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On the integrability of the supremum of stochastic volatility models and other martingales

Stefan Gerhold, Julian Pachschwöll, Johannes Ruf

Abstract

We propose a method to bound the expectation of the supremum of the price process in stochastic volatility models. It can be applied, for example, to the rough Bergomi model, avoiding the need to discuss finiteness of higher moments. Our motivation stems from the theory of American option pricing, as an integrable supremum implies the existence of an optimal stopping time for any linearly bounded payoff. Moreover, we survey the literature on martingales with non-integrable supremum, and give a new construction that yields uniformly integrable martingales with this property.

On the integrability of the supremum of stochastic volatility models and other martingales

Abstract

We propose a method to bound the expectation of the supremum of the price process in stochastic volatility models. It can be applied, for example, to the rough Bergomi model, avoiding the need to discuss finiteness of higher moments. Our motivation stems from the theory of American option pricing, as an integrable supremum implies the existence of an optimal stopping time for any linearly bounded payoff. Moreover, we survey the literature on martingales with non-integrable supremum, and give a new construction that yields uniformly integrable martingales with this property.

Paper Structure

This paper contains 12 sections, 12 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Integrability of the supremum in stochastic volatility models
  3. Applications
  4. Rough Bergomi model and multi-factor extensions
  5. Models with coefficients of linear growth
  6. Signature volatility models
  7. Martingales with non-integrable supremum
  8. General remarks
  9. A construction by Dubins and Gilat
  10. A new approach
  11. Auxiliary results for SVEs
  12. Proof of the reverse direction of Theorem \ref{['thm:rBergomi_multi']}

Key Result

Lemma 2.1

Let $\left(X_t\right)_{t \geq 0}$ be a nonnegative right-continuous submartingale and let $T > 0$. Then we have If the submartingale is additionally uniformly integrable, the extended process $\left(X_t\right)_{t \in [0,\infty]}$ is a submartingale as well by Doob's $L^1$-convergence theorem. In this case, the inequality also holds for $T=\infty$.

Theorems & Definitions (34)

  • Lemma 2.1
  • Definition 2.2: Progressive functional
  • Definition 2.3: Generic stochastic volatility model
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Corollary 2.6
  • proof
  • Definition 3.1: Mixed multi-factor rough Bergomi model lacombe2020asymptotics
  • Theorem 3.2
  • ...and 24 more