# AN OPERATOR SPLITTING METHOD FOR PRICING THE ELS OPTION

• Accepted : 2010.08.31
• Published : 2010.09.25

#### Abstract

This paper presents the numerical valuation of the two-asset step-down equitylinked securities (ELS) option by using the operator-splitting method (OSM). The ELS is one of the most popular financial options. The value of ELS option can be modeled by a modified Black-Scholes partial differential equation. However, regardless of whether there is a closedform solution, it is difficult and not efficient to evaluate the solution because such a solution would be represented by multiple integrations. Thus, a fast and accurate numerical algorithm is needed to value the price of the ELS option. This paper uses a finite difference method to discretize the governing equation and applies the OSM to solve the resulting discrete equations. The OSM is very robust and accurate in evaluating finite difference discretizations. We provide a detailed numerical algorithm and computational results showing the performance of the method for two underlying asset option pricing problems such as cash-or-nothing and stepdown ELS. Final option value of two-asset step-down ELS is obtained by a weighted average value using probability which is estimated by performing a MC simulation.

#### References

1. Y. Achdou and O. Pironneau, Computational methods for option pricing, SIAM, Philadelphia, 2005.
2. F. Black and M. Sholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (3) (1973), 637-659. https://doi.org/10.1086/260062
3. D.J. Duffy, Finite difference methods in financial engineering : a partial differential equation approach, John Wiley and Sons, New York, 2006.
4. E.G. Haug, The complete guide to option pricing formulas, McGraw-Hill, New York 1997.
5. S. Ikonen and J. Toivanen, Operator splitting methods for American option pricing, Applied Mathematics Letters, 17 (2004), 809-814. https://doi.org/10.1016/j.aml.2004.06.010
6. R. Kangro and R. Nicolaides, Far field boundary conditions for Black-Scholes equations, SIAM Journal on Numerical Analysis , 38 (4) (2000), 1357-1368. https://doi.org/10.1137/S0036142999355921
7. K.S. Lee, Y.E. Gwong, and J.H. Shin, Deravatives modeling I: Using MATLAB$^{\circledR}$, A-Jin, Seoul, 2008.
8. C.W.Oosterlee, On multigrid for linear complementarity problems with application to American-style options, Electronic Transactions on Numerical Analysis, 15 (2003), 165-185.
9. J. Persson and L. von Sydow, Pricing European multi-asset options using a space-time adaptive FD-method, Computing and Visualization in Science, 10 (2007), 173-183. https://doi.org/10.1007/s00791-007-0072-y
10. R. Seydel, Tools for computational finance, Springer Verlag, Berlin, 2003.
11. D. Tavella and C. Randall, Pricing financial instruments:the finite difference method, John Wiley and Sons, New York, 2000.
12. J. Topper, Financial engineering with finite elements, John Wiley and Sons, New York, 2005.
13. P. Wilmott, J. Dewynne and S. Howison, Option pricing : mathematical models and computation, Oxford Financial Press, Oxford, 1993.
14. R. Zvan, K. R. Vetzal and P.A. Forsyth, PDE methods for pricing barrier options, Journal of Economic Dynamics and Control, 24 (2000), 1563-1590. https://doi.org/10.1016/S0165-1889(00)00002-6