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## Trinomial Trees

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**Chris Hebert**11/22/10 Trinomial Trees**History of Trinomial Trees**• Developed by Phelim Boyle • 1986 • Binomial Trees (1979)**Recall: Binomial Trees**• The Binomial Option Pricing Model is essentially a tree that is constructed to show possible values that an underlying asset can take and the resulting value of the option at these values. • The tree is constructed by assuming the stock can only go up or down by a factor related to the length of time and the volatility of the stock.**What are Trinomial Trees?**• The Trinomial Tree is an adjusted version of the binomial tree. Rather than assuming either up or down, trinomial trees allow for a third option, that the stock price remains constant.**Up and down by how much?**• Let’s again call the upward movement u, and the downward movement d, while a constant movement will be called m. • The magnitude of this movement is based on volatility and the size of the time interval. • According to Boyle:**Boyle vs. CRR**• Boyle’s parameters: • Cox-Ross-Rubenstein:**Why does Boyle use 2t?**• Because we have 3 choices now, we can allow for a bigger change. • Boyle basically combines 2 binomial trees: one were the options are up or even and another where the options are even or down. • Because of the recombining nature of the trees, even-even is the same as up down, so all nodes will fill. • The best way to summarize Boyle’s tree is a binomial tree where we skip every other node.**Example**• S=90.00 • K=100.0 • T= 1 • σ= .3 • r= .01 • Δt= .1**S=90.00**• K=100.0 • T= 1 • σ= .3 • r= .01 • Δt= .1**Pull Back Formula**• Recall: • We use the exact same formula for the trinomial tree except we take 3 nodes into account. • Also, we use these probabilities for p:**Kamrad and Ritchken (1991)**• They discovered a different version of the trinomial tree. • They made a few simplifications to the Boyle trinomial tree and prove that it is possible to essentially skip every other step in the binomial tree as a means of generating a trinomial tree.**Kamrad and Ritchken (1991)**• The derivation of the trinomial tree from the moments of the Brownian motion yields:**So How do We Choose λ?**• We choose λ so that the option prices given by the tree converge to Black-Scholes as quickly as possible. • Speed is everything for option pricing. • Recall, the drawback for binomial trees is the speed in calculating them. • Trinomial trees allow for a better approximation, with fewer nodes to calculate, so we need less time.**What is the Optimal λ?**• Kamrad and Ritchken found that the optimal λ is 3/2 • (50,50,.2,.03,1)**How does K&R differ from Boyle?**• Kamrad and Ritchken streamline the trinomial tree. • They focus the endpoints of the tree on a smaller region of possible end values. • By doing this, they increase speed from the binomial tree, and increase accuracy from Boyle.**Recall: Binomial Advantages**• Primary advantage of Binomial trees is that American options become much easier to price. • Other advantages • Can factor in dividends • No calculus needed • Easy to make a spreadsheet to find option values**Trinomial Trees Advantages**• Trinomial trees are calculated in the same way as binomial trees, so all advantages of binomial trees are held intact with trinomial trees.**Recall: Binomial Tree Disadvantages/ Limitations**• Speed • In order to have accurate results, we want to maximize the number of time periods, but this means creating a lot of nodes which can be slow to calculate even with today’s computers. • For some extreme options, like a cash or nothing, pricing can be inaccurate until a very large number of time intervals is used.**Trinomial Tree**• Speed is improved from Binomial tree. • We can get more accurate results in fewer intervals, so fewer nodes need to be calculated. • Trinomial trees do a better job of addressing exotic options. • Trinomial trees have more end nodes producing a more accurate layout of payoffs.**Call Option: Stock Price: 100, Strike Price: 102,**Volatility: 20%, Dividend Yield: 5%, Risk Free: 8%**So Why Use Binomial Tree over the Trinomial Tree?**• Implementation of the binomial tree is easier. • For vanilla options, the binomial tree converges pretty quickly so the speed advantage and accuracy advantage of the trinomial tree is small.**So Why Use the Trinomial Tree over the Binomial Tree?**• For exotic options, The trinomial tree can be significantly more accurate and significantly quicker.**Works Cited**• Albanese, Claudio., and Giuseppe Campolieti. Advanced Derivatives Pricing and Risk Management: Theory, Tools and Hands-on Programming Application. Amsterdam ; Boston: Elsevier Academic Press, 2006. • Berg, Imme van den. Principles of Infinitesimal Stochastic and Financial Analysis: . Singapore ; River Edge, N.J.: World Scientific, 2000. • Daigler, Robert T. Advanced Options Trading: The Analysis and Evaluation of Trading Strategies, Hedging Tactics, and Pricing Models. Chicago, Ill.: Probus Pub. Co., 1994. • http://en.wikipedia.org/wiki/Trinomial_tree • http://www.hoadley.net/options/binomialtree.aspx?tree=T • http://www.global-derivatives.com/index.php/options-database-topmenu/13-options-database/15-european-options#Trinomial • http://www2.warwick.ac.uk/fac/sci/maths/people/staff/oleg_zaboronski/fm/trinomial_tree_2008.pdf • http://bejoymaliakal.com/files/Stochastic%20Calculus%20-%20Binomial%20&%20Trinomial%20Trees.pdf • http://www.haas.berkeley.edu/groups/finance/WP/rpf292.pdf