Optimal consumption and investment with bounded downside risk for power utility functions

We investigate optimal consumption and investment problems for a Black-Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall. We formulate various utility maximization problems, which can be solved explicitly. We compare the optimal solutions in form of optimal value, optimal control and optimal wealth to analogous problems under additional uniform risk bounds. Our proofs are partly based on solutions to Hamilton-Jacobi-Bellman equations, and we prove a corresponding verification theorem. This work was supported by the European Science Foundation through the AMaMeF programme.Comment: 36 page

Optimal Investment in the Development of Oil and Gas Field

Let an oil and gas field consists of clusters in each of which an investor can launch at most one project. During the implementation of a particular project, all characteristics are known, including annual production volumes, necessary investment volumes, and profit. The total amount of investments that the investor spends on developing the field during the entire planning period we know. It is required to determine which projects to implement in each cluster so that, within the total amount of investments, the profit for the entire planning period is maximum. The problem under consideration is NP-hard. However, it is solved by dynamic programming with pseudopolynomial time complexity. Nevertheless, in practice, there are additional constraints that do not allow solving the problem with acceptable accuracy at a reasonable time. Such restrictions, in particular, are annual production volumes. In this paper, we considered only the upper constraints that are dictated by the pipeline capacity. For the investment optimization problem with such additional restrictions, we obtain qualitative results, propose an approximate algorithm, and investigate its properties. Based on the results of a numerical experiment, we conclude that the developed algorithm builds a solution close (in terms of the objective function) to the optimal one

Interest Rates and Information Geometry

The space of probability distributions on a given sample space possesses natural geometric properties. For example, in the case of a smooth parametric family of probability distributions on the real line, the parameter space has a Riemannian structure induced by the embedding of the family into the Hilbert space of square-integrable functions, and is characterised by the Fisher-Rao metric. In the nonparametric case the relevant geometry is determined by the spherical distance function of Bhattacharyya. In the context of term structure modelling, we show that minus the derivative of the discount function with respect to the maturity date gives rise to a probability density. This follows as a consequence of the positivity of interest rates. Therefore, by mapping the density functions associated with a given family of term structures to Hilbert space, the resulting metrical geometry can be used to analyse the relationship of yield curves to one another. We show that the general arbitrage-free yield curve dynamics can be represented as a process taking values in the convex space of smooth density functions on the positive real line. It follows that the theory of interest rate dynamics can be represented by a class of processes in Hilbert space. We also derive the dynamics for the central moments associated with the distribution determined by the yield curve.Comment: 20 pages, 3 figure

A dynamic and multifunctional account of middle‐range theories

This article develops a novel account of middle‐range theories for combining theoretical and empirical analysis in explanatory sociology. I first revisit Robert K. Merton’s original ideas on middle‐range theories and identify a tension between his developmental approach to middle‐range theorizing that recognizes multiple functions of theories in sociological research and his static definition of the concept of middle‐range theory that focuses only on empirical testing of theories. Drawing on Merton's ideas on theorizing and recent discussions on mechanism‐based explanations, I argue that this tension can be resolved by decomposing a middle‐range theory into three interrelated and evolving components that perform different functions in sociological research: (i) a conceptual framework about social phenomena that is a set of interrelated concepts that evolve in close connection with empirical analysis; (ii) a mechanism schema that is an abstract and incomplete description of a social mechanism; and (iii) a cluster of all mechanism‐based explanations of social phenomena that are based on the particular mechanism schema. I show how these components develop over time and how they serve different functions in sociological theorizing and research. Finally, I illustrate these ideas by discussing Merton’s theory of the Matthew effect in science and its more recent applications in sociology.This article develops a novel account of middle‐range theories for combining theoretical and empirical analysis in explanatory sociology. I first revisit Robert K. Merton’s original ideas on middle‐range theories and identify a tension between his developmental approach to middle‐range theorizing that recognizes multiple functions of theories in sociological research and his static definition of the concept of middle‐range theory that focuses only on empirical testing of theories. Drawing on Merton's ideas on theorizing and recent discussions on mechanism‐based explanations, I argue that this tension can be resolved by decomposing a middle‐range theory into three interrelated and evolving components that perform different functions in sociological research: (i) a conceptual framework about social phenomena that is a set of interrelated concepts that evolve in close connection with empirical analysis; (ii) a mechanism schema that is an abstract and incomplete description of a social mechanism; and (iii) a cluster of all mechanism‐based explanations of social phenomena that are based on the particular mechanism schema. I show how these components develop over time and how they serve different functions in sociological theorizing and research. Finally, I illustrate these ideas by discussing Merton’s theory of the Matthew effect in science and its more recent applications in sociology.Peer reviewe

