Editing Trinomial tree

{{Short description|Model used in financial mathematics}}

The '''trinomial tree''' is a [[Lattice model (finance)|lattice-based]] [[computational model]] used in [[financial mathematics]] to price [[option (finance)|options]]. It was developed by [[Phelim Boyle]] in 1986. It is an extension of the [[binomial options pricing model]], and is conceptually similar. It can also be shown that the approach is equivalent to the [[Finite difference method#Explicit method|explicit]] [[finite difference methods for option pricing|finite difference method for option pricing]].<ref>[https://web.archive.org/web/20070622150346/http://www.in-the-money.com/pages/author.htm Mark Rubinstein]</ref> For [[fixed income]] and [[interest rate derivative]]s see [[Lattice model (finance)#Interest rate derivatives]].

Under the trinomial method, the [[underlying]] stock price is modeled as a recombining tree, where, at each node the price has three possible paths: an up, down and stable or middle path.<ref>[http://www.sitmo.com/article/binomial-and-trinomial-trees/ Trinomial Tree, geometric Brownian motion] {{Webarchive|url=https://web.archive.org/web/20110721155149/http://www.sitmo.com/article/binomial-and-trinomial-trees/ |date=2011-07-21 }}</ref> These values are found by multiplying the value at the current node by the appropriate factor <math> u\,</math>, <math> d\,</math> or <math> m\,</math> where

In the above formulae: <math> \Delta t \,</math> is the length of time per step in the tree and is simply time to maturity divided by the number of time steps; <math> r\,</math> is the [[risk-free interest rate]] over this maturity; <math> \sigma\,</math> is the corresponding [[volatility (finance)|volatility of the underlying]]; <math> q\,</math> is its corresponding [[dividend yield]].<ref name="JHull">[[John C. Hull (economist)|John Hull]] presents alternative formulae; see: {{cite book | last = Hull | first = John C. | edition = 5th | title = Options, Futures and Other Derivatives | year = 2002 | publisher = [[Prentice Hall]] | isbn = 978-0-13-009056-0 | url-access = registration | url = https://archive.org/details/optionsfuturesot00hull_1 }}.</ref>

As with the binomial model, these factors and probabilities are specified so as to ensure that the price of the [[underlying]] evolves as a [[Martingale (probability theory)|martingale]], while the [[Moment (mathematics)|moments{{snd}}]] considering node spacing and probabilities{{snd}} are matched to those of the [[log-normal distribution]]<ref>[http://www2.warwick.ac.uk/fac/sci/maths/people/staff/oleg_zaboronski/fm/trinomial_tree_2008.pdf Pricing Options Using Trinomial Trees]</ref> (and with increasing accuracy for smaller time-steps). Note that for <math> p_u </math>, <math> p_d </math>, and <math> p_m </math> to be in the interval <math> (0,1) </math> the following condition on <math> \Delta t </math> has to be satisfied <math> \Delta t < 2\frac{\sigma^2}{(r-q)^2} </math>.

Once the tree of prices has been calculated, the option price is found at each node largely [[Binomial options pricing model#Methodology|as for the binomial model]], by working backwards from the final nodes to the present node (<math>t_{0}</math>). The difference being that the option value at each non-final node is determined based on the three{{snd}}as opposed to ''two''{{snd}} later nodes and their corresponding probabilities.<ref>[http://icit.zuj.edu.jo/icit13/Papers%20list/Camera_ready/Applied%20Mathematics/694_final.pdf Binomial and Trinomial Trees Versus Bjerksund and Stensland Approximations for American Options Pricing]</ref>

If the length of time-steps <math> \Delta t </math> is taken as an exponentially distributed random variable and interpreted as the waiting time between two movements of the stock price then the resulting stochastic process is a [[birth–death process]]. The resulting [[Korn–Kreer–Lenssen model|model]] is soluble and there exist analytic pricing and hedging formulae for various options.

The trinomial model is considered<ref>[http://www.hoadley.net/options/calculators.htm On-Line Options Pricing & Probability Calculators]</ref> to produce more accurate results than the binomial model when fewer time steps are modelled, and is therefore used when computational speed or resources may be an issue. For [[vanilla option]]s, as the number of steps increases, the results rapidly converge, and the binomial model is then preferred due to its simpler implementation. For [[exotic option]]s the trinomial model (or adaptations) is sometimes more stable and accurate, regardless of step-size.

* [[Binomial options pricing model]]

* [[Valuation of options]]

* [[Option (finance)#Model implementation|Option: Model implementation]]

* [[Korn–Kreer–Lenssen model]]

* [[Implied trinomial tree]]

==References==

*[[Phelim Boyle]], 1986. "Option Valuation Using a Three-Jump Process", ''International Options Journal'' 3, 7–12.

*{{cite journal |last=Rubinstein |first=M. |authorlink=Mark Rubinstein |year=2000 |title=On the Relation Between Binomial and Trinomial Option Pricing Models |journal=[[Journal of Derivatives]] |volume=8 |issue=2 |pages=47&ndash;50 |url=//www.in-the-money.com/pages/author.htm |doi=10.3905/jod.2000.319149 |url-status=dead |archiveurl=https://web.archive.org/web/20070622150346/http://www.in-the-money.com/pages/author.htm |archivedate=June 22, 2007 |citeseerx=10.1.1.43.5394 }}

*Paul Clifford et al. 2010. [https://warwick.ac.uk/fac/sci/maths/people/staff/oleg_zaboronski/fm/trinomial_tree_2010_kevin.pdf Pricing Options Using Trinomial Trees], [[University of Warwick]]

*Tero Haahtela, 2010. [https://www.realoptions.org/papers2010/241.pdf "Recombining Trinomial Tree for Real Option Valuation with Changing Volatility"], [[Aalto University]], Working Paper Series.

* Ralf Korn, Markus Kreer and Mark Lenssen, 1998. "Pricing of european options when the underlying stock price follows a linear birth-death process", Stochastic Models Vol. 14(3), pp 647 – 662

* Peter Hoadley. [http://www.hoadley.net/options/binomialtree.aspx?tree=T Trinomial Tree Option Calculator (Tree Visualized)]

[[Category:Options (finance)]]

[[Category:Models of computation]]

[[Category:Trees (data structures)]]

[[Category:Financial models]]