Using an expansion of the transition density function of a one-dimensional time inhomogeneous diffusion, we obtain the first-and second-order terms in the short time asymptotics of European call option prices. The method described can be generalized to any order. We then use these option prices approximations to calculate the firstand second-order deviation of the implied volatility from its leading value and obtain approximations which we numerically demonstrate to be highly accurate.

In this paper we investigate the possible values of basket options. Instead of postulating a model and pricing the basket Option using that model, we consider the set of all models which are consistent with the observed prices of vanilla options, and, within this class, find the model for which the price of the basket option is largest. This price is an upper bound on the prices of the basket option which are consistent with no-arbitrage. In the absence of additional assumptions it is the lowest upper bound on the price of the basket option. Associated with the bound is a simple super-replicating strategy involving trading in the individual calls

The theory of ballooning modes, which are modes localized to a particular magnetic field line, is extended to corLfiguratlons relevant to space plasmas. Included are the effects of gravity and rotation and, in particular, boundary effects on magnetic field lines which intersect the plasma boundary. Three types of boundary conditions are considered, corresponding to perfectly conducting, conducting, and insulating boundaries. The interchange instability is also examined and is shown to be a special case of the ballooning instability. bution of mass in planetary magnetospheres, the subject of the interchange instability was extensively discussed, even recently [Cheng, 1985; Rogers and Sonnerup, 1986; Southwood and Kivelson, 1987, 1989]. As we later show in this article, the interchange instability is but a special case of the ballooning instability where the mode does not perturb the equilibrium magnetic field. It is, in fact, possible to derive the interchange stability criterion directly from the ballooning stability criterion. It is therefore to be expected that ballooning modes are as relevant and useful for the understanding of plasma circulation processes as are the interchange modes. The extension of ballooning modes theory to space plasmas presents us with new tasks on two different levels. On the technical level, it is desirable (and not too difficult) to incorporate additional effects which are significant in the space environment, such as the effects of gravity and of plasma rotation [Lakhina et al., 1990a, b]. Indeed, for the cold Iogenic plasma torus in the rapidly cotorating Jovian magnetosphere, these two effects may be more important than the pressure. A different level of work, requiring a more fundamental thinking, is called for in order to take 1513 E. Hameiri,

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