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VIX and European options with jumps in the short-maturity regime

Desen Guo, Dan Pirjol, Xiaoyu Wang, Lingjiong Zhu

TL;DR

This work derives leading-order short-maturity asymptotics for VIX and European option prices in local-stochastic volatility models with compound Poisson jumps, covering OTM and ATM regimes. By approximating $\mathrm{VIX}_T^2$ with $\eta^2(S_T)V_T+\kappa$ in the small-$\tau$ limit, the authors obtain closed-form limit expressions that isolate the jump components (idiosyncratic S and V, and common jumps) as the dominant drivers in the OTM regime, while diffusion dominates the ATM regime for options. They introduce and analyze three jump models—Eraker, Kou-type, and folded normal—providing explicit asymptotic predictions for European and VIX options and illustrating strong numerical agreement with MC pricing for small maturities. The results facilitate calibration and pricing of VIX derivatives in non-affine, jump-diffusion settings and demonstrate the practical relevance of jump structure in short-dated volatility products.

Abstract

We present a study of the short-maturity asymptotics for VIX and European option prices in local-stochastic volatility models with compound Poisson jumps. Both out-of-the-money (OTM) and at-the-money (ATM) asymptotics are considered. The leading-order asymptotics are obtained in closed-form. We apply our results to three examples: the Eraker model, a Kou-type model, and a folded normal model. Numerical illustrations are provided for these three examples that show the accuracy of predictions based on the asymptotic results.

VIX and European options with jumps in the short-maturity regime

TL;DR

This work derives leading-order short-maturity asymptotics for VIX and European option prices in local-stochastic volatility models with compound Poisson jumps, covering OTM and ATM regimes. By approximating with in the small- limit, the authors obtain closed-form limit expressions that isolate the jump components (idiosyncratic S and V, and common jumps) as the dominant drivers in the OTM regime, while diffusion dominates the ATM regime for options. They introduce and analyze three jump models—Eraker, Kou-type, and folded normal—providing explicit asymptotic predictions for European and VIX options and illustrating strong numerical agreement with MC pricing for small maturities. The results facilitate calibration and pricing of VIX derivatives in non-affine, jump-diffusion settings and demonstrate the practical relevance of jump structure in short-dated volatility products.

Abstract

We present a study of the short-maturity asymptotics for VIX and European option prices in local-stochastic volatility models with compound Poisson jumps. Both out-of-the-money (OTM) and at-the-money (ATM) asymptotics are considered. The leading-order asymptotics are obtained in closed-form. We apply our results to three examples: the Eraker model, a Kou-type model, and a folded normal model. Numerical illustrations are provided for these three examples that show the accuracy of predictions based on the asymptotic results.
Paper Structure (33 sections, 15 theorems, 208 equations, 4 figures, 10 tables)

This paper contains 33 sections, 15 theorems, 208 equations, 4 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Literature review
  3. Model setup
  4. Main Results
  5. VIX options
  6. European options
  7. Modeling the Jumps
  8. Eraker model
  9. Asymptotic predictions for European options
  10. Asymptotic predictions for VIX options
  11. Kou-type model
  12. Asymptotic predictions for European options
  13. Asymptotic predictions for VIX options
  14. Folded normal model
  15. Asymptotic predictions for European options
  16. ...and 18 more sections

Key Result

Proposition 2.1

If Assumption assump:bounded and assump:lip hold, and Then we have and moreover, where and

Figures (4)

  • Figure 4.1: Plots of the asymptotic results for European calls and puts with maturity $T=0.1$ in the Eraker model. The black curves show the asymptotic prediction for $1000 \frac{a_{E,C}(K)}{\lambda^{C}}$ and $1000 \frac{a_{E,P}(K)}{\lambda^{C}}$. The theoretical results are compared with the MC simulation (red dots) described in text.
  • Figure 4.2: Test for the asymptotic results for VIX calls under the Eraker model. The blue curve shows the asymptotic result for $1000 \frac{a_{V,C}(K)}{\lambda^{C}}$ and the red dots the result of an MC simulation. The test uses VIX options with maturity $T=0.1$ and $\sigma_V=0.001$ under the Eraker model.
  • Figure 4.3: Plots of the asymptotic results for European calls and puts with maturity $T=0.01$ (left) and $T=0.1$ (right) in the Kou-type model. The black curves show the asymptotic prediction for $1000 \frac{a_{E,C}(K)}{\lambda^{C}}$ and $1000 \frac{a_{E,P}(K)}{\lambda^{C}}$. The theoretical results are compared with the MC simulation (red dots) described in text.
  • Figure 4.4: Test for the asymptotic results for VIX calls in the Kou-type model. The black curve shows the asymptotic result for $1000 \frac{a_{V,C}(K)}{\lambda^{C}}$ and the red dots the result of an MC simulation. The test uses VIX options with maturity $T=0.01$ (left) and $T=0.1$ (right).

Theorems & Definitions (31)

  • Proposition 2.1
  • Corollary 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Corollary 3.1
  • Corollary 3.2
  • Corollary 3.3
  • Remark 3.1
  • ...and 21 more