Efficient inverse $Z$-transform: sufficient conditions

ArXiv ID: 2305.10725 “View on arXiv”

Authors: Unknown

Abstract

We derive several sets of sufficient conditions for applicability of the new efficient numerical realization of the inverse $Z$-transform. For large $n$, the complexity of the new scheme is dozens of times smaller than the complexity of the trapezoid rule. As applications, pricing of European options and single barrier options with discrete monitoring are considered; applications to more general options with barrier-lookback features are outlined. In the case of sectorial transition operators, hence, for symmetric Lévy models, the proof is straightforward. In the case of non-symmetric Lévy models, we construct a non-linear deformation of the dual space, which makes the transition operator sectorial, with an arbitrary small opening angle, and justify the new realization. We impose mild conditions which are satisfied for wide classes of non-symmetric Stieltjes-Lévy processes.

Keywords: Option Pricing, Lévy Models, Inverse Z-transform, Barrier Options, Numerical Methods, Derivatives

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats
• Why: The paper presents advanced mathematical analysis involving complex analysis, operator theory, and deformation of spaces to derive sufficient conditions for an efficient inverse Z-transform, earning a high math score. However, it lacks backtest data, implementation code, or empirical performance metrics, focusing instead on theoretical validity and algorithmic complexity, resulting in a low empirical rigor score.

  flowchart TD
    Start(["Research Goal:<br>Efficient Inverse Z-Transform"]) --> Step1["Derive Sufficient Conditions<br>for New Numerical Scheme"]
    
    Step1 --> Condition{"Model Type?"}
    
    Condition -- "Symmetric Lévy<br>(Sectorial Operator)" --> Case1["Straightforward Proof"]
    Condition -- "Non-Symmetric Lévy" --> Case2["Construct Non-Linear<br>Dual Space Deformation"]
    
    Case1 --> Comp1["Apply Efficient Scheme<br>Complexity ~ dozens times<br>smaller than Trapezoid Rule"]
    Case2 --> Comp1
    
    Comp1 --> Data1["European Options"]
    Comp1 --> Data2["Discrete Barrier Options"]
    Comp1 --> Data3["Barrier-Lookback Features"]
    
    Data1 --> Outcome1["Key Findings:<br>1. Mild conditions ensure<br>application to Stieltjes-Lévy processes<br>2. Efficiency gain for large n"]
    Data2 --> Outcome1
    Data3 --> Outcome1