---
title: 'Eraker Model: Stochastic Volatility with Jumps'
url: https://www.emergentmind.com/topics/eraker-model
type: topic
---

# Eraker Model: Stochastic Volatility with Jumps

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 [2601.17248].

## 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 [2601.17248].

## 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.

Source: https://www.emergentmind.com/topics/eraker-model