LIBOR market model

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.^[1] It is used for pricing interest rate derivatives, especially exotic derivatives like Bermudan swaptions, ratchet caps and floors, target redemption notes, autocaps, zero coupon swaptions, constant maturity swaps and spread options, among many others. The quantities that are modeled, rather than the short rate or instantaneous forward rates (like in the Heath-Jarrow-Morton framework) are a set of forward rates (also called forward LIBORs), 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 process under its forward measure, i.e. a Black model leading to a Black formula for interest rate caps. 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, 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.

Model dynamic

The LIBOR market model models a set of $n$ forward rates $L_{j}$ , $j=1,\ldots,n$ as lognormal processes. Under the respective $T_j$ -Forward measure $Q_{T_{j+1}}$

dL_j(t) = \sigma_j(t) L_j(t) dW^{Q_{T_{j+1}}}(t)\text{.}

Here, $L_{j}$ denotes the forward rate for the period $[T_{j},T_{j+1}]$ . For each single forward rate the model corresponds to the Black model. 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 $T$ -Forward measures. By means of the multivariate Girsanov's theorem one can show^[2]^[3] that

dW^{Q_{T_j}}(t) = \begin{cases} dW^{Q_{T_p}}(t) - \sum\limits_{k=j}^{p-1} \frac{\delta L_k(t)}{1 + \delta L_k(t)} {\sigma}_k(t) dt \qquad j < p \\ dW^{Q_{T_p}}(t) \qquad \qquad \qquad \quad \quad \quad \quad \quad \quad j = p \\ dW^{Q_{T_p}}(t) + \sum\limits_{k=p}^{j-1} \frac{\delta L_k(t)}{1 + \delta L_k(t)} {\sigma}_k(t) dt \qquad \quad j > p \\ \end{cases}

and

dL_j(t) = \begin{cases} L_j(t){\sigma}_j(t)dW^{Q_{T_{p}}}(t) - L_j(t)\sum\limits_{k=j}^{p-1} \frac{\delta L_k(t)}{1 + \delta L_k(t)} {\sigma}_j(t){\sigma}_k(t){\rho}_{jk}dt \qquad j <p\\ L_j(t){\sigma}_j(t)dW^{Q_{T_{p}}}(t) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad j = p \\ L_j(t){\sigma}_j(t)dW^{Q_{T_{p}}}(t) + L_j(t)\sum\limits_{k=p}^{j-1} \frac{\delta L_k(t)}{1 + \delta L_k(t)} {\sigma}_j(t){\sigma}_k(t){\rho}_{jk}dt \quad \qquad j > p\\ \end{cases}

External links

References

↑ M. Musiela, M. Rutkowski: Martingale methods in financial modelling. 2nd ed. New York : Springer-Verlag, 2004. Print.
↑ D. Papaioannou (2011): "Applied Multidimensional Girsanov Theorem", SSRN
↑ "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"

Literature

Brace, A., Gatarek, D. et Musiela, M. (1997): “The Market Model of Interest Rate Dynamics”, Mathematical Finance, 7(2), 127-154.
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.

Stochastic processes

Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding

Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage

Both	Branching process Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process White noise

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Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie

Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson

Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c

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Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract

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