Basic affine jump diffusion

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In mathematics probability theory, a basic affine jump diffusion (basic AJD) is a stochastic process Z of the form

where is a standard Brownian motion, and is an independent compound Poisson process with constant jump intensity and independent exponentially distributed jumps with mean . For the process to be well defined, it is necessary that and . A basic AJD is a special case of an affine process and of a jump diffusion. On the other hand, the Cox–Ingersoll–Ross (CIR) process is a special case of a basic AJD.

Basic AJDs are attractive for modeling default times in credit risk applications, [1] [2] [3] [4] since both the moment generating function

and the characteristic function

are known in closed form. [3]

The characteristic function allows one to calculate the density of an integrated basic AJD

by Fourier inversion, which can be done efficiently using the FFT.

References

  1. Darrell Duffie, Nicolae Gârleanu (2001). "Risk and Valuation of Collateralized Debt Obligations". Financial Analysts Journal. 57: 41–59. doi:10.2469/faj.v57.n1.2418. S2CID   12334040. Preprint
  2. Allan Mortensen (2006). "Semi-Analytical Valuation of Basket Credit Derivatives in Intensity-Based Models". Journal of Derivatives. 13 (4): 8–26. doi:10.3905/jod.2006.635417. Preprint
  3. 1 2 Andreas Ecker (2009). "Computational Techniques for basic Affine Models of Portfolio Credit Risk". Journal of Computational Finance. 13: 63–97. doi:10.21314/JCF.2009.200. Preprint
  4. Feldhutter, P.; Nielsen, M. S. (January 2012). "Systematic and idiosyncratic default risk in synthetic credit markets" (PDF). Journal of Financial Econometrics. 10 (2): 292–324. doi:10.1093/jjfinec/nbr011.