# Trinomial tree

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{{Short description|Model used in financial mathematics}}
The '''trinomial tree''' is a [lattice-based](/source/Lattice_model_(finance)) [computational model](/source/computational_model) used in [financial mathematics](/source/financial_mathematics) to price [options](/source/option_(finance)) on [equity](/source/Stock). It was developed by [Phelim Boyle](/source/Phelim_Boyle) in 1986. It is an extension of the [binomial options pricing model](/source/binomial_options_pricing_model), and is conceptually similar. It can also be shown that the approach is equivalent to the [explicit](/source/Finite_difference_method) [finite difference method for option pricing](/source/finite_difference_methods_for_option_pricing).<ref>[https://web.archive.org/web/20070622150346/http://www.in-the-money.com/pages/author.htm Mark Rubinstein]</ref> 

Trinomial trees [are also deployed](/source/Lattice_model_(finance)) 
<ref>M. Leippold and Z. Wiener (2003). [http://simonbenninga.com/wiener/leippold-wiener2003.pdf ''Efficient Calibration of Trinomial Trees for One-Factor Short Rate Models'']</ref> 
for [fixed income](/source/fixed_income) and [interest rate derivative](/source/interest_rate_derivative)s; see under [Lattice model (finance)](/source/Lattice_model_(finance)).

==Formula==
Under the trinomial method, the [underlying](/source/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
:<math> u = e^{\sigma\sqrt {2\Delta t}}</math>
:<math> d = e^{-\sigma\sqrt {2\Delta t}} = \frac{1}{u} \,</math> (the structure is recombining)
:<math> m = 1 \,</math>

and the corresponding probabilities are: 
:<math> p_u = \left(\frac{e^{(r - q)  \Delta t / 2}- e^{-\sigma\sqrt {\Delta t/2}}}{e^{\sigma\sqrt {\Delta t/2}}- e^{-\sigma\sqrt {\Delta t/2}}}\right)^2 \,</math> 
:<math> p_d = \left(\frac{e^{\sigma\sqrt {\Delta t/2}}-e^{(r - q)  \Delta t / 2}}{e^{\sigma\sqrt {\Delta t/2}}- e^{-\sigma\sqrt {\Delta t/2}}}\right)^2 \,</math> 
:<math> p_m = 1 - (p_u + p_d) \,</math>.

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](/source/risk-free_interest_rate) over this maturity; <math> \sigma\,</math> is the corresponding [volatility of the underlying](/source/volatility_(finance)); <math> q\,</math> is its corresponding [dividend yield](/source/dividend_yield).<ref name="JHull">[John Hull](/source/John_C._Hull_(economist)) presents alternative formulae; see: {{cite book | last = Hull | first = John C. | edition = 5th | title = Options, Futures and Other Derivatives | year = 2002 | publisher = [Prentice Hall](/source/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](/source/underlying) evolves as a [martingale](/source/Martingale_(probability_theory)), while the [moments{{snd}}](/source/Moment_(mathematics)) considering node spacing and probabilities{{snd}} are matched to those of the [log-normal distribution](/source/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 [as for the binomial model](/source/Binomial_options_pricing_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](/source/birth%E2%80%93death_process). The resulting [model](/source/Korn%E2%80%93Kreer%E2%80%93Lenssen_model) is soluble and there exist analytic pricing and hedging formulae for various options.

==Application==
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](/source/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](/source/exotic_option)s the trinomial model (or adaptations<ref>Deloire et Roth 2024  "Multi-asset and generalised Local Volatility. An efficient implementation" [https://arxiv.org/abs/2411.05425]</ref>) is sometimes more stable and accurate, regardless of step-size.

==See also==
* [Binomial options pricing model](/source/Binomial_options_pricing_model)
* [Valuation of options](/source/Valuation_of_options)
* [Option: Model implementation](/source/Option_(finance))
* [Korn–Kreer–Lenssen model](/source/Korn%E2%80%93Kreer%E2%80%93Lenssen_model)
* [Implied trinomial tree](/source/Implied_trinomial_tree)
* {{slink|Lattice model (finance)#Interest rate derivatives}}

==References==
<references/>

==External links==
*[Phelim Boyle](/source/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](/source/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](/source/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](/source/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)]

{{Derivatives market}}

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

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Adapted from the Wikipedia article [Trinomial tree](https://en.wikipedia.org/wiki/Trinomial_tree) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Trinomial_tree?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.