Applied Mathematics & Information Sciences

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The foreign exchange markets felt the necessity of using contracts written on harmonic averages. They are attractive because are cheaper than the contracts written on the arithmetic average, and make more financial sense than contracts written on geometric averages. The goal of this paper is to consider Asian options and future contracts on harmonic averages of stock values. Since the harmonic average of a set of lognormal random variables does not have an explicit representation, a close-form pricing formula for options and futures is missing. However, we obtain the value of a future contract expressed as an infinite series and provide an approximative formula for it. In the absence of a closed form formula for the value of a call, we obtain an approximation formula for the case when the stock volatility s is small. This is done by using a variable reduction technique and applying a convolution with the heat kernel of the underlying operator.