## On smoothing of the Crank-Nicolson scheme and higher order schemes for pricing barrier options.(English)Zbl 1137.91477

Summary: 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 $$(0,2m-1)$$ subdiagonal Padé schemes.

### MSC:

 91G60 Numerical methods (including Monte Carlo methods) 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65Y05 Parallel numerical computation 65Y20 Complexity and performance of numerical algorithms 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35K20 Initial-boundary value problems for second-order parabolic equations 91G20 Derivative securities (option pricing, hedging, etc.)
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