R_j(u)\).

Specifically, \(R_j(u)\) receives the additive jump contribution

\[ \lambda_1\!\left(\int e^{u^\top z}\nu(dz) - 1\right) = \lambda_1\!\left(\hat{\nu}(u) - 1\right) \]

which depends on \(u\) (through \(\hat{\nu}\)) but not on \(x\). This is exactly the form required for the Riccati system to close.

Why nonlinear intensity breaks the structure. If \(\lambda(x) = \lambda_0 + \lambda_2 x^2\), then the generator produces \(e^{u^\top x} \cdot (\lambda_0 + \lambda_2 x^2)(\hat{\nu}(u) - 1)\). The \(x^2\) term means the coefficient of \(e^{u^\top x}\) is quadratic, not affine, in \(x\). This cannot be absorbed into the Riccati system \(\frac{d\psi}{d\tau} = R(\psi)\) because \(R(\psi)\) must be independent of \(x\). The exponential-affine ansatz \(\exp(\phi + \psi^\top x)\) fails to satisfy the Kolmogorov backward equation, and no finite-dimensional ODE system determines the characteristic function.

Exercise 6. Compare the computational cost of evaluating the characteristic function for (i) pure Heston, (ii) Bates (Heston + Merton jumps), and (iii) Heston with double-exponential (Kou) jumps. In each case, is the Riccati system solvable in closed form, or is numerical ODE integration required?

Solution to Exercise 6

(i) Pure Heston. The Riccati system consists of a scalar ODE for \(\psi_1(\tau)\) (quadratic in \(\psi_1\)) and a quadrature for \(\phi(\tau)\). The ODE for \(\psi_1\) has a closed-form solution in terms of exponentials:

\[ \psi_1(\tau) = \frac{(iu - u^2)(1 - e^{-d\tau})}{2d - (d + \gamma)(1 - e^{-d\tau})} \]

where \(d = \sqrt{\gamma^2 + \xi^2(iu - u^2)}\) and \(\gamma = \kappa - \rho\xi iu\). The function \(\phi\) is also available in closed form. Cost: a few elementary function evaluations (exponentials, square root). Extremely fast.

(ii) Bates (Heston + Merton jumps). The ODE for \(\psi_1\) is identical to pure Heston --- jumps do not affect it because the jump intensity is constant (\(\lambda_1 = 0\)). The jump contribution adds \(\lambda(e^{\mu_J u_2 + \sigma_J^2 u_2^2/2} - 1)\) to the \(\phi\) equation, which is still integrated in closed form (it adds a term linear in \(\tau\)). Cost: essentially the same as Heston, plus one exponential evaluation for the jump MGF. Negligible additional cost.

(iii) Heston + Kou (double-exponential) jumps. If the jumps are only in the log-price with constant intensity (as in the Bates-Kou variant), the situation is similar to case (ii): the \(\psi_1\) ODE is unchanged, and the jump contribution to \(\phi\) is \(\lambda(p\eta_1/(\eta_1 - u_2) + (1-p)\eta_2/(\eta_2 + u_2) - 1)\), which is a rational function of \(u_2\) and adds a term linear in \(\tau\). This is also closed form. Cost: comparable to Bates.

Summary. All three models admit closed-form characteristic functions when the jumps are in the log-price with constant intensity, because the jump term modifies only \(\phi\) (not \(\psi_1\)) and the MGF of the jump distribution can be evaluated analytically. Numerical ODE integration would be required only if (a) the jump intensity were state-dependent (\(\lambda_1 \neq 0\)), making the \(\psi_1\) ODE non-standard, or (b) the jump size distribution had no closed-form MGF. In terms of computational cost, all three are fast (sub-millisecond per characteristic function evaluation), with Heston being marginally the cheapest.