Study on Models of Term Structure Finance Essay

One of the earliest stochastic models of the term structure was developed by Vasicek (1977). His model is based on the evolution of an unspecified short-term interest rate. Although the model is developed in a fairly general framework, it is most frequently applied in a more specific form that is presented here. The Vasicek model has many advantages, though it also has some shortcomings. The model assumes that the short rate follows the stochastic process dr = a(b-r)dt + sdz Here, dz = a standard process a = coefficient for the formula r = the current level of the interest rate. b= parameter for long run normal interest rate If the interest rate is above the long run mean (r > b), then the coefficient a (>0) makes the drift become negative so that the rate will be pulled down in the direction of r. Likewise, if the rate is less than the long run mean, r < b, then the coefficient a makes the drift become positive so that the rate will be pulled up in the direction of r. The coefficient a is, thus, the speed of adjustment of the interest rate towards its long run normal level. This feature is particularly attractive because without it, interest rates could drift permanently upward the way stock prices do and this is simply not observed in practice. In our century the interest rate has an important place in all transactions that include simply lending and borrowing. However, its importance has risen with developing and slight stationary economic conditions of the world. The desire to have foresight for the level of future interest rate has become crucial in the sense that to know the lending and borrowing rate and not to miss price the interest rate instruments. As a result, the modeling of interest rate has risen as a problem. As solution to this problem there have been many models proposed. Their common feature is that they all model a stochastic problem, i.e. they deal with uncertainty. The first model in 1973 was proposed by Merton. However, the pioneering one was suggested by Vasicek in 1977 and in the following years many there models that are much more analytically un tractable has come out. These stochastic models are mainly classified with respect to the number of factors which are assumed to have a stochastic evolution in the model. In this study we will use Vasicek short-rate model among some of the most fundamental one factor interest-rate models to predict the yield curve of tomorrow by using today's observed yield data It was introduced by Oldrich Vasicek in the year of 1977. It is a mathematical model that describes the interest rate growth. The Vasicek Model is a type of "one-factor model" that explains the movements of short time interest rate when it is only driven by the market risk. In Vasicek model wiener process shows the random market risk while the standard deviation shows the volatility of the interest rate. The Vasicek Model is also applied for the interest rate derivative assessment. The theory of Vasicek Model is very well adapted by the credit markets. However the interest rate may not be original future interest rate but the interest rate must be closer to the original rate. If we use correctly the Brownian motion and the volatility related to the market rate than it may give the right desired outcomes. However the major problem will occur when there will be a sudden change in market risk. Vasicek's revolutionary work is the earliest account of a bond pricing model that incorporates stochastic interest rate. The short rate dynamics is modeled as a distribution process with constant parameters. When the bond price is based on this assumption, it has the feature that on a given date, the ratio of expected excess return per unit of volatility (the market price of risk) is the same, regardless of bond's maturity.

2.6.1 Pricing Bonds with Vasicek Model

Given the term structure of interest rates (fully described in this model by the spot rate) the pricing formula for bonds is straight forward. Bond prices are determined by summing the present value of the coupons and terminal value, discounting at the discount factors P(r, 0, T). Thus, the value of a bond paying coupon c at time 1, 2, . . . , M and having face value of 1 is given by the function: Bond Price[c, r, M]:= Sum [P[r, 0, t] * c, {t, 1, M}] + P[r, 0, M] It is often convenient to consider continuously paid out coupons, in which case the above formula becomes: Bond Price[c, r, M]:= Integrate [P[r, 0, t] * c, {t, 0, M}] + P[r, 0, M] A European option on a bond can also be priced analytically with the Vasicek term structure model.

2.6.2 Advantage and disadvantages of Vasicek model

We know that nothing is perfect in the world. Every effective thing has negative effect with their positive elements. The advantages and disadvantage of Vasicek model are as follow:

a) Advantages

The method is based on reliable mathematical calculation which makes the future forecasted interest rate near to the future original interest rate. It's a short rate model which gives a future interest rate nearly to the same which will happen in future like future spot rate. It also helps in risk hedging. By using the Vasicek model we can forecast the future interest rate and if there is any uncertain risk in interest rate then we can hedge our amount which can be affected by the uncertain risk. The interest rate is calculated for the shorter time period under Vasicek model i.e. the volatility factors almost keep the same as it was taking at the time of interest rate derivation.

b) Disadvantages

shortcomings of the Vasicek model: Interest rate can become negative. It only happens when people like to hold the cash instead to invest it. Difficult to make the model fit the yield curve exactly. Like if there occur a change between the derived interest rate and the original interest rate then the yield curve of derived interest rate will vary from the original yield curve. And it affects the other estimations based on the model. Constant local volatility is not realistic, but it is easy to generalize with market rate but we know that the volatility changes as the time is passes on. So it's not good to generalize the volatility. Numerous generalizations are possible. However the more generalization input will affect the output result. The estimation is always unreliable. If we take rates of two or three parameters as general rates then it may affect our future forecasting rate. It must be different from the original one. Even with the Vasicek model, we need to solve non-linear equations to construct the tree. So its mean there is more than one variable and sometime it's very difficult to calculate the more variables. Increase in variables make chances of error high. More complicated models typically imply that we need to solve even more complicated equations. Because to solve the more variable we need to solve more equations.

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