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Trinomial tree
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<h2 id="formula">Formula</h2> <p>Under the trinomial method, the <a href="/facts/Underlying/J9S15NxX">underlying</a> 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.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> These values are found by multiplying the value at the current node by the appropriate factor u {\displaystyle u\,} , d {\displaystyle d\,} or m {\displaystyle m\,} where </p> u = e σ 2 Δ t {\displaystyle u=e^{\sigma {\sqrt {2\Delta t}}}} d = e − σ 2 Δ t = 1 u {\displaystyle d=e^{-\sigma {\sqrt {2\Delta t}}}={\frac {1}{u}}\,} (the structure is recombining) m = 1 {\displaystyle m=1\,} <p>and the corresponding probabilities are: </p> p u = ( e ( r − q ) Δ t / 2 − e − σ Δ t / 2 e σ Δ t / 2 − e − σ Δ t / 2 ) 2 {\displaystyle 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}\,} p d = ( e σ Δ t / 2 − e ( r − q ) Δ t / 2 e σ Δ t / 2 − e − σ Δ t / 2 ) 2 {\displaystyle 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}\,} p m = 1 − ( p u + p d ) {\displaystyle p_{m}=1-(p_{u}+p_{d})\,} . <p>In the above formulae: Δ t {\displaystyle \Delta t\,} is the length of time per step in the tree and is simply time to maturity divided by the number of time steps; r {\displaystyle r\,} is the <a href="/facts/Risk-free_interest_rate/BJprxn2e">risk-free interest rate</a> over this maturity; σ {\displaystyle \sigma \,} is the corresponding <a href="/facts/Volatility_(finance)/mCdus1MW">volatility of the underlying</a>; q {\displaystyle q\,} is its corresponding <a href="/facts/Dividend_yield/fRGtev1z">dividend yield</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p><p>As with the binomial model, these factors and probabilities are specified so as to ensure that the price of the <a href="/facts/Underlying/J9S15NxX">underlying</a> evolves as a <a href="/facts/Martingale_(probability_theory)/wk9qdXFn">martingale</a>, while the <a href="/facts/Moment_(mathematics)/GL7vJ1nb">moments – </a> considering node spacing and probabilities – are matched to those of the <a href="/facts/Log-normal_distribution/V2RXmmlE">log-normal distribution</a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> (and with increasing accuracy for smaller time-steps). Note that for p u {\displaystyle p_{u}} , p d {\displaystyle p_{d}} , and p m {\displaystyle p_{m}} to be in the interval ( 0 , 1 ) {\displaystyle (0,1)} the following condition on Δ t {\displaystyle \Delta t} has to be satisfied Δ t < 2 σ 2 ( r − q ) 2 {\displaystyle \Delta t<2{\frac {\sigma ^{2}}{(r-q)^{2}}}} . </p><p>Once the tree of prices has been calculated, the option price is found at each node largely <a href="/facts/Binomial_options_pricing_model/S8Jt6ENH">as for the binomial model</a>, by working backwards from the final nodes to the present node ( t 0 {\displaystyle t_{0}} ). The difference being that the option value at each non-final node is determined based on the three – as opposed to <i>two</i> – later nodes and their corresponding probabilities.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p>If the length of time-steps Δ t {\displaystyle \Delta t} 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 <a href="/facts/Birth%25E2%2580%2593death_process/ud6IWMxu">birth–death process</a>. The resulting <a href="/facts/Korn%25E2%2580%2593Kreer%25E2%2580%2593Lenssen_model/NTsd9pYC">model</a> is soluble and there exist analytic pricing and hedging formulae for various options. </p> <h2 id="application">Application</h2> <p>The trinomial model is considered<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> 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 <a href="/facts/Vanilla_option/5gJFXV3X">vanilla options</a>, as the number of steps increases, the results rapidly converge, and the binomial model is then preferred due to its simpler implementation. For <a href="/facts/Exotic_option/j2B5qzcN">exotic options</a> the trinomial model (or adaptations<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>) is sometimes more stable and accurate, regardless of step-size. