Paper: arXiv 2401.00539

Authors: Elisa Alòs, Eulalia Nualart, Makar Pravosud

Abstract

In this paper we study short-time behavior of the at-the-money implied volatility for Inverse European options with fixed strike price. The asset price is assumed to follow a general stochastic volatility process. Using techniques of the Malliavin calculus such as the anticipating It^o’s formula we first compute the level of the implied volatility of the option when the maturity converges to zero. Then, we find a short maturity asymptotic formula for the skew of the implied volatility that depends on the roughness of the volatility model. We also show that our results extend easily to Quanto-Inverse options. We apply our general results to the SABR and fractional Bergomi models, and provide some numerical simulations that confirm the accurateness of the asymptotic formula for the skew. Finally, we provide an empirical application using Bitcoin options traded on Debirit to show how our theoretical formulas can be used to model real market data of such options.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents rigorous mathematical derivations using Malliavin calculus to analyze Inverse option implied volatility under rough and stochastic volatility models. It bridges advanced theoretical quant finance with practical crypto-market applications by testing the asymptotic formulas against Deribit Bitcoin option data.

Research Flowchart

  flowchart TD
    A["Research Goal: Analyze short-time ATM Implied Volatility (IV) & Skew for Inverse Options"] --> B["Methodology: Malliavin Calculus & Anticipating Itô's Formula"]
    B --> C["Data & Models: Synthetic Simulations, SABR, fBergomi & Deribit BTC Options"]
    C --> D["Computational Analysis: Derive short-time IV level & skew formulas"]
    D --> E["Findings: Proof of model roughness impact on IV skew"]
    D --> F["Findings: Validated asymptotic formulas & extensions to Quanto-Inverse options"]

Cited References (31)

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  6. Managing Smile Risk (2002)
  7. What Does Implied Volatility Skew Measure? (2011)
  8. Pricing Cryptocurrency Options (2020) — link
and 23 more
  1. Delta Hedging Bitcoin Options with a Smile (2023)
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  4. Cryptocurrency trading: a comprehensive survey (2022) — link
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  17. Introduction to Malliavin Calculus (2018) — link
  18. Is Bitcoin a Commodity? On price jumps, demand shocks, and certainty of supply (2019) — link
  19. On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility (2007) — link
  20. On the curvature of the smile in stochastic volatility models (2017)
  21. Short maturity Asian options in local volatility models (2016)
  22. Inverse and Quanto Inverse Options in a Black-Scholes World (2023)
  23. Estimating the Hurst parameter from short term volatility swaps: a Malliavin calculus approach (2019) — link

0 of 31 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.