On smoothing of the Crank–Nicolson scheme and higher order schemes for pricing barrier options

• Published in: Journal of computational and applied mathematics Vol. 204; no. 1; pp. 144 - 158
• Main Authors: Wade, B.A., Khaliq, A.Q.M., Yousuf, M., Vigo-Aguiar, J., Deininger, R.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.07.2007; Elsevier
• Subjects: Barrier options; Black–Scholes PDE; Crank–Nicolson scheme; Exact sciences and technology; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Padé schemes; Partial differential equations; Sciences and techniques of general use; Smoothing; 65M15; Smoothing; Black–Scholes PDE; 65Y20; Padé schemes; 65M12; 65Y05; Barrier options; Crank–Nicolson scheme; Damping; Discontinuous; Parameter estimation; Scheme; Black Scholes model; Crank Nicolson method; Barrier; Partial differential equation; Pricing; Oscillation; Evolution; Discontinuity; Smoothing methods; Optimal convergence; Change; Domains; Numerical analysis; 65M12; 65M15; 65Y05; 65Y20; Applied mathematics; Price; Derivative; Strategy; Optimal approximation; Barrier options; Black-Scholes PDE; Crank-Nicolson scheme; Padé schemes; Smoothing
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2006.04.034

Most option pricing problems have nonsmooth payoffs or discontinuous derivatives at the exercise price. Discrete barrier options have not only nonsmooth payoffs but also time dependent discontinuities. In pricing barrier options, certain aspects are triggered if the asset price becomes too high or too low. Standard smoothing schemes used to solve problems with nonsmooth payoff do not work well for discrete barrier options because of discontinuities introduced in the time domain when each barrier is applied. Moreover, these unwanted oscillations become worse when estimating the hedging parameters, e.g., Delta and Gamma. We have an improved smoothing strategy for the Crank–Nicolson method which is unique in achieving optimal order convergence for barrier option problems. Numerical experiments are discussed for one asset and two asset problems. Time evolution graphs are obtained for one asset problems to show how option prices change with respect to time. This smoothing strategy is then extended to higher order methods using diagonal ( m , m ) —Padé main schemes under a smoothing strategy of using as damping schemes the ( 0 , 2 m - 1 ) subdiagonal Padé schemes.