Unexpectedly accurate and parsimonious approximations for balls in R d and related functions are given using half-spaces. Instead of a polytope (an intersection of half-spaces) which would require exponentially many half-spaces (of order (2 ) pairs of indicators of half-spaces and threshold a linear combination of them. In neural network terminology, we are using a single hidden layer perceptron approximation to the indicator of a ball. A special role in the analysis is played by probabilistic methods and approximation of Gaussian functions. The result is then applied to functions that have variation V f with respect to a class of ellipsoids. Two hidden layer feedforward sigmoidal neural nets are used to approximate such functions. The approximation error is shown to be bounded by a constant times V f ÂT 1Â2 1 +V f dÂT 1Â4 2 , where T 1 is the number of nodes in the outer layer and T 2 is the number of nodes in the inner layer of the approximation f T 1 , T 2 .

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Accurate and parsimonious approximations for indicator functions of d-dimensional balls and related functions are given using level sets associated with the thresholding of a linear combination of ramp sigmoid activation functions. In neural network terminology, we are using a single-hidden-layer perceptron network implementing the ramp sigmoid activation function to approximate the indicator of a ball. In order to have a relative accuracy , we use T = c(d 2 / 2 ) ramp sigmoids, a result comparable to that of Cheang and Barron (2000) [4], where unit step activation functions are used instead. The result is then applied to functions that have variation V f with respect to a class of ellipsoids. Two-hidden-layer feedforward neural nets with ramp sigmoid activation functions are used to approximate such functions. The approximation error is shown to be bounded by a constant times V f /T 1 2 1 + V f d/T 1 4 2 , where T 1 is the number of nodes in the outer layer and T 2 is the number of nodes in the inner layer of the approximation f T 1 ,T 2 .

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In this article, we provide representations of European and American exchange option prices under stochastic volatility jump-diffusion (SVJD) dynamics following models by Merton (1976), Heston (1993), and Bates (1996. A Radon-Nikodým derivative process is also introduced to facilitate the shift from the objective market measure to other equivalent probability measures, including the equivalent martingale measure. Under the equivalent martingale measure, we derive the integropartial differential equation that characterizes the exchange option prices. We also derive representations of the European exchange option price using the change-ofnuméraire technique proposed by Geman, El Karoui, and Rochet (1995) and the Fourier inversion formula derived by Caldana and Fusai (2013), and show that these two representations are comparable. Lastly, we show that the American exchange option price can be decomposed into the price of the European exchange option and an early exercise premium.