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No arbitrage assumption implies the differentiability of derivative pricing function

Kihun Nam, Yunxi Xu

TL;DR

The paper characterizes when a function f, used to transform a continuous Markov semimartingale X into Y_t = f(t,X_t), yields a semimartingale or Itô process under no-arbitrage (NFLVR). Using the L-derivative framework and the function spaces V^L_μ(loc) and ĥV^L_μ(loc), it establishes necessary and sufficient conditions for Y to be an Itô process or a semimartingale, respectively, along with a decomposition and a generalized gradient in x. The analysis leverages time-change techniques to extend results from Itô processes to general continuous Markov semimartingales and connects regularity of f to Malliavin differentiability, implying differentiability of derivative prices with respect to underlying noise. These findings bridge stochastic calculus and financial modeling by identifying minimal regularity requirements on pricing functions to preserve semimartingale structure under no-arbitrage.

Abstract

In this article, we show necessary and sufficient conditions for a function to transform a continuous Markov semimartingale to a semimartingale. As a result, the no-arbitrage principle guarantees the differentiability of asset prices with respect to the underlying noise, if the asset prices are continuous and the underlying noise is a continuous Markov semimartingale.

No arbitrage assumption implies the differentiability of derivative pricing function

TL;DR

The paper characterizes when a function f, used to transform a continuous Markov semimartingale X into Y_t = f(t,X_t), yields a semimartingale or Itô process under no-arbitrage (NFLVR). Using the L-derivative framework and the function spaces V^L_μ(loc) and ĥV^L_μ(loc), it establishes necessary and sufficient conditions for Y to be an Itô process or a semimartingale, respectively, along with a decomposition and a generalized gradient in x. The analysis leverages time-change techniques to extend results from Itô processes to general continuous Markov semimartingales and connects regularity of f to Malliavin differentiability, implying differentiability of derivative prices with respect to underlying noise. These findings bridge stochastic calculus and financial modeling by identifying minimal regularity requirements on pricing functions to preserve semimartingale structure under no-arbitrage.

Abstract

In this article, we show necessary and sufficient conditions for a function to transform a continuous Markov semimartingale to a semimartingale. As a result, the no-arbitrage principle guarantees the differentiability of asset prices with respect to the underlying noise, if the asset prices are continuous and the underlying noise is a continuous Markov semimartingale.

Paper Structure

This paper contains 4 sections, 6 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Functions transforming an Itô process into semimartingale
  3. Functions transforming a continuous Markov semimartingale into a semimartingale
  4. Acknowledgement of AI use

Key Result

Theorem 2.5

Suppose $X$ satisfies Assumption sde assumption. The process $(f(t,X_t))_{t\geq 0}$ is an Itô process if and only if $f\in V^{L}_\mu(loc)$ and it admits the decomposition On the other hand, the process $(f(t,X_t))_{t\geq 0}$ is a semimartingale if and only if $f\in \hat{V}^L_\mu(loc)$, and it has decomposition where $A^f$ is uniquely determined by the relation for each bounded continuous functi

Theorems & Definitions (20)

  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5
  • proof
  • Theorem 2.7
  • Remark 2.8
  • Remark 2.9
  • Lemma 2.10: Lemma 1.2.3, nualart2006malliavin.
  • proof
  • ...and 10 more