“Pricing interest rate derivatives under different interest rate modeling: a critical and empirical comparison” AUTHORS Antonio De Simone Antonio De Simone (2010). Pricing interest rate derivatives under different ARTICLE INFO interest rate modeling: a critical and empirical comparison. Investment Management and Financial Innovations, 7(2) RELEASED ON Friday, 23 April 2010 JOURNAL "Investment Management and Financial Innovations" FOUNDER LLC “Consulting Publishing Company “Business Perspectives” NUMBER OF REFERENCES NUMBER OF FIGURES NUMBER OF TABLES 0 0 0 © The author(s) 2021. This publication is an open access article. businessperspectives.org Investment Management and Financial Innovations, Volume 7, Issue 2, 2010 Antonio De Simone (Italy) Pricing interest rate derivatives under different interest rate modeling: a critical and empirical comparison Abstract This paper deals with issues related to the choice of the interest rate model to price interest rate derivatives. After the development of the market models, choosing the interest rate model has become almost a trivial task. However, their use is not always possible, so that the problem of choosing the right methodology still remains. The aim of this paper is to compare some of the most used interest rate derivatives pricing models to understand what the issues are and the drawbacks connected to each one. It is shown why and in which cases the use of each model does not give appreciable results and when, on the other hand, the opposite occurs. More exactly, it will be shown that the lack of data on the implied volatilities or the ineffi- ciencies in the financial market can prevent the use of the market models, because a satisfying calibration of the inter- est rate trajectories cannot be guaranteed. Moreover, it is shown how the smile effect in the interest rate options market can affect the price provided by each model and, more exactly, that the difference between the price provided by the models and the observed market prices gets larger, as the strike price increases. Keywords: Cox Ingersoll and Ross, Black Derman and Toy, Libor Market Model, Lognormal Forward-Libor Model, cap, interest rate risk. JEL Classification: G12, C53, C65. Introduction© think to use some other interest rate models to get a “better” price. This paper deals with the issues linked to the valua- tion of interest rate derivatives such as caps and The aim of this paper is to point out when LMM floors and, more precisely, with the question of does not give appreciable results and to show some which model practitioners should choose to obtain empirical criteria on how to choose the right meth- the fair value of Over The Counter (OTC) interest odology, when practitioners face the task of evaluat- rate derivatives. ing interest rate derivatives. To begin with, it is possible to point out that an ef- This paper focuses in particular on three interest rate fective pricing process should allow to obtain the models, which are the most famous and the most unknown price of a financial contract as consistent used in practice: the first one (model A) is the Cox, as possible with the observed prices of other instru- Ingersoll and Ross (CIR, 1985) model, and it is one ments, so that arbitrage opportunities are secluded. of the first stochastic models of the term structure For this reason, an effective pricing model should proposed in literature; the second one (model B) is replicate the observed current prices of other finan- the Black, Derman and Toy model (BDT, 1990), and it cial securities, as far as it is possible. was one of the most popular ones before the advent of Libor Market Model; the third model (model C) is the In the financial markets, after the advent of the still mentioned Libor Market Model, and more exactly “market models”, choosing the right methodology in the Lognormal Forward-Libor Model (LFM), in the pricing has become almost a trivial task, because version proposed for the first time by Brace, Gatarek these models offer an easy way to calibrate the fu- and Musiela (Brace et al., 1997). ture trajectories of interest rates, so that the current market prices can generally be replicated. Remark The paper follows an inductive approach by report- that not every interest rate model offers this possi- ing empirical evidence, whose results can suggest bility: for example, the Libor Market Model some general rules. Starting by pricing a simple (LMM, Brace et al. 1997), which actually is one of interest rate derivative (e.g., a cap) by means of the the most popular, allows to obtain prices consis- three mentioned models, some shortcomings arise, tently with the standard market practice of pricing as well as other some interesting aspects involved in caps, floors and swap-options by using the Black’s pricing derivatives. In fact, by pricing a target con- formula (Black, 1976). tract it will be evident when the use of the LMM could not give appreciable results; moreover, by However, if on the one hand the LMM seems to be a applying every model to the same target contract, powerful tool in pricing interest rate derivatives, on the quantitative differences between the prices gen- the other hand, in some circumstances, its usage erated by each one will be shown. At the end of the does not give satisfactory results, so that one could comparison, interesting suggestions as to which model practitioners should generally choose will be © Antonio De Simone, 2010. available. 49 Investment Management and Financial Innovations, Volume 7, Issue 2, 2010 The paper proceeds as follows. Section 1 offers a (2007), where one of the key issues faced by the review of the existing literature. Section 2 illustrates author is to establish criteria for model quality; issue the target contract, the data used in pricing it as well which is somewhat linked to this work. However, it as a brief description of the interest rates derivatives can be pointed out that Jacobs focuses his attention pricing models mentioned before. Moreover, this on continuous-time stochastic interest rate and sto- section highlights other important aspects concern- chastic volatility models, such as CIR model and ing the critical application of the LMM, as well as Heat, Jarrow and Morton (HJM, 1992) model, but the concept of “appreciable results”. Section 3 re- he does not take into account the LMM or any other ports the comparison of the models and the corre- discrete-time stochastic interest rate model. sponding results. In particular, this section reports Another important work related to this, which is some empirical evidence on the weaknesses of the closely linked in particular to the market models, BDT and of the CIR models, as well as some advice can be found in Plesser, de Jong and Driessen about the selection of the pricing methodology. Fi- (2001), where nevertheless, the attention is focused nally, the last section provides final remarks and on the Libor Market model and on the Swap Market conclusions. model only, and no comparison is made between 1. The background continuous-time and discrete-time interest rate mod- els; comparison that, on the other hand, is central in Although plenty of papers on pricing interest rates this work. derivatives have been written, the same does not hold for topics related to the comparison between A broader, interesting analysis on interest rate interest rate models. derivatives pricing models is carried out by Bar- one (2004), where almost each kind of model is In fact, it is remarkable that this paper originally studied, included continuous and discrete time provides a comparison of models with heteroge- models, equilibrium and arbitrage models, one neous features, because its aim is closely linked to factor and multifactor models as well. However, the necessity of choosing the pricing methodology the comparison between all these models is based in pricing interest rate derivatives, from a profes- on a theoretical point of view only, where aspects sional point of view. On the other hand, it is no- linked to the concrete application to pricing, ticeable that the literature about comparisons of hedging and risk management issues are not cen- interest rate derivatives pricing models appears tral in those work. not to be very wide; moreover, the comparison is often made among models with homogeneous This work will in fact try to use an approach similar characteristics. to that followed by Jacobs and Plesser, which is based on empirical analysis, without renounce to To begin with, it may be pointed out that a relevant report some important considerations on the finan- work that tries to compare interest rate models with cial theories on which models are based; considera- heterogeneous characteristics can be found in tions which can moreover be used to carry out some Khan et al. (2008), where the comparison in- important conclusions about the use of interest rate volves the Hull-White and the Black-Karasinski models themselves. model. However, the most popular market models are not considered in that work, also because its Finally, it is noticeable that consideration about the aim is linked to risk management rather than pric- usage of the BDT model can be found in Bali et al. ing issues. (1999), where a comparison between two different approaches in determining the volatility parameters It can be highlighted that important consideration on is offered. Also, if the approach to estimating the the drawbacks of the models, which are of funda- volatility is completely different, this issue is faced mental importance to establish whether and to what in this paper too. extent an interest rate model can be successfully used in pricing derivatives, can be found in the 2. Pricing interest rate derivatives works of the authors that for the first time developed In this section target contract, necessary data and the models themselves, and in particular some atten- applied models are presented.