From Shannon's Sampling to Optimal Stopping in Finance

Abstract: The Whittaker-Shannon-Kotel¡änikov sampling theorem and the Paley-Wiener theorem state that an entire function of exponential type (band-limited function) can be reconstructed exactly from its values at discrete sampling points through the cardinal series. For functions that are analytic in a horizontal strip, the cardinal series is still highly accurate, with the approximation error converging to zero exponentially in the length of the sampling interval. In option valuation applications in finance, the characteristic function of the log return process of an asset is naturally analytic. We explore such analyticity and propose Hilbert transform based methods for the valuation of American options. Millions of option contracts are traded daily on Chicago Board Option Exchange and Chicago Mercantile Exchange. Most of exchange traded options are of American style. The valuation of an American option reduces to an optimal stopping problem. We illustrate the effectiveness of the Hilbert transform approach for fast and accurate valuation of American options.

Bio: Liming Feng is an associate professor in the Department of Industrial and Enterprise Systems Engineering at the University of Illinois at Urbana-Champaign. He obtained his Ph.D. in Industrial Engineering and Management Sciences from Northwestern University in 2006. His main research interests are in quantitative finance. He is interested in developing theory and efficient computational methods for solving various quantitative finance problems. He helped establish the Master of Science in Financial Engineering program at the University of Illinois at Urbana-Champaign.