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Representation and Integration by Parts Formulas for Affine Processes

Arturo Kohatsu-Higa, Yuma Tamura

TL;DR

The paper develops probabilistic representations and integration-by-parts formulas for derivatives of expectations under affine diffusions using Fourier-analytic techniques, circumventing pathwise differentiability and enabling square-root diffusion models such as the CIR process. By relating derivatives to expectations under auxiliary affine processes with shifted parameters, it provides delta-type and IBP formulas for derivatives with respect to the initial value and model parameters. The approach yields concrete results in one- and simple multidimensional settings and is illustrated through CIR and population-evolution applications, offering a robust framework for sensitivity analysis and potential calibration even with non-smooth payoffs.

Abstract

Affine processes play an important role in mathematical finance and other applied areas due to their tractable structure. In the present article, we derive probabilistic representations and integration by parts (IBP) formulas for expectations involving affine processes. These formulas are expressed in terms of expectations of affine processes with modified parameters and are derived using Fourier analytic techniques and characteristic functions. Notably, our method does not require pathwise differentiability, allowing us to handle models with square-root diffusion coefficients for a large set of parameters. The methodology can be applied to the classic Cox--Ingersoll--Ross (CIR) model, a model for interest rates in mathematical finance, where the initial value derivative corresponds to one of the ``Greeks'' used in option pricing in mathematical finance. Furthermore, we illustrate the theory with an application to a population evolution model arising as a scaling limit of discrete branching processes. Our approach offers a unified and robust framework for sensitivity analysis in models where classical Malliavin calculus techniques are difficult to apply.

Representation and Integration by Parts Formulas for Affine Processes

TL;DR

The paper develops probabilistic representations and integration-by-parts formulas for derivatives of expectations under affine diffusions using Fourier-analytic techniques, circumventing pathwise differentiability and enabling square-root diffusion models such as the CIR process. By relating derivatives to expectations under auxiliary affine processes with shifted parameters, it provides delta-type and IBP formulas for derivatives with respect to the initial value and model parameters. The approach yields concrete results in one- and simple multidimensional settings and is illustrated through CIR and population-evolution applications, offering a robust framework for sensitivity analysis and potential calibration even with non-smooth payoffs.

Abstract

Affine processes play an important role in mathematical finance and other applied areas due to their tractable structure. In the present article, we derive probabilistic representations and integration by parts (IBP) formulas for expectations involving affine processes. These formulas are expressed in terms of expectations of affine processes with modified parameters and are derived using Fourier analytic techniques and characteristic functions. Notably, our method does not require pathwise differentiability, allowing us to handle models with square-root diffusion coefficients for a large set of parameters. The methodology can be applied to the classic Cox--Ingersoll--Ross (CIR) model, a model for interest rates in mathematical finance, where the initial value derivative corresponds to one of the ``Greeks'' used in option pricing in mathematical finance. Furthermore, we illustrate the theory with an application to a population evolution model arising as a scaling limit of discrete branching processes. Our approach offers a unified and robust framework for sensitivity analysis in models where classical Malliavin calculus techniques are difficult to apply.

Paper Structure

This paper contains 9 sections, 11 theorems, 55 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Derivatives of expectations of affine diffusions
  4. Derivative with respect to the initial value
  5. Other derivatives
  6. An extension to particularly simple multidimensional case
  7. Applications
  8. CIR model
  9. Model of evolution of population

Key Result

Theorem 1

Let $t \in \mathbb{R}_{>0}$ and $(\alpha,\beta,b) \in D$. Moreover, let $X^x$ and $X^{1,x}$ be affine diffusions with parameters $(\alpha,\beta,b)$ and $(\alpha,\beta,b+\alpha/2)$ respectively and initial value $x \in \mathbb{R}_{>0}$. Then, for $f \in L^1(\mathbb{R})$ such that $\widehat{f}\in L^1(

Theorems & Definitions (25)

  • Definition 1
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • proof
  • Remark 4
  • Remark 5
  • ...and 15 more