International Journal of Theoretical and Applied Finance

The recent development of the securitisation of funds of private equity funds poses the question of the individual and joint modelling of the underlying funds. Private equity funds are different from other managed funds because of their particular bounded life cycle: when the fund starts, the investment partners make an initial capital commitment, the fund managers gradually draw down the committed capital into investments, returns and proceeds are distributed as the investments are realised and the fund is eventually liquidated as the final investment horizon is reached. Modelling private equity funds therefore requires three stages: the modelling of the commitment drawdowns, the modelling of the investment value and the modelling of the return repayments. A standard lognormal process is utilised for the dynamics of the investment value. Squared Bessel processes are utilised for the dynamics of the rates of drawdowns and repayments.

Résumé: Le récent développement de la titrisation de fonds de fonds de placements privés pose la question de la modélisation individuelle et jointe des fonds sous-jacents. Les fonds de placements privés sont différents des autres sociétés d'investissement à cause de leur cycle de vie particulier et limité: au démarrage du fonds, les associés s'engagent sur un apport initial en capital; puis les gérants du fonds opèrent des tirages progressifs sur le capital apporté pour procéder à des investissements; les revenus et les profits sont distribués à mesure que les investissements sont réalisés; enfin, le fonds est liquidé lorsque l'horizon d'investissement est atteint. La modélisation d'un fonds doit donc se faire en trois étapes: la modélisation des tirages sur l'apport en capital, la modélisation de la valeur des investissements et enfin la modélisation des paiements et remboursements des dividendes et retours sur investissements. Un processus lognormal standard est utilisé pour la dynamique de la valeur des investissements. Des processus de Bessel carré sont utilisés pour la dynamique des taux de tirage et de remboursement.

The empirical distribution of common stock returns is a subject of interest to many researchers, as it often determines the validity of theoretical models proposed in the economics and finance studies. This paper brings to the attention the availability of two flexible systems of distributions for fitting data: the Johnson system of distributions and the Tadikamalla–Johnson system of distributions. We explore the feasibility of fitting the empirical distributions of several financial series to these two systems of distributions. Both systems of distributions are highly flexible and capable of accommodating all possible skewness and kurtosis values. The probability density function and the cumulative distribution function take on simple closed forms and appropriate transformations of the data lead to normal/logistic distributions. In addition, the parameter estimation procedures are easy to implement. When the results are compared with those of other data fitting models, in all cases tested, the proposed distributions provide a good fit to the empirical distribution of data.

In the context of quantitative analysis of the arbitrage pricing theory (APT) model, conventional factor analytic approaches such as maximum likelihood factor analysis (MLFA) cannot provide satisfactory answers to two important questions. The first one concerns the correct identification of factor number while the second one is related to the rotation indeterminacy of factor loadings. In the literature, MLFA followed by likelihood ratio (LR) test and the analysis of eigenvalues of sample covariance matrix were two popular approaches used to determine the appropriate number of factors. However, it was shown empirically that both of them suffered from different kinds of biases. We find the recently developed non-gaussian factor analysis (NFA) model by Xu [24] provides a new perspective for the determination of the appropriate factor number k in APT, with promising empirical results demonstrated.

The aim of this work is to develop a nonparametric tool for detecting dependence in the tails of financial data. We provide a simple method to locate and measure serial dependence in the tails, based on runs tests. Our empirical investigations on many financial time series reveal a strong departure from independence for daily logreturns, which is not filtered out by usual Garch models.

We find evidence of "pure" contagion effects in international banking arising from the collapse of BCCI. A Markov regime-switching approach is employed to allow for the uncertainty surrounding the date of BCCI's collapse. The results indicate that there are shortcomings in the supervision of internationally spread banking groups like BCCI, and carry implications for the EU single market programme in financial services.

We consider the American put with finite time horizon T, assuming that, under an EMM chosen by the market, the stock returns follow a regular Lévy process of exponential type. We formulate the free boundary value problem for the price of the American put, and develop the non-Gaussian analog of the method of lines and Carr's randomization method used in the Gaussian option pricing theory. The result is the (discretized) early exercise boundary and prices of the American put for all strikes and maturities from 0 to T. In the case of exponential jump-diffusion processes, a simple efficient pricing scheme is constructed. We show that for many classes of Lévy processes, the early exercise boundary is separated from the strike price by a non-vanishing margin on the interval [0, T), and that as the riskless rate vanishes, the optimal exercise price goes to zero uniformly over the interval [0, T), which is in the stark contrast with the Gaussian case.

This paper develops a spectral expansion approach to the valuation of contingent claims when the underlying state variable follows a one-dimensional diffusion with the infinitesimal variance a²(x), drift b(x) and instantaneous discount (killing) rate r(x). The Spectral Theorem for self-adjoint operators in Hilbert space yields the spectral decomposition of the contingent claim value function. Based on the Sturm–Liouville (SL) theory, we classify Feller's natural boundaries into two further subcategories: non-oscillatory and oscillatory/non-oscillatory with cutoff Λ≥0 (this classification is based on the oscillation of solutions of the associated SL equation) and establish additional assumptions (satisfied in nearly all financial applications) that allow us to completely characterize the qualitative nature of the spectrum from the behavior of a, b and r near the boundaries, classify all diffusions satisfying these assumptions into the three spectral categories, and present simplified forms of the spectral expansion for each category. To obtain explicit expressions, we observe that the Liouville transformation reduces the SL equation to the one-dimensional Schrödinger equation with a potential function constructed from a, b and r. If analytical solutions are available for the Schrödinger equation, inverting the Liouville transformation yields analytical solutions for the original SL equation, and the spectral representation for the diffusion process can be constructed explicitly. This produces an explicit spectral decomposition of the contingent claim value function.