Review of Derivatives Research

, Volume 14, Issue 1, pp 37–65

# A binomial approximation for two-state Markovian HJM models

Article

## Abstract

This article develops a lattice algorithm for pricing interest rate derivatives under the Heath et al. (Econometrica 60:77–105, 1992) paradigm when the volatility structure of forward rates obeys the Ritchken and Sankarasubramanian (Math Financ 5:55–72) condition. In such a framework, the entire term structure of the interest rate may be represented using a two-dimensional Markov process, where one state variable is the spot rate and the other is an accrued variance statistic. Unlike in the usual approach based on the Nelson-Ramaswamy (Rev Financ Stud 3:393–430) transformation, we directly discretize the heteroskedastic spot rate process by a recombining binomial tree. Further, we reduce the computational cost of the pricing problem by associating with each node of the lattice a fixed number of accrued variance values computed on a subset of paths reaching that node. A backward induction scheme coupled with linear interpolation is used to evaluate interest rate contingent claims.

## Keywords

Interest rate options Contingent claims Binomial algorithms Discrete-time models

C63 G12

## Preview

Unable to display preview. Download preview PDF.

## References

1. Aingworth, D., Motwani, R., & Oldham, J. D. (2000). Accurate approximations for Asian options. In Proceedings of the 11th annual ACM-SIAM symposium on discrete algorithms.Google Scholar
2. Caverhill A. (1994) When is the short rate Markovian?. Mathematical Finance 4: 305–312
3. Chalasani P., Jha S., Egriboyun F., Varikooty A. (1999) A refined binomial lattice for pricing American Asian options. Review of Derivatives Research 3: 85–105
4. Costabile M., Massabò I. (2010) A simplified approach to approximate diffusion processes widely used in finance. Journal of Derivatives, 17(3): 65–85
5. Cox J. C., Rubinstein M. (1985) Option markets. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar
6. Forsyth P. A., Vetzal K. R., Zvan R. (2002) Convergence of numerical methods for valuing path-dependent options using interpolation. Review of Derivatives Research 5: 273–314
7. Heath D., Jarrow R., Morton A. (1990) Bond pricing and the term structure of interest rates: A discrete time approximation. Journal of Financial and Quantitative Analysis 25: 419–440
8. Heath D., Jarrow R., Morton A. (1992) Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation. Econometrica 60: 77–105
9. Hull J., White A. (1993) Efficient procedures for valuing European and American path-dependent options. Journal of Derivatives 6: 21–31
10. Hull J., White A. (1993) Bond option pricing on a model for the evolution of bond prices. Advances in Options and Futures Research 6: 1–13Google Scholar
11. Jiang L., Dai M. (2004) Convergence of binomial methods for European/American path-dependent options. SIAM Journal on Numerical Analysis 42(3): 1094–1109
12. Kramin M., Kramin T., Young S., Dharan V. (2005) A simple induction approach and an efficient trinomial lattice for multi-state variable interest rate derivatives models. Review of Quantitative Finance and Accounting 24: 199–226
13. Li A., Ritchken P., Sankarasubramanian L. (1995) Lattice models for pricing American interest rate claims. Journal of Finance 2: 719–737
14. Lin J., Ritchken P. (2006) On pricing derivatives in the presence of auxiliary state variables. Journal of Derivatives 14: 29–46
15. Nelson D., Ramaswamy K. (1990) Simple binomial processes as diffusion approximations in financial models. Review of Financial Studies 3: 393–430
16. Ritchken P., Sankarasubramanian L. (1995) Volatility structures of forward rates and the dynamics of the term structure. Mathematical Finance 5: 55–72
17. Tian Y. (1994) A reexamination of lattice procedures for interest rate-contingent claims. Advances in Futures and Options Research 7: 87–111Google Scholar