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Pricing and delta computation in jump-diffusion models with stochastic intensity by Malliavin calculus

Ayub Ahmadi, Mahdieh Tahmasebi

TL;DR

The paper addresses derivative pricing and delta computation in a jump-diffusion model with CIR-type stochastic intensity using Malliavin calculus. It derives Wiener- and Poisson-space Malliavin weights to express price and delta via duality formulas, accounting for the Malliavin derivative of the intensity. The convergence of the Euler scheme for both the asset price and the delta is established and validated through numerical tests in Gaussian-jump and Kou-type models, demonstrating accuracy and hedging potential. The results provide practical Malliavin-based tools for risk management in markets with jumps and stochastic intensity dynamics.

Abstract

This paper investigates the pricing of financial derivatives and the calculation of their delta Greek when the underlying asset is a jump-diffusion process in which the stochastic intensity component follows the CIR process. Utilizing Malliavin derivatives for pricing financial derivatives and challenging to find the Malliavin weight for accurately calculating delta will be established in such models. Because asset prices rely on information from the intensity process, the moments of the Malliavin weights and the underlying asset must be bound. We apply the Euler scheme to show the convergence of the approximated solution, a financial derivative, and its delta Greeks and we have established the convergence analysis. Our approach has been validated through numerical experiments, highlighting its effectiveness and potential for risk management and hedging strategies in markets characterized by jump and stochastic intensity dynamics.

Pricing and delta computation in jump-diffusion models with stochastic intensity by Malliavin calculus

TL;DR

The paper addresses derivative pricing and delta computation in a jump-diffusion model with CIR-type stochastic intensity using Malliavin calculus. It derives Wiener- and Poisson-space Malliavin weights to express price and delta via duality formulas, accounting for the Malliavin derivative of the intensity. The convergence of the Euler scheme for both the asset price and the delta is established and validated through numerical tests in Gaussian-jump and Kou-type models, demonstrating accuracy and hedging potential. The results provide practical Malliavin-based tools for risk management in markets with jumps and stochastic intensity dynamics.

Abstract

This paper investigates the pricing of financial derivatives and the calculation of their delta Greek when the underlying asset is a jump-diffusion process in which the stochastic intensity component follows the CIR process. Utilizing Malliavin derivatives for pricing financial derivatives and challenging to find the Malliavin weight for accurately calculating delta will be established in such models. Because asset prices rely on information from the intensity process, the moments of the Malliavin weights and the underlying asset must be bound. We apply the Euler scheme to show the convergence of the approximated solution, a financial derivative, and its delta Greeks and we have established the convergence analysis. Our approach has been validated through numerical experiments, highlighting its effectiveness and potential for risk management and hedging strategies in markets characterized by jump and stochastic intensity dynamics.
Paper Structure (20 sections, 20 theorems, 132 equations, 6 figures, 5 tables)

This paper contains 20 sections, 20 theorems, 132 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. A review on Malliavin calculus
  3. Malliavin calculus concepts on Wiener space
  4. The Malliavin calculus on Poisson space
  5. First approach
  6. Second approach
  7. Stochastic jump processes with stochastic intensity
  8. Malliavin derivative of the solution on Wiener space
  9. Pricing and Delta calculation
  10. Delta with Wiener-Malliavin weight
  11. Delta with Poisson-Malliavin weight
  12. The convergence of the Euler scheme
  13. Numerical Example
  14. Pricing and convergence rate
  15. Delta in the first and second approach
  16. ...and 5 more sections

Key Result

Lemma 2.1

n3 Let $F,G\in\mathbb{D}_N^{1,2}$ Suppose that $FG\in L^{2}(\Omega)$ and $(F + D^{N}F)(G + D^{N}G)\in L^{2}(\Omega\times [0,T] \times \mathbb{R}_0)$.Then the product FG also belongs to $\mathbb{D}_N^{1,2}$ and

Figures (6)

  • Figure 1: Error of asset price and European call option with normal jumps and 100 paths simulation.
  • Figure 2: Pricing of European call option for $T=1, S_0=5$ and the function $J_{t,z}=z$ with Gaussian jump distribution and 1000 paths simulation.
  • Figure 3: Error of price model and European option with Double Exponential Distribution jumps (Kou model) and 100 paths simulation.
  • Figure 4: sensitivity of price with respect to k
  • Figure 5: sensitivity of price with respect to sigma2
  • ...and 1 more figures

Theorems & Definitions (34)

  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • ...and 24 more