Exploratory semiparametric analysis of two-dimensional diffusions in finance

We examine bivariate extensions of Aït-Sahalia’s approach to the estimation of univariate diffusions. Our message is that extending his idea to a bivariate setting is not straightforward. In higher dimensions, as opposed to the univariate case, the elements of the Itô and Fokker-Planck representations do not coincide; and, even imposing sensible assumptions on the marginal drifts and volatilities is not sufficient to obtain direct generalisations. We develop exploratory estimation and testing procedures, by parametrizing the drifts of both component processes and setting restrictions on the terms of either the Itô or the Fokker-Planck covariance matrices. This may lead to highly nonlinear ordinary differential equations, where the definition of boundary conditions is crucial. For the methods developed, the Fokker-Planck representation seems more tractable than the Itô’s. Questions for further research include the design of regularity conditions on the time series dependence in the data, the kernels actually used and the bandwidths, to obtain asymptotic properties for the estimators proposed. A particular case seems promising: “causal bivariate models” in which only one of the diffusions contributes to the volatility of the other. Hedging strategies which estimate separately the univariate diffusions at stake may thus be improved.

On Ponzi schemes in infinite horizon collateralized economies with default penalties

Araujo, Páscoa and Torres-Martínez (2002) showed that, without imposing any debt constraint, Ponzi schemes are ruled out in infinite horizon economies with limited commitment when collateral is the only mechanism that partially secures loans. Páscoa and Seghir (2009) presented two examples in which they argued that Ponzi schemes may reappear if, additionally to the seizure of the collateral, there are sufficiently harsh default penalties assessed (directly in terms of utility) against the defaulters. Moreover, they claimed that if default penalties are moderate then Ponzi schemes are ruled out and existence of a competitive equilibrium is restored. This paper questions the validity of the claims made in Páscoa and Seghir (2009). First, we show that it is not true that harsh default penalties lead to Ponzi schemes in the examples they have proposed. A competitive equilibrium with no trade can be supported due to unduly pessimistic expectations on asset deliveries. We subsequently refine the equilibrium concept in the spirit of Dubey, Geanakoplos and Shubik (2005) in order to rule out spurious inactivity on asset markets due to irrational expectations. Our second contribution is to provide a specific example of an economy with moderate default penalties in which Ponzi schemes reappear when overpessimistic beliefs on asset deliveries are ruled out. Our finding shows that, contrary to what is claimed by Páscoa and Seghir (2009), moderate default penalties do not always prevent agents to run a Ponzi scheme.

Duality with time-changed Lévy processes

In this paper we study the pricing problem of derivatives written in terms of a two dimensional time{changed L¶evy processes. Then, we examine an existing relation between prices of put and call options, of both the European and the American type. This relation is called put{call duality. It includes as a particular case, the relation known as put{call symmetry. Necessary and su±cient conditions for put{call symmetry to hold are shown, in terms of the triplet of local charac- teristic of the Time{changed L¶evy process. In this way we extend the results obtained in Fajardo and Mordecki (2004) to the case of time{changed Lévy processes.