A theoretical and empirical analysis of the Libor Market Model and its application in the South African SAFEX Jibar Market

摘要

Instantaneous rate models, although theoretically satisfying, are lessso in practice. Instantaneous rates are not observable and calibra-tion to market data is complicated. Hence, the need for a marketmodel where one models LIBOR rates seems imperative. In thismodeling process, we aim at regaining the Black-76 formula[7] forpricing caps and °oors since these are the ones used in the market.To regain the Black-76 formula we have to model the LIBOR ratesas log-normal processes. The whole construction method meanscalibration by using market data for caps, °oors and swaptionsis straightforward. Brace, Gatarek and Musiela[8] and, Miltersen,Sandmann and Sondermann[25] showed that it is possible to con-struct an arbitrage-free interest rate model in which the LIBORrates follow a log-normal process leading to Black-type pricing for-mulae for caps and °oors. The key to their approach is to startdirectly with modeling observed market rates, LIBOR rates in thiscase, instead of instantaneous spot rates or forward rates. There-after, the market models, which are consistent and arbitrage-free[6],[22], [8], can be used to price more exotic instruments. This modelis known as the LIBOR Market Model.In a similar fashion, Jamshidian[22] (1998) showed how to con-struct an arbitrage-free interest rate model that yields Black-typepricing formulae for a certain set of swaptions. In this particularcase, one starts with modeling forward swap rates as log-normalprocesses. This model is known as the Swap Market Model.Some of the advantages of market models as compared to othertraditional models are that market models imply pricing formulae forcaplets, °oorlets or swaptions that correspond to market practice.Consequently, calibration of such models is relatively simple[8].The plan of this work is as follows. Firstly, we present an em-pirical analysis of the standard risk-neutral valuation approach, theforward risk-adjusted valuation approach, and elaborate the pro-cess of computing the forward risk-adjusted measure. Secondly, wepresent the formulation of the LIBOR and Swap market modelsbased on a ¯nite number of bond prices[6], [8]. The technique usedwill enable us to formulate and name a new model for the SouthAfrican market, the SAFEX-JIBAR model.In [5], a new approach for the estimation of the volatility of theinstantaneous short interest rate was proposed. A relationship between observed LIBOR rates and certain unobserved instantaneousforward rates was established. Since data are observed discretely intime, the stochastic dynamics for these rates were determined un-der the corresponding risk-neutral measure and a ¯ltering estimationalgorithm for the time-discretised interest rate dynamics was pro-posed.Thirdly, the SAFEX-JIBAR market model is formulated based onthe assumption that the forward JIBAR rates follow a log-normalprocess. Formulae of the Black-type are deduced and applied to thepricing of a Rand Merchant Bank cap/°oor. In addition, the corre-sponding formulae for the Greeks are deduced. The JIBAR is thencompared to other well known models by numerical results.Lastly, we perform some computational analysis in the followingmanner. We generate bond and caplet prices using Hull's [19] stan-dard market model and calibrate the LIBOR model to the cap curve,i.e determine the implied volatilities ¾i's which can then be usedto assess the volatility most appropriate for pricing the instrumentunder consideration. Having done that, we calibrate the Ho-Leemodel to the bond curve obtained by our standard market model.We numerically compute caplet prices using the Black-76 formula for caplets and compare these prices to the ones obtained using thestandard market model. Finally we compute and compare swaptionprices obtained by our standard market model and by the LIBORmodel.