Functional Itô calculus†

We extend some results of the Itô calculus to functionals of the current path of a process to reflect the fact that often the impact of randomness is cumulative and depends on the history of the process, not merely on its current value. We express the differential of the functional in terms of adequately defined partial derivatives to obtain an Itô formula. We develop an extension of the Feynman-Kac formula to the functional case and an explicit expression of the integrand in the Martingale Representation Theorem. We establish that under certain conditions, even path dependent options prices satisfy a partial differential equation in a local sense. We exploit this fact to find an expression of the price difference between two models and compute variational derivatives with respect to the volatility surface.     

Dynamic programming for valuing American options under a variance-gamma process

Lévy processes provide a solution to overcome the shortcomings of the lognormal hypothesis. A growing literature proposes the use of pure-jump Lévy processes, such as the variance-gamma (VG) model. In this setting, explicit solutions for derivative prices are unavailable, for instance, for the valuation of American options. We propose a dynamic programming approach coupled with finite elements for valuing American-style options under an extended VG model. Our numerical experiments confirm the convergence and show the efficiency of the proposed methodology. We also conduct a numerical investigation that focuses on American options on S&P 500 futures contracts.     

Road transport productivity in the sixteenth‐century Low Countries: the case of Brabant, 1450–1650†

Though there is a consensus that transport plays a central role in economic development, for the period before the eighteenth century there is a lack of strategic information for assessing the importance of road transport productivity changes in economic development. Transport prices in particular are crucial missing pieces of the puzzle. Sources rarely reveal information that meets the standards of reliable price history. However, it is possible to create a reliable transport price series on the basis of the transport of millstones to ducal mills in Brabant. Assessing the impact of the ‘transport productivity changes’ that can be inferred from this transport price series is a hazardous exercise. Moreover, as Masschaele has observed, land transport prices closely match general agricultural price trends. Land transport was essentially an agricultural service, determined both by cost (especially horse provender) and income effects. Transport price inflation was not demand‐led. However, while transport did not impede urbanization and economic growth, conversely, in sixteenth‐century Brabant—a highly urbanized region that experienced considerable growth in the volume of land transport—no significant land transport productivity gains were achieved.     

Horizontal mergers between multisided platforms: Insights from Cournot competition

This paper discusses the literature on horizontal mergers between multisided platforms and argues that the Cournot model can provide useful insights into the welfare effects of such mergers. To illustrate those insights, we develop a simple model in which two‐sided platforms offer a homogeneous service and compete à la Cournot, and derive the effects of “average‐marginal‐cost‐preserving” mergers on consumers on both sides of the market. We conclude with a discussion of several research avenues that could be explored to understand better the impact of horizontal mergers between multisided platforms.     

Immigration and the path dependence of education: the case of German‐speakers in São Paulo,Brazil (1840–1920)

This article studies the path dependence of human capital accumulation in the state of São Paulo, Brazil. It focuses on the impacts of German‐speaking immigrants on education through three channels: their share of the population in the nineteenth century, their on‐the‐job skills, and the schools they founded. Using a new dataset based on almanacs from 1873 and 1888, these effects are evaluated for the nineteenth, early twentieth, and early twenty‐first centuries. The article shows that the institutionalized demand for education of these immigrants, reflected by the establishment of schools, was their main contribution to the accumulation of human capital. The effect of German schools on educational levels required a period to mature and dissipated over time. Nevertheless, its influence was substantial at the beginning of the twentieth century, affecting enrolment levels in private and state schools, a result that suggests the existence of spillover and contagion effects. Moreover, current indicators for stocks and flows of human capital in São Paulo are strongly associated with their historical levels. At the same time, this path dependence is conditional on the type of school: while a positive persistence is found for the private system throughout the twentieth century, convergence occurred in state schools.