Pitman's Theorem, BlackScholes Equation, and Derivative Pricing for Fundraisers
Abstract
We propose a financial market model that comprises a savings account and a stock, where the stock price process is modeled as a onedimensional diffusion, wherein 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 extra 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 threedimensional Bessel process. Two contributions are presented: First, the prices of European options for the fundraiser are derived. Second, a numerical scheme is proposed for call option prices in a market with a bubble, where multiple solutions exist for the BlackScholes 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2023
 DOI:
 10.48550/arXiv.2303.13956
 arXiv:
 arXiv:2303.13956
 Bibcode:
 2023arXiv230313956T
 Keywords:

 Quantitative Finance  Mathematical Finance;
 60J60(Primary);
 60J65(Secondary)
 EPrint:
 25 pages