Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local risk-minimization for exponential additive processes

Takuji Arai

TL;DR

This work advances local risk-minimization (LRM) in incomplete markets by deriving explicit LRM representations for asset models driven by exponential additive processes with time-dependent Lévy measures. Building on a corrected extension of Handa et al., it uses the minimal martingale measure and Föllmer-Schweizer decomposition to obtain a computable formula for the LRM strategy of a call payoff, and it casts the result into a numerically tractable Carr-Madan FFT framework. The VGSSD process is introduced as a primary additive-process example, and numerical experiments illustrate how LRM hedges respond to time-varying jump dynamics. The findings provide a practical hedging toolkit for non-stationary jump models and establish groundwork for applying LRM to broader exponential additive market models.

Abstract

We explore local risk-minimization, a quadratic hedging method for incomplete markets, in exponential additive models. The objectives are to derive explicit mathematical expressions and to conduct numerical experiments. While local risk-minimization is well studied for Lévy processes, little is known for the additive process case because, unlike Lévy processes, the Lévy measure for an additive process depends on time, which significantly complicates the mathematical framework. This paper shall provide a set of necessary conditions for deriving expressions for LRM strategies in exponential additive models, as integrability conditions on the Lévy measure, which allow us to confirm whether these conditions are satisfied for given concrete models. In the final section, we introduce the variance-gamma scaled self-decomposable process, a Sato process that generalizes the variance-gamma process, as a primary example, and perform numerical experiments.

Local risk-minimization for exponential additive processes

TL;DR

This work advances local risk-minimization (LRM) in incomplete markets by deriving explicit LRM representations for asset models driven by exponential additive processes with time-dependent Lévy measures. Building on a corrected extension of Handa et al., it uses the minimal martingale measure and Föllmer-Schweizer decomposition to obtain a computable formula for the LRM strategy of a call payoff, and it casts the result into a numerically tractable Carr-Madan FFT framework. The VGSSD process is introduced as a primary additive-process example, and numerical experiments illustrate how LRM hedges respond to time-varying jump dynamics. The findings provide a practical hedging toolkit for non-stationary jump models and establish groundwork for applying LRM to broader exponential additive market models.

Abstract

We explore local risk-minimization, a quadratic hedging method for incomplete markets, in exponential additive models. The objectives are to derive explicit mathematical expressions and to conduct numerical experiments. While local risk-minimization is well studied for Lévy processes, little is known for the additive process case because, unlike Lévy processes, the Lévy measure for an additive process depends on time, which significantly complicates the mathematical framework. This paper shall provide a set of necessary conditions for deriving expressions for LRM strategies in exponential additive models, as integrability conditions on the Lévy measure, which allow us to confirm whether these conditions are satisfied for given concrete models. In the final section, we introduce the variance-gamma scaled self-decomposable process, a Sato process that generalizes the variance-gamma process, as a primary example, and perform numerical experiments.
Paper Structure (9 sections, 11 theorems, 78 equations, 4 figures)

This paper contains 9 sections, 11 theorems, 78 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Exponential additive processes
  4. LRM strategies for exponential additive processes
  5. On the result of Handa et al. HSS
  6. Main theorem
  7. Numerically tractable expression
  8. Numerical experiments for VGSSD
  9. Conclusion

Key Result

Lemma 3.1

Under Assumption (A3), ${\int_0^T\int_{{\mathbb R}_0}(e^x-1)^2\overline{\nu}(dt,dx)<\infty}$ holds.

Figures (4)

  • Figure 1: Experiment (A) for Model (1)
  • Figure 2: Experiment (A) for Model (2)
  • Figure 3: Experiment (B) for Model (1)
  • Figure 4: Experiment (B) for Model (2)

Theorems & Definitions (18)

  • Definition 2.2
  • Remark 2.3
  • Lemma 3.1
  • Proposition 3.2
  • Remark 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Remark 3.7
  • Proposition 3.8
  • ...and 8 more