Discussion on the Vasicek Model

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  • Topic: Short rate model, Bond, Mathematical finance
  • Pages : 8 (2612 words )
  • Download(s) : 98
  • Published : September 24, 2008
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Introduction
Movements of interest rates are vital in the decision making process of investments and risk management in financial markets (Zeytun and Gupta, 2007). A method that can help to predict interest rate movements are term structure modelling is based on the theory that describes the behaviour on interest rate though the interest rate. These models can help to seek to identify the elements or factors that are believed to explain the dynamics of interest rates (Choudhry et al, 2002). Interest rate models can help in predicting the movement of the interest rates which in turn can help in predicting the pricing of financial derivatives movements. Term structure models can be used for valuing interest rate derivatives such as American style swap options, callable bonds and structured notes (Hull, 2006). A number of theoretical equilibrium models have been proposed in the recent past to describe the term structure of interest rates, such as Brennan and Schwartz [1979], Cox, Ingersoll, and Ross [1981], Langetieg [1980], and Vasicek [1977]. These models postulate alternative assumptions about the nature of the stochastic process driving interest rates, and deduct a characterization of the term structure implied by these assumptions in an efficiently operating market. These models can be categorised into two main models, arbitrage free models and equilibrium models. Arbitrage free Models

These models match the observed prices in the market. These models does not allowed for arbitrage profits to be realized in the markets, by basing an arbitrage strategy on the on the values generated by the model and actual market prices. Models under this category include the Ho-Lee model and the Heath-Jarrow-Morton model.

Equilibrium Models.
The equilibrium models explain the term structure, based on economic fundamentals that affect the term structure. The models usually start off with the assumptions about economic variables leading to the derivation of a process for the short term rate r. The short term rate is than tested in order to observe the implications on bond prices and option prices (Hull, 2006). Restrictions are placed in order to derive bond and derivative prices to be derived (Fabozzi, 2007). Models under this category include the Cox-Ingersoll-Ross mode and the Vasicek model. This paper will attempt to explore the Vasicek, looking at the contribution, limitations as well as looking at how the Vasicek model was employed in other empirical studies.

The Vasicek model
Vasicek in 1977 developed a model which derives a general form of the interest rate structure term which is a yield based one factor equilibrium model. This model assumes that the short rate process follows a normal distribution, which incorporates mean reversion, is popular with certain practitioners as well as academics because it is tractable. The development of the model is based on an arbitrage argument similar to the works of Black and Scholes (1973) for option pricing. The model is formulated in continuous time, although some implications for discrete interest rate series were noted.

The Vasicek model assumes that the risk neutral process for r is:- dr+a(b-r) dt + σ dz
a, b and σ are constants. The short rate is pulled to a level b at rate a. This pull is a normally distributed stochastic term σ dz. If interest rates are high, this part of the equation will become negative and therefore, lead to a decrease in interest rates, and thus keeping interest rates at a certain targeted level (Zeytun and Gupta, 2007). If interest rates are too low, this part of the equation becomes larger, pushing interest rates higher, and avoiding a freefall in interest rates (Zeytun and Gupta, 2007). The left hand part of the equation is a disturbance term that affects the process of r(t). There are also compelling economic arguments in favour of mean reversions, that when interest rates are high, the economy tends to slow down and borrowers require less...
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