Paper: arXiv 2408.15416

Abstract

This paper presents a new model for options pricing. The Black-Scholes-Merton (BSM) model plays an important role in financial options pricing. However, the BSM model assumes that the risk-free interest rate, volatility, and equity premium are constant, which is unrealistic in the real market. To address this, our paper considers the time-varying characteristics of those parameters. Our model integrates elements of the BSM model, the Heston (1993) model for stochastic variance, the Vasicek model (1977) for stochastic interest rates, and the Campbell and Viceira model (1999, 2001) for stochastic equity premium. We derive a linear second-order parabolic PDE and extend our model to encompass fixed-strike Asian options, yielding a new PDE. In the absence of closed-form solutions for any options from our new model, we utilize finite difference methods to approximate prices for European call and up-and-out barrier options, and outline the numerical implementation for fixed-strike Asian call options.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper involves high-level mathematics, including a multi-factor stochastic SDE system and the derivation of a complex 6-variable parabolic PDE with correlations, requiring advanced stochastic calculus. However, it presents theoretical derivations and outlines numerical methods (finite difference) without providing actual backtesting, dataset usage, or empirical results on real market data.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Create a new options pricing model<br>addressing constant parameter limitations of BSM"] --> B["Methodology<br>Integrate Heston, Vasicek, & Campbell-Viceira models<br>Derive linear 2nd-order parabolic PDE"]
  
  B --> C{"Data/Inputs"}
  C --> D["Stochastic Volatility<br>(Heston Model)"]
  C --> E["Stochastic Interest Rates<br>(Vasicek Model)"]
  C --> F["Stochastic Equity Premium<br>(Campbell-Viceira Model)"]
  
  D & E & F --> G["Computational Process<br>Finite Difference Methods"]
  
  G --> H["Outcomes<br>Approximated Prices for:<br>European Call<br>Up-and-Out Barrier<br>Fixed-strike Asian Call"]
  
  H --> I["Key Findings<br>Extended model captures real market dynamics<br>Effective numerical implementation via FDM"]

Cited References (15)

  1. The Pricing of Options and Corporate Liabilities (1973)
  2. A Theory of the Term Structure of Interest Rates (1985) — link
  3. Consumption and Portfolio Decisions when Expected Returns are Time Varying (1999)
  4. Who Should Buy Long-Terms Bonds? (2001)
  5. Stochastic calculus for finance (2014)
  6. ADI finite difference schemes for the Heston - Hull - White PDE (2011) — link
  7. Theory of Rational Option Pricing (1973)
  8. Finite Difference Schemes for Heston Model (2008) — link
and 7 more
  1. Completeness and Hedging (2004) — link
  2. A Closed - Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options (1993) — link
  3. Pricing and hedging in incomplete markets with model uncertainty (2020) — link
  4. Second order parabolic differential equations (2005)
  5. On the Partial Difference Equations of Mathamtical Physics (2018)
  6. On Cross - Currency Models with Stochastic Volatility and Correlated Interest Rates (2010) — link
  7. An equilibrium characterization of the term structure (1977)

0 of 15 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.