In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, a basic affine jump diffusion (basic AJD) is a
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
Z of the form
:
where
is a standard
Brownian motion
Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
, and
is an independent
compound Poisson process
A compound Poisson process is a continuous-time stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution. To be precise, a compound ...
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
Jump diffusion is a stochastic process that involves jump process, jumps and diffusion process, diffusion. It has important applications in magnetic reconnection, coronal mass ejections, condensed matter physics, and pattern theory and computationa ...
. 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
Credit risk is the chance that a borrower does not repay a loan
In finance, a loan is the tender of money by one party to another with an agreement to pay it back. The recipient, or borrower, incurs a debt and is usually required to pay ...
applications,
[ since both the moment generating function
:
and the ]characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
:
are known in closed form.[
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
{{Reflist, refs=
[{{cite journal , author = Darrell Duffie, Nicolae Gârleanu , year = 2001 , title = Risk and Valuation of Collateralized Debt Obligations , journal = Financial Analysts Journal , volume = 57 , pages = 41–59 , doi=10.2469/faj.v57.n1.2418, s2cid = 12334040 }]
Preprint
/ref>
[{{cite journal , author = Allan Mortensen , year = 2006 , title = Semi-Analytical Valuation of Basket Credit Derivatives in Intensity-Based Models , journal = Journal of Derivatives , volume = 13 , issue = 4 , pages = 8–26 , doi=10.3905/jod.2006.635417}]
Preprint
/ref>
[{{cite journal , author = Andreas Ecker , year = 2009 , title = Computational Techniques for basic Affine Models of Portfolio Credit Risk , journal = Journal of Computational Finance , volume = 13 , pages = 63–97 , doi=10.21314/JCF.2009.200}]
Preprint
/ref>
[{{cite journal
, last1 = Feldhutter , first1 = P.
, last2 = Nielsen , first2 = M. S.
, date = January 2012
, doi = 10.1093/jjfinec/nbr011
, issue = 2
, journal = Journal of Financial Econometrics
, pages = 292–324
, title = Systematic and idiosyncratic default risk in synthetic credit markets
, url = http://www.feldhutter.com/CDOpaper070710.pdf
, volume = 10]
Stochastic processes