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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Trinomial tree ： ウィキペディア英語版 Trinomial tree The trinomial tree is a lattice based computational model used in financial mathematics to price options. It was developed by Phelim Boyle in 1986. It is an extension of the binomial options pricing model, and is conceptually similar.〔(Trinomial Method (Boyle) 1986 )〕 It can also be shown that the approach is equivalent to the explicit finite difference method for option pricing.〔(Mark Rubinstein )〕 ==Formula== 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.〔(Trinomial Tree, geometric Brownian motion )〕 These values are found by multiplying the value at the current node by the appropriate factor $u\backslash ,$, $d\backslash ,$ or $m\backslash ,$ where :$u\; =\; e^\}\; =\; \backslash frac\; \backslash ,$ (the structure is recombining) :$m\; =\; 1\; \backslash ,$ and the corresponding probabilities are: :$p\_u\; =\; \backslash left(\backslash frac\}\}\}\}\backslash right)^2\; \backslash ,$ :$p\_d\; =\; \backslash left(\backslash frac\}\}\}\backslash right)^2\; \backslash ,$ :$p\_m\; =\; 1\; -\; (p\_u\; +\; p\_d)\; \backslash ,$. In the above formulae: $\backslash Delta\; t\; \backslash ,$ is the length of time per step in the tree and is simply time to maturity divided by the number of time steps; $r\backslash ,$ is the risk-free interest rate over this maturity; $\backslash sigma\backslash ,$ is the corresponding volatility of the underlying; $q\backslash ,$ is its corresponding dividend yield.〔John Hull presents alternative formulae; see: .〕 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, while the moments - considering node spacing and probabilities - are matched to those of the log normal distribution〔(Pricing Options Using Trinomial Trees )〕 (and with increasing accuracy for smaller time-steps). Note that for $p\_u$, $p\_d$, and $p\_m$ to be in the interval $(0,1)$ the following condition on $\backslash Delta\; t$ has to be satisfied . Once the tree of prices has been calculated, the option price is found at each node largely as for the binomial model, by working backwards from the final nodes to today. The difference being that the option value at each non-final node is determined based on the three - as opposed to ''two'' - later nodes and their corresponding probabilities. The model is best understood visually - see, for example (Trinomial Tree Option Calculator ) (Peter Hoadley). If the length of time-steps $\backslash Delta\; t$ 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 birth-death process. The resulting model is soluble and there exist analytic pricing and hedging formulae for various options. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Trinomial tree」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.