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. / Economics / D.Phil. (Operations Research)

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. / Economics / D.Phil. (Operations Research)

Thesis (MComm (Mathematics))--University of Stellenbosch, 2007. / The word numéraire refers to the unit of measurement used to value a portfolio of assets. The change of numéraire technique involves converting from one measurement to another. The foreign exchange markets are natural settings for interpreting this technique (but are by no means the only examples). This dissertation includes elementary facts about the change of numeraire technique. It also discusses the mathematical soundness of the technique in the abstract setting of Delbaen and Schachermayer’s Mathematics of Arbitrage. The technique is then applied to financial pricing problems. The right choice of numéraire could be an elegant approach to solving a pricing problem or could simplify computation and modelling.