The asset pricing when the interest rates are differentiable stochastic processes

Abstract

This paper considers a problem of asset pricing for case when the short-term interest rate process does not have the markovian property. In this case the price can be determined also by state variables some of that are not observable. In the same time from the practical point of view, the mathematical expression for the asset price is acceptable for the partici-pants of the market if it includes only observable variables. Therefore procedure of elimina-tion from this mathematical expression of all not observable components of vector of state variables should be developed. In stochastic problems it is assumed to eliminate not observ-able indexes by taking the conditional expectation. Such approach is used in this paper. It is supposed that the interest rate process is differentiable but its mathematical derivative of some order is a diffusion process. In this case the values of this process at future times de-pend on values of process and its derivatives at present time. It means that there is a depend-ence on the process path. It is derived the expression for determination the asset price under these conditions. In this relation to usual formula for price some multiplier is added that de-pends on stochastic properties of mathematical derivatives of interest rate process. Extension of the Vasicek model on the differentiable processes is introduced. The comparison the bond price for this extension with bond price of standard Vasicek model is made. The plan of paper is as follow. In Introduction the problem substitution is made and the state variables are determined. In Section 2 the no arbitrage condition for multi-factor model of term structure is given. The equation for the asset price at general multi-factor model is derived in Section 3. In next section it is shown as to eliminate not observable components of state variables. Section 5 contains the analysis of differentiable short-term interest rate proc-esses. Special case of interest rate processes with one derivative is given in detail in Section 6. Equation for the asset price when the short-term interest rate process is differentiable is derived in Section 7. In Section 8 the extension of the Vasicek model is considered.