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

Òscar Burés

TL;DR

This work analyzes the short-maturity behavior of up-and-in barrier options under general stochastic volatility using Malliavin calculus. By studying the law of the supremum of the log-price, the authors derive a concentration bound and a density estimate for the maximum, leading to an upper bound on barrier option prices that decays faster than any polynomial as the maturity $T$ tends to zero. The framework is applied to the Rough Bergomi model, with a truncation argument ensuring validity under unbounded volatility, and numerical experiments corroborate the theoretical findings that barrier options decay more rapidly than European calls at short maturities. These results provide rigorous insights into zero-maturity pricing of path-dependent options in rough-volatility settings, informing both pricing and risk management in practice.

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.

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

TL;DR

This work analyzes the short-maturity behavior of up-and-in barrier options under general stochastic volatility using Malliavin calculus. By studying the law of the supremum of the log-price, the authors derive a concentration bound and a density estimate for the maximum, leading to an upper bound on barrier option prices that decays faster than any polynomial as the maturity tends to zero. The framework is applied to the Rough Bergomi model, with a truncation argument ensuring validity under unbounded volatility, and numerical experiments corroborate the theoretical findings that barrier options decay more rapidly than European calls at short maturities. These results provide rigorous insights into zero-maturity pricing of path-dependent options in rough-volatility settings, informing both pricing and risk management in practice.

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.
Paper Structure (9 sections, 23 theorems, 114 equations, 3 figures)

This paper contains 9 sections, 23 theorems, 114 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Model
  3. Malliavin Calculus Tools
  4. Main Result
  5. Short-Time Behaviour Of An Up-And-In Barrier Call Option
  6. A First Asymptotic Result
  7. Estimation Of The Density Of $M_T$
  8. Application To The Rough Bergomi Model
  9. Conclusions

Key Result

Proposition 3.1

Let $F \in \mathbb{D}^{1,2}$ and let $u \in \operatorname{Dom}(\delta)$. Then, where $\langle \cdot, \cdot \rangle$ denotes the usual scalar product in $L^2([0,T])$.

Figures (3)

  • Figure 1: ITM European and up-and-in call options
  • Figure 2: ATM European and up-and-in call options
  • Figure 3: ATM European and up-and-in call options

Theorems & Definitions (48)

  • Proposition 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 4.1
  • Lemma 5.1
  • proof
  • Definition 5.2
  • Lemma 5.3
  • ...and 38 more