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| The '''LIBOR market model''', also known as the '''BGM Model''' ('''Brace Gatarek Musiela Model''', in reference to the names of some of the inventors) is a financial model of interest rates.<ref>M. Musiela, M. Rutkowski: Martingale methods in financial modelling. 2nd ed. New York : Springer-Verlag, 2004. Print.</ref> It is used for pricing [[interest rate derivative]]s, especially exotic derivatives like Bermudan swaptions, ratchet caps and floors, target redemption notes, autocaps, zero coupon swaptions, [[constant maturity swap]]s and spread options, among many others. The quantities that are modeled, rather than the [[short rate model|short rate]] or instantaneous forward rates (like in the [[Heath-Jarrow-Morton framework]]) are a set of [[forward rates]] (also called forward [[LIBOR]]s), which have the advantage of being directly observable in the market, and whose volatilities are naturally linked to traded contracts. Each forward rate is modeled by a [[lognormal distribution|lognormal]] process under its [[forward measure]], i.e. a [[Black model]] leading to a Black formula for [[interest rate cap]]s. This formula is the market standard to quote cap prices in terms of implied volatilities, hence the term "market model". The LIBOR market model may be interpreted as a collection of forward LIBOR dynamics for different forward rates with spanning tenors and maturities, each forward rate being consistent with a Black interest rate caplet formula for its canonical maturity. One can write the different rates dynamics under a common pricing [[Measure (mathematics)|measure]], for example the [[forward measure]] for a preferred single maturity, and in this case forward rates will not be lognormal under the unique measure in general, leading to the need for numerical methods such as [[monte carlo simulation]] or approximations like the frozen drift assumption.
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| == Model dynamic ==
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| The LIBOR market model models a set of <math>n</math> forward rates <math>L_{j}</math>, <math>j=1,\ldots,n</math> as [[lognormal distribution | lognormal]] processes. Under the respective <math>T_j</math> -Forward measure <math>Q_{T_j}</math>
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| :<math>
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| dL_j(t) = \sigma_j(t) L_j(t) dW^{Q_{T_j}}(t)\text{.}
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| </math>
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| Here, <math>L_{j}</math> denotes the forward rate for the period <math>[T_{j},T_{j+1}]</math>. For each single forward rate the model corresponds to the Black model.
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| The novelty is that, in contrast to the [[Black model]], the LIBOR market model describes the dynamic of a whole family of forward rates under a common measure. The question now is how to switch between the different <math>T</math>-Forward measures.
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| By means of the multivariate [[Girsanov's theorem]] one can show<ref>[http://www.yetanotherquant.de/libor/tutorial.pdf "An accompaniment to a course on interest rate modeling: with discussion of Black-76, Vasicek and HJM models and a gentle introduction to the multivariate LIBOR Market Model"]</ref>
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| that
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| :<math>
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| dW^{Q_{T_j}}(t) =
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| \begin{cases}
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| dW^{Q_{T_p}}(t) - \sum\limits_{k=j+1}^{p} \frac{\delta L_k(t)}{1 + \delta
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| L_k(t)} {\sigma}_k(t) dt \qquad j < p \\
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| dW^{Q_{T_p}}(t) \qquad \qquad \qquad \quad \quad \quad \quad \quad \quad j = p \\
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| dW^{Q_{T_p}}(t) + \sum\limits_{k=p}^{j-1} \frac{\delta L_k(t)}{1 + \delta
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| L_k(t)} {\sigma}_k(t) dt \qquad \quad j > p \\
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| \end{cases}
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| </math>
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| and
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| :<math>
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| dL_j(t) =
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| \begin{cases}
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| L_j(t){\sigma}_j(t)dW^{Q_{T_{p}}}(t) - L_j(t)\sum\limits_{k=j+1}^{p} \frac{\delta
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| L_k(t)}{1 + \delta L_k(t)} {\sigma}_j(t){\sigma}_k(t){\rho}_{jk}dt \qquad j <p\\
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| L_j(t){\sigma}_j(t)dW^{Q_{T_{p}}}(t) \qquad \qquad \qquad \qquad \qquad \qquad
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| \qquad \qquad \quad \quad j = p \\
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| L_j(t){\sigma}_j(t)dW^{Q_{T_{p}}}(t) + L_j(t)\sum\limits_{k=p}^{j-1} \frac{\delta
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| L_k(t)}{1 + \delta L_k(t)} {\sigma}_j(t){\sigma}_k(t){\rho}_{jk}dt \quad \qquad j > p\\
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| \end{cases}
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| </math>
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| ==External links==
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| * [http://www.christian-fries.de/finmath/applets/ Java applets for pricing under a LIBOR market model and Monte-Carlo methods]
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| * [http://www.finmath.net/topics/libormarketmodel/ Jave source code and spreadsheet of a LIBOR market model, including calibration to swaption and product valuation]
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| * [http://www.damianobrigo.it/bocconi.html Damiano Brigo's lecture notes on the LIBOR market model for the Bocconi University fixed income course]
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| ==References==
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| {{reflist}}
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| * Brace, A., Gatarek, D. et Musiela, M. (1997): “The Market Model of Interest Rate Dynamics”, Mathematical Finance, 7(2), 127-154.
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| * Miltersen, K., Sandmann, K. et Sondermann, D., (1997): „Closed Form Solutions for Term Structure Derivates with Log-Normal Interest Rates“, Journal of Finance, 52(1), 409-430.
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| {{Stochastic processes}}
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| [[Category:Mathematical finance]]
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| [[Category:Interest rates]]
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| [[Category:Fixed income analysis]]
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| [[Category:Finance theories]]
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Nice to satisfy you, my title is Ling and I completely dig that title. Interviewing is what she does in her day occupation but soon her husband and her will start their own company. One of my preferred hobbies is camping and now I'm trying to make money with it. My house is now in Kansas.
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