Bonds futures: Delta? No gamma!

Henrard, Marc (2006): Bonds futures: Delta? No gamma!

Abstract

Bond futures are liquid but complex instruments. Here they are analysed in a one-factor Gaussian HJM model. The in-the-model delta and out-of-the-model delta and gamma are studied. An explicit formula is provided for in-the-model delta. The out-of-the-model delta and gamma are equivalent to partial derivatives with respect to discount factors. In particular cases the derivative can not be obtained by standard techniques. The same situations lead to cases where the gammas (second order partial derivatives) do not exists.

• Item Type: MPRA Paper
• Institution: BIS
• Language: English
• Keywords: Bond future; delivery option; delta; gamma; HJM gaussian model; in-the-model; out-of-the-model
• Subjects: G - Financial Economics > G1 - General Financial Markets > G13 - Contingent Pricing ; Futures Pricing; E - Macroeconomics and Monetary Economics > E4 - Money and Interest Rates > E43 - Interest Rates: Determination, Term Structure, and Effects

References

Brody, D.~C. and Hughston, L.~P. (2004). Chaos and coherence: a new framework for interest-rate modelling. Proc. R. Soc. Lond. A., 460:85--110.

Carverhill, A. (1994). When is the short rate Markovian. Mathematical Finance, 4(4):305--312.

Heath, D., Jarrow, R., and Morton, A. (1992). Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica, 60(1):77--105.

Henrard, M. (2003). Explicit bond option and swaption formula in Heath-Jarrow-Morton one-factor model. International Journal of Theoretical and Applied Finance, 6(1):57--72.

Henrard, M. (2006a). Bonds futures and their options: more than the cheapest-to-deliver; quality option and marginning. Technical report, SSRN.

Henrard, M. (2006b). A semi-explicit approach to Canary swaptions in HJM one-factor model. Applied Mathematical Finance, 13(1):1--18.

Hunt, P.~J. and Kennedy, J.~E. (2004). Financial Derivatives in Theory and Practice. Wiley series in probability and statistics. Wiley, second edition.

Lamberton, D. and Lapeyre, B. (2000). Introduction to stochastic calculus applied to finance. Capman \& Hall / CRC.

Nunes, J. and de~Oliveira, L. (2004). Quasi-analytical multi-factor valuation of treasury bond futures with and embedded quality option. Technical Report 2493, EFA 2004 Maastricht Meeting.