Option Pricing with Stochastic Volatility, Equity Premium, and Interest Rates

ArXiv ID: 2408.15416 “View on arXiv”

Authors: Unknown

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.

Keywords: Option pricing, Stochastic volatility, Stochastic interest rates, Black-Scholes-Merton, Finite difference methods

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: 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.

  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"]