An Optimal Execution Problem with Market Impact

We study an optimal execution problem in a continuous-time market model that considers market impact. We formulate the problem as a stochastic control problem and investigate properties of the corresponding value function. We find that right-continuity at the time origin is associated with the strength of market impact for large sales, otherwise the value function is continuous. Moreover, we show the semi-group property (Bellman principle) and characterise the value function as a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. We introduce some examples where the forms of the optimal strategies change completely, depending on the amount of the trader's security holdings and where optimal strategies in the Black-Scholes type market with nonlinear market impact are not block liquidation but gradual liquidation, even when the trader is risk-neutral.Comment: 36 pages, 8 figures, a modified version of the article "An optimal execution problem with market impact" in Finance and Stochastics (2014

The dynamics of financial stability in complex networks

We address the problem of banking system resilience by applying off-equilibrium statistical physics to a system of particles, representing the economic agents, modelled according to the theoretical foundation of the current banking regulation, the so called Merton-Vasicek model. Economic agents are attracted to each other to exchange `economic energy', forming a network of trades. When the capital level of one economic agent drops below a minimum, the economic agent becomes insolvent. The insolvency of one single economic agent affects the economic energy of all its neighbours which thus become susceptible to insolvency, being able to trigger a chain of insolvencies (avalanche). We show that the distribution of avalanche sizes follows a power-law whose exponent depends on the minimum capital level. Furthermore, we present evidence that under an increase in the minimum capital level, large crashes will be avoided only if one assumes that agents will accept a drop in business levels, while keeping their trading attitudes and policies unchanged. The alternative assumption, that agents will try to restore their business levels, may lead to the unexpected consequence that large crises occur with higher probability

Eroding market stability by proliferation of financial instruments

We contrast Arbitrage Pricing Theory (APT), the theoretical basis for the development of financial instruments, with a dynamical picture of an interacting market, in a simple setting. The proliferation of financial instruments apparently provides more means for risk diversification, making the market more efficient and complete. In the simple market of interacting traders discussed here, the proliferation of financial instruments erodes systemic stability and it drives the market to a critical state characterized by large susceptibility, strong fluctuations and enhanced correlations among risks. This suggests that the hypothesis of APT may not be compatible with a stable market dynamics. In this perspective, market stability acquires the properties of a common good, which suggests that appropriate measures should be introduced in derivative markets, to preserve stability.Comment: 26 pages, 8 figure

Pricing Exotic Options in a Path Integral Approach

In the framework of Black-Scholes-Merton model of financial derivatives, a path integral approach to option pricing is presented. A general formula to price European path dependent options on multidimensional assets is obtained and implemented by means of various flexible and efficient algorithms. As an example, we detail the cases of Asian, barrier knock out, reverse cliquet and basket call options, evaluating prices and Greeks. The numerical results are compared with those obtained with other procedures used in quantitative finance and found to be in good agreement. In particular, when pricing at-the-money and out-of-the-money options, the path integral approach exhibits competitive performances.Comment: 21 pages, LaTeX, 3 figures, 6 table

Engineering serendipity: When does knowledge sharing lead to knowledge production?

Research Summary We investigate how knowledge similarity between two individuals is systematically related to the likelihood that a serendipitous encounter results in knowledge production. We conduct a field experiment at a medical research symposium, where we exogenously varied opportunities for face‐to‐face encounters among 15,817 scientist‐pairs. Our data include direct observations of interaction patterns collected using sociometric badges, and detailed, longitudinal data of the scientists\u27 postsymposium publication records over 6 years. We find that interacting scientists acquire more knowledge and coauthor 1.2 more papers when they share some overlapping interests, but cite each other\u27s work between three and seven times less when they are from the same field. Our findings reveal both collaborative and competitive effects of knowledge similarity on knowledge production outcomes. Managerial Summary Managers often try to stimulate innovation by encouraging serendipitous interactions between employees, for example by using office space redesigns, conferences and similar events. Are such interventions effective? This article proposes that an effective encounter depends on the degree of common knowledge shared by the individuals. We find that scientists who attend the same conference are more likely to learn from each other and collaborate effectively when they have some common interests, but may view each other competitively when they work in the same field. Hence, when designing opportunities for face‐to‐face interactions, managers should consider knowledge similarity as a criteria for fostering more productive exchanges