On the short-time behaviour of up-and-in barrier options using Malliavin calculus

Paper: arXiv 2510.15423

Authors: Òscar Burés

Abstract

In this paper we study the short-maturity asymptotics of up-and-in barrier options under a broad class of stochastic volatility models. Our approach uses Malliavin calculus techniques, typically used for linear stochastic partial differential equations, to analyse the law of the supremum of the log-price process. We derive a concentration inequality and explicit bounds on the density of the supremum in terms of the time to maturity. These results yield an upper bound on the asymptotic decay rate of up-and-in barrier option prices as maturity vanishes. We further demonstrate the applicability of our framework to the rough Bergomi model and validate the theoretical results with numerical experiments.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical treatment of short-time asymptotics for up-and-in barrier options using advanced Malliavin calculus techniques across stochastic and rough volatility models. While mathematically sophisticated and novel in its asymptotic decay bounds, it is predominantly theoretical with minimal empirical validation mentioned in the text.

Research Flowchart

  flowchart TD
    A[Research Goal: Derive short-time asymptotic bounds for Up-and-In barrier options] --> B[Inputs: Stochastic Volatility Models e.g., Rough Bergomi]
    B --> C[Apply Malliavin Calculus to Log-Price Process Supremum]
    C --> D[Derive Concentration Inequality & Density Bounds for Supremum]
    D --> E[Validate Theoretical Asymptotic Decay Rates via Numerical Experiments]
    E --> F[Outcomes: Explicit Upper Bounds on Option Decay Rates at Short Maturities]

Cited References (12)

1. Analysis on Wiener space and applications (2010) — link
2. An asymptotic expansion formula for up-and-out barrier option price under stochastic volatility model (2013)
3. The stochastic heat equation: hitting probabilities and the probability density function of the supremum via Malliavin calculus (2018)
4. The pricing of options on assets with stochastic volatilities (1987)
5. Option values under stochastic volatilities (1987)
6. Stock price distributions with stochastic volatility: An analytic approach (1991)
7. Short-time at-the-money skew and rough fractional volatility (2017)
8. Long memory in continuous-time stochastic volatility models (1998)

0 of 12 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.