Averaged bond prices for Fong-Vasicek and the generalized Vasicek interest rates models

In short rate interest rate models, the behaviour of the short rate is given by a stochastic dierential equation (1-factor models) or a system of stochastic dierential equations (mul- tifactor models). Interest rates with dierent maturities are determined by bond prices, which are solutions of the parabolic partial dierential equation. We consider the gener- alized 2-factor Vasicek model and Fong-Vasicek model with stochastic volatility. In the 2-factor Vasicek model, the short rate is a sum of two independent Ornstein-Uhlenbeck processes. The bond price is a function of maturity and level of each of the components of the short rate. In Fong-Vasicek model, the volatility of the short rate is stochastic. The bond price is a function of maturity, short rate and volatility. In both cases, we do not observe all values necessary to obtain a bond price. Therefore, we propose the averaging of the bond prices. We consider the limiting probability distribution of unobservable vari- ables. In this way, we obtain the averaged bond prices depending only on the maturity and short rate. We prove that there is no 1-factor model yielding the same bond prices as are the averaged values described above. them, for the instanteneous interest rate r (short rate). The bond prices, and hence the term structures of the interest rates, are then obtained by solving the partial dierential equation. We consider 2-factor generalized Vasicek and Fong-Vasicek two factor models. In the generalized Vasicek model, the bond price is a function of maturity, and the level of the short rate components, which are unobservable in the market separately. In Fong-Vasicek model, the bond price is a function of maturity, the level of the short rate and the level of volatility. The volatility is an unobservable parameter. Since each of these model contains unobservable quantities, the interesting questions are the properties of the averaging of the bond prices with respect to the distribution of these unobservale variables. This is motived by papers (10) about averaging in stochastic volatility models of stock prices and (7) about averaging in stochastic volatility models of bond prices which are used in the series expansion of the prices. The asymptotic distribution of the hidden process is used. It can be justified if the processes have been evolving for a suciently