A Space Mapping approach for the calibration of financial models with the application to the Heston model

ArXiv ID: 2501.14521 “View on arXiv”

Authors: Unknown

Abstract

We present a novel approach for parameter calibration of the Heston model for pricing an Asian put option, namely space mapping. Since few parameters of the Heston model can be directly extracted from real market data, calibration to real market data is implicit and therefore a challenging task. In addition, some of the parameters in the model are non-linear, which makes it difficult to find the global minimum of the optimization problem within the calibration. Our approach is based on the idea of space mapping, exploiting the residuum of a coarse surrogate model that allows optimization and a fine model that needs to be calibrated. In our case, the pricing of an Asian option using the Heston model SDE is the fine model, and the surrogate is chosen to be the Heston model PDE pricing a European option. We formally derive a gradient descent algorithm for the PDE constrained calibration model using well-known techniques from optimization with PDEs. Our main goal is to provide evidence that the space mapping approach can be useful in financial calibration tasks. Numerical results underline the feasibility of our approach.

Keywords: Heston model, Space mapping, Asian option, PDE constrained optimization, Parameter calibration

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is mathematically dense with PDE-constrained optimization, adjoint derivations, and theoretical analysis, but it lacks empirical evidence such as backtested performance on historical data or real market data, presenting only conceptual feasibility with synthetic or theoretical results.

  flowchart TD
    Start["Research Goal: Calibrate Heston Model<br>for Asian Put Option Pricing"] --> Method["Key Methodology: Space Mapping Approach"]

    subgraph Input ["Data & Model Inputs"]
        I1["Market Data: Asian Put Options"]
        I2["Fine Model: Heston SDE<br>for Asian Options"]
        I3["Coarse Model: Heston PDE<br>for European Options"]
    end

    Method --> Input

    subgraph Process ["Computational Process"]
        P1["Initialize Parameters"]
        P2["Solve Coarse PDE Model"]
        P3["Calculate Residuum"]
        P4["Update Parameters<br>via Gradient Descent"]
        P1 --> P2 --> P3 --> P4 --> P2
    end

    Input --> Process

    Process --> Result["Key Findings & Outcomes"]
    
    Result --> O1["Feasibility of Space Mapping<br>in Financial Calibration"]
    Result --> O2["Efficient Gradient Descent<br>Optimization"]
    Result --> O3["Validated on Asian Put Option<br>Pricing Application"]