2023/03/24 by Yukihiro Tsuzuki, Tsuzuki, Yukihiro
Economics, Econometrics and Finance · #60J60(Primary) #60J65(Secondary) #Complex Systems and Time Series Analysis #Economic theories and models #FOS: Economics and business #Mathematical Finance (q-fin.MF) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2303.13956
openalex publication_date 2023/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.