The value of the distant future: Discounting in random environments

We analyze how future costs must be balanced against present costs. This is traditionally done using an exponential function with a constant discount rate. The choice of discount rate can dramatically effect the question on what is the value of the future. This is specially critical for environmental problems such as global warming, and it has generated a controversy as to the urgency for immediate action (Stern, 2006; Nordhaus, 2007a,b). We briefly review the issue for the nonspecialist and take into account the randomness of the economic evolution by studying the discount function of three widely used processes for the dynamics of interest rates: OrnsteinUhlenbeck, Feller and log-normal. We also outline our previous empirical survey on 14 countries over time spans ranging up to more than 300 years We estimate the parameters of one of the models studied (the Ornstein-Uhlenbeck process) and obtain the long-run discount rate for all these countries. The long-run discount obtained for stable countries (countries that have not suffered periods of destabilizing inflation) supports the low discounting rate proposed by Stern (2006) over higher rates that have been advocated by others (Nordhaus, 2007a,b)

A correlated stochastic volatility model measuring leverage and other stylized facts

We present a stochastic volatility market model where volatility is correlated with return and is represented by an Ornstein-Uhlenbeck process. With this model we exactly measure the leverage effect and other stylized facts, such as mean reversion, leptokurtosis and negative skewness. We also obtain a close analytical expression for the characteristic function and study the heavy tails of the probability distribution.Comment: 22 pages, 2 figures and 2 table

Three-dimensional telegrapher's equation and its fractional generalization

We derive the three-dimensional telegrapher's equation out of a random walk model. The model is a threedimensional version of the multistate random walk where the number of different states form a continuum representing the spatial directions that the walker can take. We set the general equations and solve them for isotropic and uniform walks which finally allows us to obtain the telegrapher's equation in three dimensions. We generalize the isotropic model and the telegrapher's equation to include fractional anomalous transport in three dimensions

A comparison between several correlated stochastic volatility models

We compare the most common SV models such as the Ornstein-Uhlenbeck (OU), the Heston and the exponential OU (expOU) models. We try to decide which is the most appropriate one by studying their volatility autocorrelation and leverage effect, and thus outline the limitations of each model. We add empirical research on market indices confirming the universality of the leverage and volatility correlations.Comment: 4 pages, 2 figures, APFA 4 conferences contribution (13-15 november, 2003, Warsaw

Multiple time scales in volatility and leverage correlations: An stochastic volatility model

Financial time series exhibit two different type of non linear correlations: (i) volatility autocorrelations that have a very long range memory, on the order of years, and (ii) asymmetric return-volatility (or `leverage') correlations that are much shorter ranged. Different stochastic volatility models have been proposed in the past to account for both these correlations. However, in these models, the decay of the correlations is exponential, with a single time scale for both the volatility and the leverage correlations, at variance with observations. We extend the linear Ornstein-Uhlenbeck stochastic volatility model by assuming that the mean reverting level is itself random. We find that the resulting three-dimensional diffusion process can account for different correlation time scales. We show that the results are in good agreement with a century of the Dow Jones index daily returns (1900-2000), with the exception of crash days.Comment: 19 pages, 5 figure