Eraker Model: Stochastic Volatility with Jumps

• Eraker model is a jump–diffusion framework with stochastic volatility that incorporates both idiosyncratic and correlated jumps to capture short-maturity option pricing dynamics.
• It employs distinct Poisson jump processes for asset prices and variances, enabling accurate calibration of European and VIX derivative contracts.
• The model’s closed-form asymptotic approximations allow efficient numerical validation and its application in high-frequency trading environments.

The Eraker model, often denoted “SVCJ” (Stochastic Volatility with Correlated Jumps), is a jump–diffusion financial model that specifies risk-neutral dynamics for asset prices and their variances. It is characterized by both idiosyncratic and correlated Poisson jumps superimposed on a continuous local–stochastic volatility background. The Eraker model provides a tractable framework for pricing derivatives such as European options and volatility index (VIX) contracts, particularly in regimes with short maturities where jump contributions dominate the leading-order behavior (Guo et al., 24 Jan 2026).

1. Risk-Neutral Dynamics and Jump Structure

In the formulation of Guo et al. (2026), under the risk-neutral measure $\mathcal{Q}$, the Eraker model defines asset price $S_t$ and variance $V_t$ as follows:

• Continuous components:

$$\frac{dS_t}{S_t} = \sqrt{V_t}\,dW_t + (r - q - \lambda^S\mu^S - \lambda^C\mu^{C,S})\,dt$$

$\frac{dV_t}{V_t} = \mu_V\,dt + \sigma_V\,dZ_t$

$W_t$ and $Z_t$ are independent Brownian motions.

• Jump constituents:
  • Idiosyncratic jumps in $S_t$: Arriving via $N_t^S\sim$ Poisson$(\lambda^S)$, jump sizes $Y_i^S\sim \mathcal{N}(\alpha_S, \sigma_S^2)$.
  • Idiosyncratic jumps in $V_t$: $N_t^V\sim$ Poisson$(\lambda^V)$, jump sizes $Y_i^V\sim$ Exp$(\eta_V)$, $\eta_V>1$, producing upward-only jumps.
  • Common (co-jumps): $N_t^C\sim$ Poisson$(\lambda^C)$; variance jump sizes $Y_1^{C,V}\sim$ Exp$(\eta_{C,V})$, with conditional price jump $Y_1^{C,S}=\mu^S+\rho_J Y_1^{C,V}+\sigma_{C,S} Z$, $Z\sim \mathcal{N}(0,1)$.

The co-jump joint density is: $$p^C(x,y) = \eta_{C,V} e^{-\eta_{C,V}y} (2\pi \sigma_{C,S})^{-1/2} e^{-\frac{(x-\mu^S-\rho_J y)^2}{2\sigma_{C,S}^2}}$$

Key compensators for each jump type:

• $\mu^S=E[e^{Y_1^S}-1]=e^{\alpha_S+\sigma_S^2/2}-1$, $m^S=E[Y_1^S]=\alpha_S$
• $\mu^V=1/(\eta_V-1)$
• $$\mu^{C,S}=E[e^{Y_1^{C,S}}-1]=\frac{\eta_{C,V}}{\eta_{C,V}-\rho_J}e^{\mu^S+\sigma_{C,S}^2/2}-1$$, $m^{C,S}=\mu^S+\rho_J/\eta_{C,V}$

The offset parameter for jumps is

$$\kappa = 2[\lambda^S(\mu^S-m^S) + \lambda^C(\mu^{C,S}-m^{C,S})] \geq 0$$

2. Short-Maturity Asymptotic Formulas for Option Prices

In the regime $T\to 0$ and local volatility $\eta(\cdot)\equiv 1$, short-maturity pricing for VIX and European options can be expressed via closed-form asymptotics, distinguishing between out-of-the-money (OTM) and at-the-money (ATM) cases.

VIX Options

• OTM Calls: If $V_0+\kappa<K^2$, as $T\to 0$:

$C_V(K,T) \sim T \cdot a_{V,C}(K)$

with

$$a_{V,C}(K) = \lambda^C \int_0^\infty [\sqrt{V_0 e^y + \kappa} - K]^+ \eta_{C,V} e^{-\eta_{C,V}y} dy + \lambda^V \int_0^\infty [\sqrt{V_0 e^x + \kappa} - K]^+ \eta_V e^{-\eta_V x} dx$$

Closed form is obtained via $y_0 = \log\left(\frac{K^2-\kappa}{V_0}\right)$ and

$$I_1(a,b,\eta)=\frac{2\sqrt{b}}{2\eta-1} {}_2F_1(-\tfrac12, \eta-\tfrac12; \eta+\tfrac12; -\frac{a}{b})$$

yielding

$$\int_0^\infty [\sqrt{b e^y + a} - K]^+ \eta e^{-\eta y} dy = e^{-\eta y_0} [\eta I_1(a, b e^{y_0}, \eta) - K]$$

All jumps in $V$ are upward; the OTM put limit vanishes: $P_V(K,T) = o(T)$.

