Jan 13, 2006 ... Interest Rate Models ... The interest rate market. .... J. James and N. Webber,
Interest Rate Modelling, Wiley 2000 [Emphasis on computational ...
Mark H.A.Davis
Stochastic Differential Equations and Interest Rate Models MSc Course in Mathematics and Finance Imperial College London January 13, 2006
Department of Mathematics Imperial College London South Kensington Campus London SW7 2AZ
Contents
1
The interest rate market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fixed-income securities and market conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Market-traded contracts and the yield curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 1
2
Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Existence and uniqueness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Weak and strong solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The gaussian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 3 3 3
3
Econometric analysis of the Yield Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Principal components analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 5
4
Short Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 General approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Futures markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Hull-White model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The CIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 7 7 7 7
5
The HJM approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 HJM and the drift condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Finite-dimensional realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Short-rate models as a special case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9 9 9 9
6
Market models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Changes of num´eraire and the forward measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Outline of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The formal construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Application to the Libor Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Algorithm for constructing the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Deflators and Forward Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Deflators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 The swap market model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 11 11 12 13 14 14 15 15
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
References
1. T. Bj¨ ork, Arbitrage Theory in Continuous Time, 2nd ed. Oxford University Press 2004 [Well written general account.] 2. A. Brace, D. G¸atarek and M. Musiela, The market model of interest rate dynamics, Mathematical Finance 7 (1997) 127-155 [Introduced Libor Market Model] 3. D. Brigo and F. Mercurio, Interest Rate Models: Theory and Practice, Springer 2001 [Main course textbook; 2nd edition available soon!] 4. A. Cairns, Interest Rate Models: an Introduction, Princeton University Press 2004. [Good alternative reference.] 5. J. Cox, J. Ingersoll and S. Ross, A theory of the term structure of interest rates, Econometrica 53 (1985) 385-408 [Introduced CIR model] 6. D. Duffie and R. Kan, A yield-factor model of interest rates, Mathematical Finance 6 (1996) 379-406 [Introduced ‘afffine’ models.] 7. D. Heath, R.A. Jarrow and A. Morton, Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation, Econometrica 60 (1992) 77-105 [Origin of ‘HJM’.] 8. J. Hull, Options, Futures and Other Derivatives, 6th ed. Prentice Hall 2005 [or earlier editions. The practitioners’ bible.] 9. P.J. Hunt and J. Kennedy, Financial derivatives in Theory and Practice, Wiley 2000 [Good ‘market’ coverage in 2nd half; heavy maths in first half.] 10. J. James and N. Webber, Interest Rate Modelling, Wiley 2000 [Emphasis on computational finance.] 11. F. Jamshidian, Libor and swap market models and measures, Finance and Stochastics 1 (1997) 293-330 [Introduces swap market model.] 12. I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed. Springer 1991 [ A standard reference. Somewhat more advanced than [15].] 13. R.C. Merton, Continuous-time Finance, Blackwell 1992 [Collection ofclassi papers.] 14. M. Musiela and M. Rutkowski, Martingale Methods in Financial Modelling, 2nd ed. Springer 2004 [Encyclopaedic overage at deteiled mathematical level.] 15. B. Øksendal, Stochastic Differential Equations, 6th ed. Springer 2003 [Good reference for SDEs etc.]
