A financial market with singular drift and no arbitrage

Area of Science:

• Quantitative Finance
• Mathematical Finance
• Financial Modeling

Background:

• Financial markets can exhibit complex dynamics, including jumps and delays, which are crucial for realistic modeling.
• Previous models with singular drifts, like geometric Itô-Lévy processes, have raised concerns about market arbitrage, especially in continuous settings.
• The impact of jumps and information flow delays on market arbitrage in these models remained unclear.

Purpose of the Study:

• To develop and analyze a jump diffusion market model incorporating a singular drift term and a delay in information flow.
• To investigate the existence of arbitrage opportunities in such a delayed market model.
• To compute optimal consumption and portfolio strategies and determine the market's maximal value.

Main Methods:

• Utilizing white noise calculus to analyze the financial market model.
• Modeling the risky asset with a geometric Itô-Lévy process featuring a singular drift term.
• Incorporating both Brownian motion and a Poisson random measure for jumps, alongside a delay parameter.

Main Results:

• The maximal value of the market is finite when the delay parameter is non-zero, indicating no arbitrage.
• As the delay approaches zero, the maximal value tends to infinity, suggesting the re-emergence of arbitrage.
• Optimal consumption rates and portfolios were explicitly computed using the developed mathematical framework.

Conclusions:

• The presence of delays in information flow can prevent arbitrage in jump diffusion financial markets.
• The model provides a framework for understanding arbitrage dynamics in markets with and without delays.
• The findings are relevant for high-frequency trading, where infinitesimal time scales can lead to singular drifts.