Paper: arXiv 2303.13956

Authors: Yukihiro Tsuzuki

Abstract

We propose a financial market model that comprises a savings account and a stock. The stock price process is modeled as a one-dimensional diffusion, in which two types of agents exist: an ordinary investor and a fundraiser who buys or sells stocks as funding activities. Although the investor information is the natural filtration of the diffusion, the fundraiser possesses additional information regarding the funding, as well as additional cash flows as a result of the funding. This concept is modeled using Pitman’s theorem for the three-dimensional Bessel process. Two contributions are presented: First, the prices of European options for the fundraiser are derived. Second, a numerical scheme for call option prices in a market with a bubble is proposed, where multiple solutions exist for the Black–Scholes equation and the derivative prices are characterized as the smallest nonnegative supersolution. More precisely, the call option price in such a market is approximated from below by the prices for the fundraiser. This scheme overcomes the difficulty that stems from the discrepancy that the payoff shows linear growth, whereas the price function shows strictly sublinear growth.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper relies heavily on advanced stochastic calculus and probability theory, specifically utilizing Pitman’s 2M-X theorem and 3D Bessel processes to handle asymmetric information and financial bubbles. However, it is purely theoretical and numerical, lacking empirical validation or historical market backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Price European options & solve Black-Scholes for fundraisers in market bubbles] --> B[Market Inputs: Stock price as 1D diffusion, investor filtration, fundraiser cash flows]
    B --> C[Methodology: Apply Pitman's theorem for 3D Bessel process to model asymmetric information]
    C --> D[Computational Scheme: Derive option prices for fundraisers & approximate bubble call prices from below]
    D --> E[Outcome 1: Analytical European option pricing formula for fundraisers]
    D --> F[Outcome 2: Numerical scheme selecting smallest nonnegative supersolution for Black-Scholes with bubbles]