In this paper we study the numerical method for a time-fractional Black-Scholes equation, which is used for option pricing. The solution of the fractional-order differential equation may be singular near certain domain boundaries, which leads to numerical difficulty. In order to capture the singular phenomena, a numerical method based on an adaptive moving mesh is developed. A finite difference method is used to discretize the time-fractional Black-Scholes equation and error analysis for the discretization scheme is derived. Then, an adaptive moving mesh based on an a priori error analysis is established by equidistributing monitor function. Numerical experiments support these theoretical results.
MSC: 65M06; 65M12; 65M15
Manuscript Type: EmpiricalResearch Question/Issue: What is the impact of bankruptcy risk on whether listed corporations are likely to be bought out by private equity firms and on the subsequent exit, including bankruptcy, of private equity backed public to private buyouts? Research Findings/Insights: Using a sample of 246 UK companies that went from public to private (P2P) company status from 1997 to 2005, we find that companies going private have a significantly higher default probability. Private equity firms sponsoring P2P deals acquire firms with a higher risk of bankruptcy than non-acquired firms that remain public. We find evidence that high receivership risk at going private increases the chance that the target will end up in receivership, but post-P2P bankruptcy likelihood is less when the P2P is a management buyout rather than any other form of buyout. Independent boards of pre-P2P targets promote P2P deals and reduce the chances of bankruptcy after the buyout, suggesting a good corporate governance structure makes a positive contribution to bankruptcy avoidance after going private transactions.c org_854 240..258 Theoretical/Academic Implications: Our finding that P2P deals involve targets with a higher risk of bankruptcy adds to theoretical insights about private equity. In contrast to previous research, it suggests that PE firms are not deterred by the risk of financial distress but consider it a value creating opportunity. Our use of the option pricing framework represents a first and novel attempt at measuring bankruptcy risk and its impact on the ability of private equity firms to achieve effective turnaround. We find a link between better governance of the target pre-P2P and lower bankruptcy risk since where the PE investor inherits a strong governance structure, manifested in independent boards, chances of subsequent bankruptcy are reduced. Similarly, where the P2P acquisition is a management buyout, the probability of bankruptcy, post-P2P, is reduced, suggesting lower informational asymmetries and better alignment of interests between managerial and private equity investors. Although, due to the small number of receivership exits in our sample of P2P firms, the results are not as strong as we would like, a more extended analysis involving a larger sample over a longer period, in particular of firms exiting through bankruptcy is expected to produce stronger results. Our results provide a sufficient basis to warrant such further analysis. Practitioner/Policy Implications: Private equity backed P2Ps of listed corporations with high bankruptcy risk augment the market for corporate control as they provide an alternative purchaser to traditional acquirers. Our finding that high bankruptcy risk at going private increases the chance the target will end up in receivership suggests a need for caution on the part of private equity firms since the turnaround of P2P targets appears to depend on how seriously distressed they are at the P2P stage. Private equity firms therefore need to engage in careful due dil...
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