Parodi, Bernhard R. (2014): A Ponzi scheme exposed to volatile markets.
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Abstract
The PGBM model for a couple of counteracting, exponentially growing capital flows is presented: the available capital stock $X(t)$ evolves according to a variant of inhomogeneous geometric Brownian motion (GBM) with timedependent drift, in particular, to the stochastic differential equation $dX(t)=[pX(t)+\rho_1\exp(q_1 t)+\rho_2\exp(q_2 t)]dt+\sigma X(t) dW(t)$, where $W(t)$ is a Wiener process. As a paragon, we study a continuoustime model for a nineparameter Ponzi scheme with an exponentially growing number of investors. Investors either maintain their investment or withdraw it after some fixed investment span and quit the system. The first two moments of the process and hence a closedform solution for the mean path are given. The capital stock exhibits a dynamic lognormal probability distribution as long as the system remains solvent. The assumed stochastic performance allows for earlier or later collaps of the investment system as compared to the deterministic analogy ($\sigma = 0$). Allowing also for negative capital values the system's default probability can be calculated at any time by numerically solving the corresponding Kolmogorov forward equation. We use the finite difference method and obtain results in accordance with those of simple MonteCarlo simulations. Finally, a minor modification of the payout function provides a toy model for a social security system exhibiting critical behaviour. Depending on whether some parameter value violates a weak noPonzi game condition or not, the system represents either a nonlasting Ponzi game or a lasting noPonzi game in the weak sense.
Item Type:  MPRA Paper 

Original Title:  A Ponzi scheme exposed to volatile markets 
Language:  English 
Keywords:  Ponzi scheme; geometric brownian motion; probability density; Kolmogorov forward equation; default probability; critical behaviour 
Subjects:  A  General Economics and Teaching > A2  Economic Education and Teaching of Economics > A20  General C  Mathematical and Quantitative Methods > C5  Econometric Modeling > C50  General G  Financial Economics > G0  General > G00  General G  Financial Economics > G2  Financial Institutions and Services > G20  General 
Item ID:  60584 
Depositing User:  Bernhard R. Parodi 
Date Deposited:  13 Dec 2014 08:18 
Last Modified:  05 Oct 2019 21:45 
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URI:  https://mpra.ub.unimuenchen.de/id/eprint/60584 
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