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HJM: A Unified Approach to Dynamic Models for Fixed Income, Credit and Equity Markets

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Paris-Princeton Lectures on Mathematical Finance 2004

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1919))

The purpose of this paper is to highlight some of the key elements of the HJM approach as originally introduced in the framework of fixed income market models, to explain how the very same philosophy was implemented in the case of credit portfolio derivatives and to show how it can be extended to and used in the case of equity market models. In each case we show how the HJM approach naturally yields a consistency condition and a no-arbitrage condition in the spirit of the original work of Heath, Jarrow and Morton. Even though the actual computations and the derivation of the drift condition in the case of equity models seems to be new, the paper is intended as a survey of existing results, and as such, it is mostly pedagogical in nature.

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References

  1. . Y. Achdou and O. Pironneau, Computational methods for option pricing, SIAM.

    Google Scholar 

  2. T. Aven, A theorem for determining the compensator of a counting process, Scandinavian Journal of Statistics 12(1) (1985), 69-72.

    MATH  MathSciNet  Google Scholar 

  3. . N. Bennani, The forward loss model: a dynamic term structure approach for the pricing of portfolios of credit derivatives, Tech. report, November 2005.

    Google Scholar 

  4. D. Blackwell, Equivalent comparisons of experiments, Annals of Mathematical Statistics 24 (1953),265-272.

    Article  MATH  MathSciNet  Google Scholar 

  5. . H. Buehler, Consistent variance curve models, Tech. report, 2007.

    Google Scholar 

  6. . _____, Expensive martingales, Quantitative Finance (2007), (to appear).

    Google Scholar 

  7. . R. Carmona and S. Nadtochiy, HJM dynamics for equity models, Tech. report, Princeton University, 2007.

    Google Scholar 

  8. R. Carmona and M. Tehranchi, A characterization of hedging portfolios for interest rate contingent claims, The Annals of Applied Probability 14 (2004), 1267-1294.

    Article  MATH  MathSciNet  Google Scholar 

  9. . _____, Interest rate models: an infinite dimensional stochastic analysis perspective, Springer Verlag, 2006.

    Google Scholar 

  10. . P. Carr and D. Madan, Toward a theory of volatility trading, vol. Volatility, Risk Publications, pp. 417-427.

    Google Scholar 

  11. . A. Carter and J.P. Fouque, Review of SemiParametric Modeling of Implied Volatility by matthisas fengler, SIAM Reviews (2007), (to appear).

    Google Scholar 

  12. R. Cont and J. da Fonseca, Dynamics of implied volatility surfaces, Quantitative Finance 2 (2002),45-60.

    Article  MathSciNet  Google Scholar 

  13. L. Cousot, Necessary and sufficient conditions for no-static arbitrage among European calls, Tech. report, Courant Institute, New York University, October 2004.

    Google Scholar 

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

    Article  MATH  Google Scholar 

  15. . M.H.A. Davis and D.G. Hobson, The range of traded option prices, Tech. report, Princeton University, July 2005.

    Google Scholar 

  16. E. Derman and I. Kani, The volatility smile and its implied tree, Tech. report, Quantitative Research Notes, Goldman Sachs, 1994.

    Google Scholar 

  17. . _____, Stochastic implied trees: Arbitrage pricing with stochastic term and strike structure of volatility, International Journal of Theoretical and Applied Finance 1 (1998), 61-110.

    Google Scholar 

  18. . D. Duffie and K. Singleton, Credit risk, Princeton University Press, 2003.

    Google Scholar 

  19. B. Dupire, Pricing with a smile, Risk 7 (1994), 32-39.

    Google Scholar 

  20. . V. Durrleman, From implied to spot volatility, Tech. report, Stanford University, April 2005.

    Google Scholar 

  21. . V. Durrleman and N. El Karoui, Coupling smiles, Tech. report, Stanford University, November 2006.

    Google Scholar 

  22. . E.B Dynkin, Diffusions, superdiffusions and partial differential equations, American Mathematical Society, 2002.

