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Some PDE results in Heston model with applications

Edoardo Lombardo

TL;DR

We study the logHeston PDE arising from a stochastic volatility model, focusing on degenerate diffusion at the boundary $y=0$ and parameter regimes where the Feller condition is not satisfied. The authors develop a viscosity-solution framework that yields existence and uniqueness without relying on the Feller condition, and they prove a verification theorem linking the PDE solution to a Feynman–Kac representation even for discontinuous final data, including digital options. They also introduce a hybrid numerical scheme that combines a finite-difference discretization in the log price with a tree/Markov-chain approximation of the volatility and prove convergence in $\ell^{\infty}$ with rate $O(h+\Delta x)$ under mild regularity. Together, these results broaden the parameter range for rigorous pricing and provide a practical algorithm for computing prices under the logHeston model. The work thereby advances both the theory of degenerate parabolic equations in finance and their numerical implementation.

Abstract

We present here some results for the PDE related to the logHeston model. We present different regularity results and prove a verification theorem that shows that the solution produced via the Feynman-Kac theorem is the unique viscosity solution for a wide choice of initial data (even discontinuous) and source data. In addition, our techniques do not use Feller's condition at any time. In the end, we prove a convergence theorem to approximate this solution by means of a hybrid (finite differences/tree scheme) approach.

Some PDE results in Heston model with applications

TL;DR

We study the logHeston PDE arising from a stochastic volatility model, focusing on degenerate diffusion at the boundary and parameter regimes where the Feller condition is not satisfied. The authors develop a viscosity-solution framework that yields existence and uniqueness without relying on the Feller condition, and they prove a verification theorem linking the PDE solution to a Feynman–Kac representation even for discontinuous final data, including digital options. They also introduce a hybrid numerical scheme that combines a finite-difference discretization in the log price with a tree/Markov-chain approximation of the volatility and prove convergence in with rate under mild regularity. Together, these results broaden the parameter range for rigorous pricing and provide a practical algorithm for computing prices under the logHeston model. The work thereby advances both the theory of degenerate parabolic equations in finance and their numerical implementation.

Abstract

We present here some results for the PDE related to the logHeston model. We present different regularity results and prove a verification theorem that shows that the solution produced via the Feynman-Kac theorem is the unique viscosity solution for a wide choice of initial data (even discontinuous) and source data. In addition, our techniques do not use Feller's condition at any time. In the end, we prove a convergence theorem to approximate this solution by means of a hybrid (finite differences/tree scheme) approach.
Paper Structure (9 sections, 21 theorems, 161 equations)

This paper contains 9 sections, 21 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Existence and uniqueness of classical solutions
  3. Existence and uniqueness of viscosity solutions
  4. Continuous initial data
  5. Comparison principle and uniqueness for continuous initial data
  6. Discontinuous initial data
  7. Application to finance: a hybrid approximation scheme for the viscosity solution
  8. The hybrid procedure
  9. Convergence in $\ell^\infty$

Key Result

Lemma 2.1

Let $u$ be defined in u_gen_sol, with $f$ and $h$ such that, as $j=0,1$, $\partial_x^{2j}g\in C^{1-j}_{{\mathbf{pol}}}({\mathbb R}\times{\mathbb R}_+)$, $\partial_x^{2j}h\in \mathcal{C}^{1-j}_{{\mathbf{pol}},T}({\mathbb R}\times{\mathbb R}_+)$ along with $h$ and $\partial_y h$ locally Hölder continu where $(X^{1,t,x,y}_s,Y^{1,t,x,y}_s)$ solves logHeston_model_sde_intro with new parameters $\rho_1=

Theorems & Definitions (49)

  • Lemma 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • proof
  • Proposition 2.6
  • Proposition 2.7
  • proof
  • ...and 39 more