Short Rate Models

Abstract

The research work presented actually tries to compare and contrast the performance and robustness of the short rate models, Vasicek, Cox-Ingersoll-Ross and Hull White to be precise. The models are tested for their accuracy to price the bonds and produce the yield curves as close to the original market yields as possible. Concept of Mean Square errors has been used to evaluate the degree of errors and to nullify any negative sign effects. An example of 1- month and 3-month MIBOR (Mumbai Interbank Offer Rates) has been taken and worked upon for calculating the yield curves from three models. The errors have been found taking the original Bond prices as the bench marks. Based on the various calculations done, an effort is made to find any arbitrage opportunities.

Literature Review

Financial research always promotes the quantitative approach towards the term structure of interest rates. The most critical component of the study over the time has been the term structure and the various factors affecting the same. Term structure has been defined as the relationship between the yield (payoff) and the maturity time. Generally, term structures are plotted graphically. To get a meaningful output from the term structures, factors like the taxation rate, default risk etc that affect the interest rates should be kept constant. The term structure, also known as yield curve can take any shape but the typical shape of the curves are: normal curve, flat curve, steep or humped curve and inverted curve.

Equilibrium V/s No Arbitrage:

Equilibrium models start with an equilibrium condition and do not take bond prices as mere exogenous inputs. These models contain specific assumptions about the dynamics of diving factors. They determine the different shapes that yield curves can attain and also try to apportion the contribution of curvatures of yield curves to future expectations of rates and volatility (Rebonato, 2000). No Arbitrage models start with no arbitrage condition, which is implied in the term structure else market forces would quickly eliminate the opportunities of arbitrage. Such models take current term structure as the input to the model and hence the model unlike equilibrium models gives the current term structure of interest rates as a solution for time t=0 (Schulmerich, 2005). No-arbitrage models are calibrated to fit the observed forward curve perfectly. This approach to modelling the term-structure was pioneered by Ho and Lee (1986). Hull and White (1990) also built a no-arbitrage model that extended the Vasicek model to fit the initial term-structure

One Factor Model V/s Multi Factor Models:

In one factor model a single factor influences the shape and development of term structure of interest rates over the time. Hence these models can only model a parallel shift but not the twist or butterfly movements in the term structure. A perfect correlation (parallel shift) between the short and long end of yield curve is the basic assumption of one factor models. However, the situation stated above is a very idealistic situation. In reality it is observed that the short and long end of yield curves are influenced by more than one risk factors that are not perfectly correlated and we need at least two factors to quantify term structure models. Minimum of two factors are needed to allow uncorrelated and unparallel shifts of short and long ends of curve. Multi factor models are the ones with two or more factors into consideration. Amin and Morton (1994) test different parsimonious one and two factor parameterizations and find that the number of parameters has a stronger effect on the behaviour of the model than does the form of the models used. Two-parameter models tend to fit prices better.

Pai & Penderson (2002) say that there is another way to improve the empirical fit of the model without using the second factor. This is possible by using the threshold model. The single factor threshold model is locally subject to many shortcomings of the traditional one factor models. Chief amongst them is the perfect correlation of yield rate movements for changes in the short rate which remain within the same striations. This perfect correlation is broken as soon as the short rate process is near the boundaries. One factor threshold model can pick up the changes in the volatility of the short rates that appear to depend on the level of short rate. This is one of the compelling reasons to study the said models.

Mean Reversion models V/s No mean reversion:

Mean reversion models are the ones in which the interest rate gets back or more technically is pulled back to a mean interest rate value. Mean reversion models find a place not only in interest rate modelling but also in commodity pricing (Lieskowsky etal., 2000)

There are various versions of the expectations hypothesis. But in simplest form it postulates that bonds are priced so that implied forward rates are equal to the expected spot rates (Cox etal, 1985). Hicks (1946) proposed the liquidity preference hypothesis and agreed that risk aversion will cause forward rates to be symmetrically greater than expected spot rates, usually by an amount increasing with maturity. Culbertson (1957) proposed the market segmentation hypothesis which says that demand and supply of bonds of a particular maturity are supposedly less affected by the prices of bonds of different maturities. But since all the previous studies were based on hypothesis and they all explained a little more than that forward rates should or need not equal the spot rates, a need for the better understanding of term structures was felt.

To define the complete probabilistic model for the term structure, we need to specify both the current term structure and the yield curve. Therefore, the short-term rate is very important to specify the same. The prices of bonds and interest rate derivatives can be calculated by constructing the term structure of interest rates. The problem then arose to define and predict the movement of interest rates. For the same academicians like Vasicek, Cox, Hull etc proposed that the under the risk neutral world, the interest rate, r, follows a diffusion process given by

where   is the drift, is the standard deviation of rate r and is the Weiner process. This is the equation of a one factor model where both the drift and the standard deviation is independent of time but depends on the rate.

With the advent of first model by Oldrich Vasicek (1977), many researchers developed various types of interest rate models. In this thesis, three models of term structure of interest rates will be presented in detail. They are

• Vasicek Model
• Cox-Ingersoll-Ross model
• Hull-White Model

In the same direction the first model for the term structure of interest rates was developed by Oldrich Vasicek in 1977.Later on the developments happened on the shortcomings of this and further models. One of the key points that have been put across is the way these models capture the actual short-term risk less rate.
References

• Amin, K.I., and A.J. Morton. (1994), “Implied Volatility Functions in Arbitrage-Free Term Structure Models,” Journal of Financial Economics, 141-180.
• Cox, J.C., J.E. Ingersoll and S.A. Ross (1985), “A Theory of the Term Structure of Interest Rates,” Econometrica, 53 , 385-407.
• Culbertson J,M.(1957) . “The term structure of Interest Rates”, Quarterly’ Journal of Economics, 71 , 485-517
• Hicks, J.R. (1946). “Value and Capita”, 2nd edition , Oxford University press, London.
• Ho, T.S.Y., and S.B. Lee. (1986), “Term Structure Movements and Pricing Interest Rate Contingent Claims,” Journal of Finance, 41, 1011-1029.
• Hull, J., and A. White. (1990), “Numerical Procedures for Implementing Term Structure Models II: Two-Factor Models,” Journal of Derivatives , 2 ,37-47.
• Lieskovsky, J.R. , R.Onkey, M.Schulmerich , C.C. Teng & J.Wee,(2000) “Pricing and Hedging Asian Options on Copper” , Working paper , MIT Sloan School of Management, Laboratory of Financial Engineering, December 2000.
• Pai, J. and H.W. Penderson, (2002), “Threshold Models of the Term Structure of Interest Rates”, (www.actuaries.org/AFIR/colloquia/Tokyo/Pai_Pedersen.pdf)
• Rebonato, R., “Interest Rate Option Models, 2nd Edition, John Wiley & Sons, 2000”
• Schulmerich , M.(2005) “Real Options Valuation: The importance of interest Rate Modeling in Theory and Practice, Springer –Verlag Berlin Heidelberg, 2005”
• Vasicek, O. (1977), “An Equilibrium Characterization of the Term Structure,” Journal of Financial Economics, 5 177-188.