# An Improved Binomial Method using Cell Averages for Option Pricing

• Received : 2011.01.19
• Accepted : 2011.04.29
• Published : 2011.06.01
• 46 8

#### Abstract

We present an improved binomial method for pricing financial deriva-tives by using cell averages. After non-overlapping cells are introduced around each node in the binomial tree, the proposed method calculates cell averages of payoffs at expiry and then performs the backward valuation process. The price of the derivative and its hedging parameters such as Greeks on the valuation date are then computed using the compact scheme and Richardson extrapolation. The simulation results for European and American barrier options show that the pro-posed method gives much more accurate price and Greeks than other recent lattice methods with less computational effort.

#### Keywords

Option Pricing;Binomial Method;Barrier Options;Cell Averages

#### References

1. Amin, K. and Khanna, A. (1994), Convergence of american option values from discrete-to continuous-time financial models, Mathematical Finance, 4, 289-304. https://doi.org/10.1111/j.1467-9965.1994.tb00059.x
2. Boyle, P. A. (1988), lattice framework for option pricing with two state variables, Journal of Financial and Quantitative Analysis, 23(1), 1-12. https://doi.org/10.2307/2331019
3. Boyle, P., Evnine, J., and Gibbs, S. (1989), Numerical evaluation of multivariate contingent claims, The Review of Financial Studies, 2(2), 241-250. https://doi.org/10.1093/rfs/2.2.241
4. Boyle, P. P. and Lau, S. H. (1994), Bumping up against the barrier with the binomial method, Journal of Derivatives, 1, 6-14.
5. Broadie, M. and Detemple, J. (1996), American option valuation: new bounds, approxi-mations, and a comparison of existing methods. Review of Financial Studies, 9, 1211-1250. https://doi.org/10.1093/rfs/9.4.1211
6. Cheuk, T. H. F. and Vorst, T. C. F. (1996), Complex barrier options, Journal of Derivatives, 4, 8-22. https://doi.org/10.3905/jod.1996.407958
7. Clewlow, L. and Strickland, C. (1998), Implementing Derivatives Models, John Wiley & Sons, Chichester, UK.
8. Cox, J., Ross, S., and Rubinstein, M. (1979), Option pricing: A simplified approach, Journal of Financial Economics, 7, 229-263. https://doi.org/10.1016/0304-405X(79)90015-1
9. Derman, E., Kani, I., Ergener, D., and Bardhan, I. (1995), Enhanced numerical methods for options with barriers, Financial Analysis Journal, Nov-Dec, 65-74.
10. Gaudenzi, M. and Pressacco, F. (2003), An efficient binomial method for pricing american options, Decisions in Economics and finance, 26, 1-17. https://doi.org/10.1007/s102030300000
11. Haug, E. G. (1997), The complete guide to option pricing formulas, McGraw-Hill.
12. Higham, D. J. (2004), An introduction to financial option valuation, Cambridge University Press.
13. Kamrad, B. and Ritchken, P. (1991), Multinomial approximating models for options with k state variables, Management science, 37(12), 1640-1652. https://doi.org/10.1287/mnsc.37.12.1640
14. Oksendal, B. (1998), Stochastic differential equations, Springer, Berlin.
15. Kwok, Y. K. (1998), Mathematical models of financial derivatives, Springer-Verlag, Singapore.
16. Lele, S. K. (1992), Compact finite difference schemes with spectral-like resolution, Journal of Computational Physics, 103, 16-42. https://doi.org/10.1016/0021-9991(92)90324-R
17. Lyuu, Y. D. (2002), Financial Engineering and Computation, Cambridge.
18. Reimer, E. and Rubinstein, M. (1991), Unscrambling the binary code, Risk Magazine, 4.
19. Richardson, L. (1927), The deferred approach to the limit, Philosophical Transactions of the Royal Society of London, Series A, 226, 299-349. https://doi.org/10.1098/rsta.1927.0008
20. Ritchken, P. (1995), On pricing barrier options, Journal of Derivatives, 3, 19-28. https://doi.org/10.3905/jod.1995.407939
21. Wilmott, P., Howison, S., and Dewynne, J. (1995), The mathematics of financial derivatives, Cambridge University Press.

#### Acknowledgement

Supported by : NRF, Kyungwon University