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

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