Consistent asset modelling with random coefficients and switches between regimes

Paper: arXiv 2401.09955

Abstract

We explore a stochastic model that enables capturing external influences in two specific ways. The model allows for the expression of uncertainty in the parametrisation of the stochastic dynamics and incorporates patterns to account for different behaviours across various times or regimes. To establish our framework, we initially construct a model with random parameters, where the switching between regimes can be dictated either by random variables or deterministically. Such a model is highly interpretable. We further ensure mathematical consistency by demonstrating that the framework can be elegantly expressed through local volatility models taking the form of standard jump diffusions. Additionally, we consider a Markov-modulated approach for the switching between regimes characterised by random parameters. For all considered models, we derive characteristic functions, providing a versatile tool with wide-ranging applications. In a numerical experiment, we apply the framework to the financial problem of option pricing. The impact of parameter uncertainty is analysed in a two-regime model, where the asset process switches between periods of high and low volatility imbued with high and low uncertainty, respectively.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is highly theoretical, featuring advanced stochastic calculus, Markov modulation, and the derivation of characteristic functions, indicating high mathematical complexity. However, the empirical component is limited to a conceptual numerical experiment on option pricing without backtests or implementation details, resulting in low empirical rigor.

Research Flowchart

  flowchart TD
  A["Research Goal:<br/>Model Asset Returns with<br/>Parameter Uncertainty & Regime Switching"] --> B
  subgraph B ["Key Methodology"]
      direction TB
      B1["Construct Stochastic Model<br/>with Random Coefficients"]
      B2["Incorporate Regime Switching<br/>(Random or Deterministic)"]
      B3["Derive Characteristic Functions<br/>for all Models"]
  end
  B --> C["Input: Financial Market Data<br/>(Asset Prices & Volatility)"]
  C --> D{"Computational Process"}
  D --> D1["Calibrate Model to Market Data"]
  D --> D2["Solve PDE / Use CF for<br/>Option Pricing"]
  D --> D3["Analyze Impact of<br/>Parameter Uncertainty"]
  D1 --> E
  D2 --> E
  D3 --> E
  subgraph E ["Key Findings / Outcomes"]
      E1["Mathematically Consistent Framework<br/>Expressible via Local Volatility"]
      E2["High Interpretability of<br/>Stochastic Regime Dynamics"]
      E3["Numerical Validation:<br/>Quantified Uncertainty Effects<br/>on Option Prices"]
  end

Cited References (43)

1. Option pricing and E sscher transform under regime switching (2005) — link
2. Mean-variance hedging of options on stocks with M arkov volatilities (1995) — link
3. American options with regime switching (2002) — link
4. A mixed-up smile (2000)
5. Pricing bond options under a Markovian regime-switching Hull–White model (2013)
6. A novel pricing method for E uropean options based on F ourier-cosine series expansions (2009) — link
7. Applied Stochastic Control of Jump Diffusions (2007) — link
8. Pricing energy quanto options in the framework of Markov-modulated additive processes (2021) — link

0 of 43 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.