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J Eng Math (2007) 59:373384 DOI 10.1007/s10665-007-9176-0

Summary: J Eng Math (2007) 59:373­384
DOI 10.1007/s10665-007-9176-0
Pricing financial claims contingent upon an underlying asset
monitored at discrete times
Ross Green · I. David Abrahams ·
Gianluca Fusai
Received: 29 June 2007 / Accepted: 11 July 2007 / Published online: 8 August 2007
© Springer Science+Business Media B.V. 2007
Abstract Exotic option contracts typically specify a contingency upon an underlying asset price monitored at a
discrete set of times. Yet, techniques used to price such options routinely assume continuous monitoring leading
to often substantial price discrepancies. A brief review of relevant option-pricing methods is presented. The pric-
ing problem is transformed into one of Wiener­Hopf type using a z-transform in time and a Fourier transform in
the logarithm of asset prices. The Wiener­Hopf technique is used to obtain probabilistic identities for the related
random walks killed by an absorbing boundary. An accurate and efficient approximation is obtained using Padé
approximants and an approximate inverse z-transform based on the trapezoidal rule. For simplicity, European bar-
rier options in a Gaussian Black­Scholes framework are used to exemplify the technique (for which exact analytic
expressions are obtained). Extensions to different option contracts and options driven by other Lévy processes are
Keywords Discrete monitoring · Fourier transform · Option pricing · Padé approximants · Wiener­Hopf
technique · z-transform


Source: Abrahams, I. David - Department of Mathematics, University of Manchester


Collections: Mathematics