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Binomial_options_pricing_model/S8Jt6ENH">Binomial options pricing model</a></li> <li><a href="/facts/Valuation_of_options/11Ft9oEA">Valuation of options</a></li> <li><a href="/facts/Option_(finance)/5gJFXV3X">Option: Model implementation</a></li> <li><a href="/facts/Korn%25E2%2580%2593Kreer%25E2%2580%2593Lenssen_model/NTsd9pYC">Korn–Kreer–Lenssen model</a></li> <li><a href="/facts/Implied_trinomial_tree/sotPp1xA">Implied trinomial tree</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Phelim_Boyle/pTamzmM1">Phelim Boyle</a>, 1986. "Option Valuation Using a Three-Jump Process", <i>International Options Journal</i> 3, 7–12.</li> <li><a href="/facts/Mark_Rubinstein/HEF1r0M6">Rubinstein, M.</a> (2000). <a href="https://web.archive.org/web/20070622150346/http://www.in-the-money.com/pages/author.htm">"On the Relation Between Binomial and Trinomial Option Pricing Models"</a>. <i>Journal of Derivatives</i>. 8 (2): 47–50. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.5394">10.1.1.43.5394</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.3905%2Fjod.2000.319149">10.3905/jod.2000.319149</a>. Archived from <a href="//www.in-the-money.com/pages/author.htm">the original</a> on June 22, 2007.</li> <li>Paul Clifford et al. 2010. <a href="https://warwick.ac.uk/fac/sci/maths/people/staff/oleg_zaboronski/fm/trinomial_tree_2010_kevin.pdf">Pricing Options Using Trinomial Trees</a>, <a href="/facts/University_of_Warwick/A88IPplp">University of Warwick</a></li> <li>Tero Haahtela, 2010. <a href="https://www.realoptions.org/papers2010/241.pdf">"Recombining Trinomial Tree for Real Option Valuation with Changing Volatility"</a>, <a href="/facts/Aalto_University/m1oSw1Cq">Aalto University</a>, Working Paper Series.</li> <li>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</li> <li>Peter Hoadley. <a href="http://www.hoadley.net/options/binomialtree.aspx?tree=T">Trinomial Tree Option Calculator (Tree Visualized)</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Mark Rubinstein <a href="https://web.archive.org/web/20070622150346/http://www.in-the-money.com/pages/author.htm" target="_blank">https://web.archive.org/web/20070622150346/http://www.in-the-money.com/pages/author.htm</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Trinomial Tree, geometric Brownian motion Archived 2011-07-21 at the Wayback Machine <a href="http://www.sitmo.com/article/binomial-and-trinomial-trees/" target="_blank">http://www.sitmo.com/article/binomial-and-trinomial-trees/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>John Hull presents alternative formulae; see: Hull, John C. (2002). Options, Futures and Other Derivatives (5th ed.). Prentice Hall. ISBN 978-0-13-009056-0.. <a href="978-0-13-009056-0" target="_blank">978-0-13-009056-0</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Pricing Options Using Trinomial Trees <a href="http://www2.warwick.ac.uk/fac/sci/maths/people/staff/oleg_zaboronski/fm/trinomial_tree_2008.pdf" target="_blank">http://www2.warwick.ac.uk/fac/sci/maths/people/staff/oleg_zaboronski/fm/trinomial_tree_2008.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Binomial and Trinomial Trees Versus Bjerksund and Stensland Approximations for American Options Pricing <a href="http://icit.zuj.edu.jo/icit13/Papers%20list/Camera_ready/Applied%20Mathematics/694_final.pdf" target="_blank">http://icit.zuj.edu.jo/icit13/Papers%20list/Camera_ready/Applied%20Mathematics/694_final.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>On-Line Options Pricing & Probability Calculators <a href="http://www.hoadley.net/options/calculators.htm" target="_blank">http://www.hoadley.net/options/calculators.htm</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Deloire et Roth 2024 "Multi-asset and generalised Local Volatility. An efficient implementation" [1] <a href="https://arxiv.org/abs/2411.05425" target="_blank">https://arxiv.org/abs/2411.05425</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> </ol>