A fourth-order smoothing scheme for pricing barrier options under stochastic volatility

Abstract

A discontinuity is introduced each time a barrier is applied, and standard smoothing schemes do not work well when used to solve problems with non-smooth payoff. An improved smoothing strategy was introduced in [M. Yousuf, Efficient Smoothing of Crank–Nicolson Method for Pricing Barrier Options Under Stochastic Volatility, Published Online: Jul 10, 2008, DOI: 10.1002/pamm.200700249, 1081101–1081102.] for smoothing A-stable Cranck–Nicolson scheme at each time a barrier is applied. In the present work, we are extending the results to develop a fourth order smoothing scheme for pricing barrier options under stochastic volatility. Partial differential equation approach is utilized for the valuation of complex option pricing models under stochastic volatility which brings major mathematical and computational challenges for estimation of stability of the estimates. Efficient parallel version of the scheme is constructed using splitting technique. Convergence table and graphs are given to demonstrate the performance of the smoothing scheme.

Keywords:

• barrier options
• padé approximations
• smoothing schemes
• non-smooth data
• positivity

2000 AMS Subject Classifications :

• 35K57
• 78M20
• 46N20
• 34K20
• 35B65
• 65M20

Acknowledgements

This work is supported by the Junior Faculty Project # JF 070005, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia.