Heuristic Segmentation of a Nonstationary Time Series

Many phenomena, both natural and human-influenced, give rise to signals whose statistical properties change under time translation, i.e., are nonstationary. For some practical purposes, a nonstationary time series can be seen as a concatenation of stationary segments. Using a segmentation algorithm, it has been reported that for heart beat data and Internet traffic fluctuations--the distribution of durations of these stationary segments decays with a power law tail. A potential technical difficulty that has not been thoroughly investigated is that a nonstationary time series with a (scale-free) power law distribution of stationary segments is harder to segment than other nonstationary time series because of the wider range of possible segment sizes. Here, we investigate the validity of a heuristic segmentation algorithm recently proposed by Bernaola-Galvan et al. by systematically analyzing surrogate time series with different statistical properties. We find that if a given nonstationary time series has stationary periods whose size is distributed as a power law, the algorithm can split the time series into a set of stationary segments with the correct statistical properties. We also find that the estimated power law exponent of the distribution of stationary-segment sizes is affected by (i) the minimum segment size, and (ii) the ratio of the standard deviation of the mean values of the segments, and the standard deviation of the fluctuations within a segment. Furthermore, we determine that the performance of the algorithm is generally not affected by uncorrelated noise spikes or by weak long-range temporal correlations of the fluctuations within segments.Comment: 23 pages, 14 figure

Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels

Credit risk, Endogenous bankruptcy, Scale functions, Fluctuation identity, Continuous and smooth pasting principles, Wiener–Hopf factorization, C61, 91B28, 91B99, 91B72,

The valuation of a firm’s investment opportunities: a reduced form credit risk perspective

Present value, Credit risk, Reduced form models, G31, G13,

Demand Function and Location Theory of the Firm under Price Uncertainty: A Reply

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Bankruptcy triggering asset value–continuous time finance approach

This paper utilizes means of game theory and option pricing to compute a bankruptcy triggering asset value. Combination of these two fields of economic study serves to separating the given problem into valuation of the payoffs, where we use option pricing and the analysis of strategic interactions between parties of a contract which could be designed and solved with the use of game theory. First of all, we design a contract between three parties each having a stake in the company, but with different rights reflected in the boundary conditions of the Black-Scholes equation. Then we will compute the values of debts and the whole value of the company. From here we directly compute the value of the firm’s equity and optimize it from the point of view of managing shareholders. The theoretically computed bankruptcy triggering asset value is then compared to the actual stock price. Depending on this relation, we may say whether the company is likely to go under or not. In addition, this article also provides reader with a real-life case study of the investment bank Bear Stearns and the optimal bankruptcy strategy in this particular case. As we will observe, the bankruptcy trigger computed in this example could have served as a good guide for predicting fall of this investment bank