• ATM Calls and Puts: If $V_0+\kappa=K^2$,

$C_V(K,T), P_V(K,T) = O(\sqrt{T})$

and

$$\lim_{T \to 0} \frac{C_V}{\sqrt{T}} = \lim_{T \to 0} \frac{P_V}{\sqrt{T}} = \frac{1}{\sqrt{2\pi}} \frac{[\eta^2 V_0]^{1/2}}{\sqrt{\eta^2 V_0 + \kappa}} \sqrt{\left(\tfrac12 \eta \sigma_V \sqrt{V_0} + \eta' \eta S_0 V_0 \rho\right)^2 + \left(\eta' \eta S_0 V_0 \sqrt{1-\rho^2}\right)^2}$$

European Options

• OTM Calls: If $S_0<K$,

$C_E(K,T) = T \cdot a_{E,C}(K) + o(T)$

with

$$a_{E,C}(K) = \lambda^C \int_0^\infty c_{BS}(K, F(y), \sigma_{C,S}) \eta_{C,V} e^{-\eta_{C,V}y} dy + \lambda^S \left\{ S_0 e^{\alpha_S + \frac12 \sigma_S^2} \Phi\left(\frac{-k + \alpha_S + \sigma_S^2}{\sigma_S}\right) - K \Phi\left(\frac{-k + \alpha_S}{\sigma_S}\right)\right\}$$

where $k = \log(K/S_0)$ and $$c_{BS}(K,F,v)=F N(-\log(K/F)/v + v/2) - K N(-\log(K/F)/v - v/2)$$.

Put prices (for $S_0>K$) analogously employ $p_{BS}$.

• ATM Calls and Puts: If $S_0=K$,

$C_E(K, T), P_E(K, T) = O(\sqrt{T})$

and

$$\lim_{T\to 0} \frac{C_E}{\sqrt{T}} = \lim_{T\to 0} \frac{P_E}{\sqrt{T}} = \eta(S_0)\frac{\sqrt{V_0}}{\sqrt{2\pi}}$$

3. Approximations and Asymptotic Regimes

The leading-order short-maturity asymptotics rely on several key approximations:

• VIX averaging: As $\tau\to 0$, $VIX_t \approx \sqrt{V_t+\kappa}$, replacing the standard 30-day averaging with its instantaneous limit.
• OTM option regime: For $T\to 0$, one-jump events (with probability $\sim T$) produce the dominant price contribution, with $\geq 2$ jumps contributing $O(T^2)$ and neglected.
• ATM option regime: No-jump diffusion dominates (order $\sqrt{T}$); jumps (order $T$) are negligible.
• Local-volatility and variance boundedness/Lipschitz: This ensures validity of expansions and uniform control on remainder terms.
• Large deviations: Used to show diffusion-only OTM terms decay exponentially for small $T$.

4. Numerical Validation and Calibration

Empirical validation of the Eraker model’s leading-order asymptotics is performed through Monte Carlo (MC) simulation, using parameters from Lian & Zhu (2013): $\lambda^C=0.47$, $\eta_{C,V}=20$, $\mu^S=-0.0869$, $\rho_J=-0.38$, $\sigma_{C,S}=0.1$, $V_0=0.0076$, with $\lambda^S=\lambda^V=0$ (pure common-jump setting) and $\sigma_V=0.01$.

Observations:

• OTM European calls/puts ($T=0.1$): Rescaled MC prices $1000\cdot C_E/(\lambda^C T)$ match asymptotic $a_{E,C}/\lambda^C$ away from ATM; discrepancies at ATM indicate $\sqrt{T}$ diffusive correction omitted by $O(T)$ term. For $T=0.01$, discrepancies shrink, confirming convergence.
• OTM VIX calls ($T=0.1$): $1000\cdot C_V/(\lambda^C T)$ closely tracks $a_{V,C}/\lambda^C$; OTM VIX puts MC $\approx 0$ as predicted.
• A plausible implication is that for realistic Eraker-type jumps and maturities under several weeks, the analytic $T\to 0$ formulas capture the slope of OTM European/VIX option prices with sufficient accuracy for calibration purposes (Guo et al., 24 Jan 2026).

5. Relation to Other Jump Models and Applications

The Eraker model is analyzed alongside related jump–diffusion structures, including Kou-type and folded normal models, illustrating the flexibility of compound Poisson jumps with tractable short-maturity expansions. Its regime is most relevant for pricing equity–volatility derivatives with rapid expiry, and for calibration in high-frequency financial environments where jump events rather than Brownian diffusions determine primary risk and pricing trajectories. The explicit separation of common and idiosyncratic jumps enables detailed modeling of event-driven co-movement between assets and their variances.

The consistency of analytic short-maturity formulas and simulation validates its utility for both pricing and calibration of short-maturity option and VIX markets.