Paper: arXiv 2503.22282

Authors: Elisa Alòs, Òscar Burés, Josep Vives

Abstract

In this paper we use Malliavin Calculus techniques in order to obtain expressions for the short-time behavior of the at-the-money implied volatility (ATM-IV) level and skew for a jump-diffusion stock price. The diffusion part is assumed to be the stochastic volatility Bachelier model and the jumps are modeled by a pure-jump Lévy process with drift so that the stock price is a martingale. Regarding the level, we show that the short-time behavior of the ATM-IV level is the same for all pure-jump Lévy processes and, regarding the skew, we give conditions on the law of the jumps for the skew to exist. We also give several numerical examples of stochastic volatilities and Lévy processes that confirm the theoretical results found in the paper.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical extension of short-time ATM implied volatility behavior to a jump-diffusion Bachelier model using Malliavin calculus. While the mathematical derivation is complex and well-structured, empirical testing is limited to numerical simulations confirming the theoretical asymptotic limits rather than real-world data validation.

Research Flowchart

  flowchart TD
    A["Research Question: Short-time ATM-IV Level & Skew in Jump-Diffusion Bachelier Model"] --> B["Model Inputs: SV Bachelier + Pure-Jump Lévy Process with Drift"]
    B --> C["Apply Malliavin Calculus Techniques"]
    C --> D["Derive Analytical Expressions for ATM-IV Level & Skew"]
    D --> E["Perform Numerical Simulations Across Various SV & Lévy Models"]
    E --> F["Key Outcomes: Level behavior universal for pure-jump Lévy; Skew existence conditions defined"]

Cited References (23)

  1. \textit On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility (2007)
  2. \textit Malliavin Calculus in Finance: Theory and Practice (2021)
  3. \textit L 'e vy Processes and Infinitely Divisible Distributions, revised edition (2013)
  4. \textit A Hull and White formula for a stochastic volatility L 'e vy model with infinite activity (2013)
  5. \textit On the implied volatility of European and Asian call options under the stochastic volatility Bachelier model — → 2308.15341
  6. \textit Implied Normal Volatility (2017)
  7. \textit Financial Modelling with Jump Processes (2004)
  8. \textit Estimating the Hurst parameter from short-term volatility swaps: a Malliavin calculus approach (2019)
and 15 more
  1. \textit Malliavin Calculus and Related Topics (2006)
  2. \textit Tempering stable processes (2007) — link
  3. \textit Processes of normal inverse Gaussian type (1997)
  4. \textit Explicit option valuation in the exponential NIG model (2021) — link
  5. \textit A Black–Scholes user’s guide to the Bachelier model (2022)
  6. \textit Managing smile risk (2002)
  7. \textit Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche mark options (1996)
  8. \textit Short-term asymptotics for the implied volatility skew under a stochastic volatility model with L 'e vy jumps (2016)
  9. \textit Short-time expansions for close-to-the-money options under a L 'e vy jump model with stochastic volatility (2015)
  10. An atlas of functions (1987)
  11. Short time behavior of the ATM implied skew in the ADO-Heston model (2024) — link
  12. Asymptotic analysis for stochastic volatility: martingale expansion (2011)
  13. \textit A Hull and White formula for a general stochastic volatility jump-diffusion model with applications to the study of the short-time behavior of the implied volatility (2008)
  14. \textit Small-maturity asymptotics for the at-the-money implied volatility slope in L 'e vy models (2016)
  15. \textit Pricing under rough volatility (2016)

1 of 23 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.