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We derive closed form solutions to the discounted optimal stopping problems related to the pricing of the perpetual American standard put and call options in an extension of the Black–Merton–Scholes model with piecewise-constant dividend and volatility rates. The method of proof is based on the reduction of the initial optimal stopping problems to the associated free-boundary problems and the subsequent martingale verification using a local time-space formula. We present explicit algorithms to determine the constant hitting thresholds for the underlying asset price process, which provide the optimal exercise boundaries for the options.

In this paper we analyse the properties of hierarchical Archimedean copulas. This class is a generalisation of the Archimedean copulas and allows for general non-exchangeable dependency structures. We show that the structure of the copula can be uniquely recovered from all bivariate margins. We derive the distribution of the copula values, which is particularly useful for tests and constructing confidence intervals. Furthermore, we analyse dependence orderings, multivariate dependence measures, and extreme value copulas. We pay special attention to the tail dependencies and derive several tail dependence indices for general hierarchical Archimedean copulas.

Consider the regression problem with a response variable Y and with a d -dimensional feature vector X . For the regression function m(x)  = E{ Y|X  =  x }, this paper investigates methods for estimating the density of the residual Y  −  m(X) from independent and identically distributed data. If the density is twice differentiable and has compact support then we bound the rate of convergence of the kernel density estimate. It turns out that for d  ≤ 3 and for partitioning regression estimates, the regression estimation error has no influence on the rate of convergence of the density estimate.

In this paper we consider the empirical process of the errors appearing in a generalized autoregressive conditional heteroskedastic with stochastic mean (GARCH-SM) model. Various functional tests of conditional symmetry can be built on the basis of the limiting distribution of this process. In particular, a Cramér–von Mises-type test is considered. Its theoretical power is studied under fixed and local alternatives. Using the Karhunen–Loève decomposition, the limiting law of the latter is approximated by a chi-square distribution under both null and alternative hypotheses. The local power under a sequence of alternatives is also computed.