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Barrier options are considered for Asian options using a differential equation method. Solutions are obtained in the form of Fourier series for barriers which expand or contract as they approach maturity. Rigorous bounds are obtained. It is shown that by differentiating with respect to a parameter, solutions for more general payoffs can be obtained. [ABSTRACT FROM AUTHOR]

The present paper addresses the problem of computing implied volatilities of options written on a domestic asset based on implied volatilities of options on the same asset expressed in a foreign currency and the exchange rate. It proposes an original method together with explicit formulae to compute the at-the-money implied volatility, the smile's skew, convexity, and term structure for short maturities. The method is completely free of any model specification or Markov assumption; it only assumes that jumps are not present. We also investigate how the method performs on the particular example of the currency triplet dollar, euro, yen. We find a very satisfactory agreement between our formulae and the market at one week and one month maturities. [ABSTRACT FROM AUTHOR]