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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 Black–Derman–Toy model ： ウィキペディア英語版 Black–Derman–Toy model In mathematical finance, the Black–Derman–Toy model (BDT) is a popular short rate model used in the pricing of bond options, swaptions and other interest rate derivatives; see Lattice model (finance) #Interest rate derivatives. It is a one-factor model; that is, a single stochastic factor – the short rate – determines the future evolution of all interest rates. It was the first model to combine the mean-reverting behaviour of the short rate with the lognormal distribution, () and is still widely used. ()() The model was introduced by Fischer Black, Emanuel Derman, and Bill Toy. It was first developed for in-house use by Goldman Sachs in the 1980s and was published in the ''Financial Analysts Journal'' in 1990. A personal account of the development of the model is provided in one of the chapters in Emanuel Derman's memoir "My Life as a Quant."() Under BDT, using a binomial lattice, one calibrates the model parameters to fit both the current term structure of interest rates (yield curve), and the volatility structure for interest rate caps (usually as implied by the Black-76-prices for each component caplet); see aside. Using the calibrated lattice one can then value a variety of more complex interest-rate sensitive securities and interest rate derivatives. Although initially developed for a lattice-based environment, the model has been shown to imply the following continuous stochastic differential equation:()() :$d\backslash ln(r)\; =\; (+\; \backslash frac\backslash ln(r))dt\; +\; \backslash sigma\_t\backslash ,\; dW\_t$ ::where, ::$r\backslash ,$ = the instantaneous short rate at time t ::$\backslash theta\_t\backslash ,$ = value of the underlying asset at option expiry ::$\backslash sigma\_t\backslash ,$ = instant short rate volatility ::$W\_t\backslash ,$ = a standard Brownian motion under a risk-neutral probability measure; $dW\_t\backslash ,$ its differential. For constant (time independent) short rate volatility, $\backslash sigma\backslash ,$, the model is: :$d\backslash ln(r)\; =\; \backslash theta\_t\backslash ,\; dt\; +\; \backslash sigma\; \backslash ,\; dW\_t$ One reason that the model remains popular, is that the "standard" Root-finding algorithms – such as Newton's method (the secant method) or bisection – are very easily applied to the calibration.() Relatedly, the model was originally described in algorithmic language, and not using stochastic calculus or martingales. () ==References== * * * * * 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「Black–Derman–Toy model」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.