The affine LIBOR models

The nature of interbank loans, as well as the daily calculation of LIBOR rates as the trimmed arithmetic average of interbank quoted rates (see www.bbalibor.com ), yield that LIBOR rates should be non-negative. A requirement from mathematical finance is that LIBOR rates should be martingales with respect to their own forward measure IP T k . That is, when B(•, T k ) is considered as numéraire of the model, then discounted bond prices B(t,T k-1 ) B(t,T k ) , 0 ≤ t ≤ T k-1 should be martingales. An additional basic requirement motivated by applications is the tractability of the model, since otherwise one cannot calibrate to the market data. Therefore the LIBOR rate processes L(•, T k-1 ) should have tractable stochastic dynamics with respect to their forward measure IP T k , for k = 1, . . . , N ; for instance of exponential Lévy type along the discrete tenor of dates T 0 < . . . < T N . Here the terminus “analytically tractable” is used in the sense that either the density of the stochastic factors driving the LIBOR rate process is known explicitly, or the characteristic function. In both cases, the numerical evaluation, which is needed for calibration to the market, is easily done.

In applications, the stochastic factors have to be evaluated with respect to different numéraires. In order to describe the dynamics with respect to a suitable martingale measure, for instance the terminal forward measure IP T N , we have to perform a change of measure. Usually this change of measure destroys the tractable structure of L(•, T k-1 ) with respect to its forward measure. This well-known phenomenon makes LIBOR market models based on Brownian motions or Lévy processes quite delicate to apply for multi-LIBOR-dependent payoffs: either one performs expensive Monte Carlo simulations or one has to approximate the equation (the keyword here is “freezing the drift”, see e.g. Siopacha and Teichmann 2011).

In order to overcome this natural intractability, forward price models have been considered, where the tractability with respect to other forward measures is pertained when changing the measure. Hence, modeling forward prices F (•, T k-1 , T k ) = 1 + δL(•, T k ) produces a very tractable model class. However negative LIBOR rates can occur with positive probability, which contradicts any economic intuition.

In this work, we propose a new approach to modeling LIBOR rates based on affine processes. The approach follows the footsteps of forward price models, however, we are able to circumvent their drawback: in our approach LIBOR rates are almost surely non-negative. Moreover, the model remains analytically tractable with respect to all possible forward measures, hence the calibration and evaluation of derivatives is fairly simple. In fact, this is the first LIBOR model where the following are satisfied simultaneously:

• LIBOR rates are non-negative;

• caps and swaptions can be priced easily using Fourier methods, for several affine factor processes; • closed-form valuation formulas for caps and swaptions are derived for the CIR process, in 1-and 2-factor models.

A particular feature of our approach is that the factor process is a timehomogenous Markov process when we consider the model with respect to the terminal measure IP T N . With respect to forward measures the factor processes will show time-inhomogeneities due to the nature of the change of measure. When we compare our approach to an affine factor setting within the HJM-methodology, we observe that in both cases one can choose -with respect to the spot measure in the HJM setting or with respect to the terminal measure in our setting -a time-homogeneous factor process. LIBOR rates have in both cases a typical dependence on time-to-maturity T N -t.

The remainder of the article is organized as follows: in Section 2 we formulate basic axioms for LIBOR market models. In Section 3 we recapitulate the literature on LIBOR models. In Section 4 we introduce affine processes which are applied in Section 5 to the construction of certain martingales. In Section 6 we present our new approach to LIBOR market models, which is applied in Section 7 to derivative pricing. In Section 8 several examples, including the CIR-based models, are presented and in Section 9 we show prototypical volatility surfaces generated by the models.

Let us denote by L(t, T ) the time-t forward LIBOR rate that is settled at time T and received at time T + δ; here T denotes some finite time horizon. The LIBOR rate is related to the prices of zero coupon bonds, denoted by B(t, T ), and the forward price, denoted by F (t, T, T + δ), by the following equations:

1 + δL(t, T ) = B(t, T ) B(t, T + δ)

= F (t, T, T + δ).

(2.1)

One postulates that the LIBOR rate should satisfy the following axioms, motivated by economic theor

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