    Google Scholar 

  23. . M.R. Fengler, Semiparametric modeling of implied volatility, Lecture Notes in Statistics, Springer Verlag, 2005.

    Google Scholar 

  24. D. Filipovic, Exponential-polynomial families and the term structure of interest rates, Bernoulli 6(6) (2000), 1-27.

    MathSciNet  Google Scholar 

  25. . _____, Consistency problems for Heath-Jarrow-Morton interest rate models, Lecture Notes in Mathematics, vol. 1760, Springer-Verlag, 2002.

    Google Scholar 

  26. . V. Piterbarg J. Sidenius and L. Andersen, A new framework for dynamic credit portfolio loss modelling, Tech. report, October 2005.

    Google Scholar 

  27. . J. Jacod and P. Protter, Risk neutral compatibility with option prices, Tech. report, Université de Paris VI and Cornell University, April 2006.

    Google Scholar 

  28. H. Kellerer, Markov-komposition und eine anwendung auf martingale, Mathematische Annalen 198 (1972), 99-122.

    Article  MATH  MathSciNet  Google Scholar 

  29. . D. Lando, Credit risk, Princeton University Press, 2004.

    Google Scholar 

  30. . J.P. Laurent and D. Leisen, Building a consistent pricing model from observed option prices, World Scientific, 2000.

    Google Scholar 

  31. R. Lee, Implied volatility: Statics, dynamics, and probabilistic interpretation, Interna- tional Journal of Theoretical and Applied Finance 4 (2001), 45-89.

    MATH  Google Scholar 

  32. . _____, Implied and local volatilities under stochastic volatility, Recent Advances in Applied Probability, Springer Verlag, 2004.

    Google Scholar 

  33. . _____, The moment formula for implied volatility at extreme strikes, Mathematical Finance 14 (2004), 469-480.

    Google Scholar 

  34. . J. da Fonseca, R. Cont and V. Durrleman, Stochastic models of implied volatility surfaces, Economic Notes 31 (2002), no. 2.

    Google Scholar 

  35. . D. Brigo R. Torresetti and A. Pallavicini, Implied expected tranche loss surface from cdo data, Tech. report, Banca IMI, March 2007.

    Google Scholar 

  36. . _____, Term structure, tranche structure, and loss distributions, Tech. report, University of Toronto, January 2007.

    Google Scholar 

  37. . C. Rebonato, Volatility and correlation: the perfect hedger and the fox., 2nd ed., Wiley, 2004.

    Google Scholar 

  38. . D. Revuz and M. Yor, Continuous martingales and brownian motion, 2nd ed., Springer- Verlag, 1990.

    Google Scholar 

  39. P. Schönbucher, A market model for stochastic implied volatility, Phil. Trans. of the Royal Society, Series A 357 (1999), 2071-2092.

    Article  MATH  Google Scholar 

  40. . _____, Credit derivatives pricing models, Wiley, 2003.

    Google Scholar 

  41. . _____, Portfolio losses and the term structure of loss transitition rates: a new methodology for the pricing of portfolio  redit derivatives, Tech. report, ETH Z December 2005.

    Google Scholar 

  42. . M. Schweizer and J. Wissel, Term structures of implied volatilities: Absence of arbitrage and existence results.

    Google Scholar 

  43. S. Sherman, On a theorem of Hardy, Littlewood, Polya and Blackwell, Proc. National Academy of Sciences USA 37 (1951), 826-831.

    Article  MATH  MathSciNet  Google Scholar 

  44. . C. Stein, Notes on the comparison of experiments, Tech. report, University of Chicago, 1951.

    Google Scholar 

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Authors and Affiliations

  1. Department of Operations Research and Financial Engineering, Princeton University, 08544, Princeton, NJ, USA

    René A. Carmona

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  1. René A. Carmona
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© 2007 Springer-Verlag Berlin Heidelberg

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Carmona, R.A. (2007). HJM: A Unified Approach to Dynamic Models for Fixed Income, Credit and Equity Markets. In: Paris-Princeton Lectures on Mathematical Finance 2004. Lecture Notes in Mathematics, vol 1919. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73327-0_1

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