Abstract:
Instantaneous rate models, although theoretically satisfying, are less
so in practice. Instantaneous rates are not observable and calibra-
tion to market data is complicated. Hence, the need for a market
model where one models LIBOR rates seems imperative. In this
modeling process, we aim at regaining the Black-76 formula[7] for
pricing caps and °oors since these are the ones used in the market.
To regain the Black-76 formula we have to model the LIBOR rates
as log-normal processes. The whole construction method means
calibration by using market data for caps, °oors and swaptions
is 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 LIBOR
rates follow a log-normal process leading to Black-type pricing for-
mulae for caps and °oors. The key to their approach is to start
directly with modeling observed market rates, LIBOR rates in this
case, 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 model
is 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-type
pricing formulae for a certain set of swaptions. In this particular
case, one starts with modeling forward swap rates as log-normal
processes. This model is known as the Swap Market Model.
Some of the advantages of market models as compared to other
traditional models are that market models imply pricing formulae for
caplets, °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, the
forward risk-adjusted valuation approach, and elaborate the pro-
cess of computing the forward risk-adjusted measure. Secondly, we
present the formulation of the LIBOR and Swap market models
based on a ¯nite number of bond prices[6], [8]. The technique used
will enable us to formulate and name a new model for the South
African market, the SAFEX-JIBAR model.
In [5], a new approach for the estimation of the volatility of the
instantaneous short interest rate was proposed. A relationship between observed LIBOR rates and certain unobserved instantaneous
forward rates was established. Since data are observed discretely in
time, the stochastic dynamics for these rates were determined un-
der the corresponding risk-neutral measure and a ¯ltering estimation
algorithm for the time-discretised interest rate dynamics was pro-
posed.
Thirdly, the SAFEX-JIBAR market model is formulated based on
the assumption that the forward JIBAR rates follow a log-normal
process. Formulae of the Black-type are deduced and applied to the
pricing of a Rand Merchant Bank cap/°oor. In addition, the corre-
sponding formulae for the Greeks are deduced. The JIBAR is then
compared to other well known models by numerical results.
Lastly, we perform some computational analysis in the following
manner. 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 used
to assess the volatility most appropriate for pricing the instrument
under consideration. Having done that, we calibrate the Ho-Lee
model 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 the
standard market model. Finally we compute and compare swaption
prices obtained by our standard market model and by the LIBOR
model.
