View source for Korn–Kreer–Lenssen model

The '''Korn–Kreer–Lenssen model''' ('''KKL model''') is a discrete [[Trinomial tree|trinomial model]] proposed in 1998 by Ralf Korn, Markus Kreer and Mark Lenssen<ref>{{cite journal|doi=10.1080/15326349808807493 | volume=14 | issue=3 | title=Pricing of european options when the underlying stock price follows a linear birth–death process | journal=Communications in Statistics. Stochastic Models | pages=647–662| year=1998 | last1=Korn | first1=Ralf | last2=Kreer | first2=Markus | last3=Lenssen | first3=Mark }}</ref> to model [[illiquid securities]] and to value [[Derivative (finance)|financial derivatives]] on these. 

It generalizes the binomial [[Cox-Ross-Rubinstein model]] in a natural way as the stock in a given time interval can either rise one unit up, fall one unit down or remain unchanged. In contrast to [[Black–Scholes]] or [[Cox-Ross-Rubinstein model]] the [[Market (economics)|market]] consisting of stock and cash is not complete yet. To value and replicate a financial derivative an additional traded security related to the original security needs to be added. This might be a Low Exercise Price Option (or short [[LEPO]]). The mathematical proof of arbitrage free pricing is based on [[Martingale representation theorem|martingale representations]] for point processes pioneered in the 1980s and 1990 by [[Albert Shiryaev]], Robert Liptser and [[Marc Yor]].

The dynamics is based on continuous time linear [[birth–death process]]es and analytic formulae for option prices and Greeks can be stated. Later work looks at market completion with general calls or puts.<ref>{{Cite web |url=http://www.wilmott.com/pdfs/101130_chen.pdf |title=Archived copy |access-date=2011-06-21 |archive-url=https://web.archive.org/web/20110929020112/http://www.wilmott.com/pdfs/101130_chen.pdf |archive-date=2011-09-29 |url-status=dead }}</ref> A comprehensive introduction may be found in the attached MSc-thesis.<ref>http://resources.aims.ac.za/archive/2010/obeng.pdf{{Dead link|date=February 2020 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>

The model belongs to the class of [[Trinomial tree|trinomial models]] and the difference to the standard [[trinomial tree]] is the following: if <math> \Delta t </math> denotes the waiting time between two movements of the stock price then in the KKL-model <math> \Delta t </math> remains finite and exponentially distributed whereas in [[trinomial tree]]s the time is discrete and the limit <math> \Delta t \rightarrow 0</math> is taken by numerical extrapolation afterwards.

==See also==
*[[Binomial options pricing model]]
*[[Trinomial tree]]
*[[Valuation of options]]
*[[Option (finance)#Model implementation|Option: Model implementation]]

== References ==
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== Literature ==
* Ralf Korn, Markus Kreer and Mark Lenssen: "Pricing of european options when the underlying stock price follows a linear birth–death process", Stochastic Models Vol. 14(3), 1998, pp. 647–662
* Xiong Chen: "The Korn–Kreer–Lenssen Model as an alternative for option pricing", Willmott Magazine June 2004, pp. 74–80

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[[Category:Financial models]]
[[Category:Mathematical finance]]
[[Category:Options (finance)]]
[[Category:Models of computation]]
[[Category:Trees (data structures)]]