The Vasicek model is a one-factor equilibrium model on interest rates based on a Brownian geometric process that takes into account the mean reversion and the time structure of interest rates.
In other words, the Vasicek model is used to predict long-term interest rates by simulating short-term interest rates. In addition, it takes into account that interest rates are different in different periods of time (time structure of interest rates).
Equilibrium interest rate models use shorter-term interest rates to calculate future interest rates taking into account the term structure of interest rates.
To construct the yield curve we need the short-term interest rates and the parameters of the model. Once we have the short-term interest rates and the parameters, we can calculate the long-term interest rates.
So to calculate future zero coupon bond prices, we need short-term zero coupon interest rates. In this way, we can also build the curve or time structure of zero coupon interest rates. Once we have the curve, we will determine the evolution of long-term interest rates given the short-term interest rates.
Vasicek Model Formula: Zero Coupon Bond Price.
Analytical solution to find the price of a zero coupon bond that pays € 1 at maturity (T) in any period of time (t) and at a short-term interest rate (r (t)).
Do not panic!
We just need:
- The period of time in which we want to know the interest rates, that is, T.
- The moment of time in which we are now or the starting moment we want, that is, t.
- The short-term interest curve, that is, r (T) or T. If we wanted to express interest rates in the starting period, we would use r (T) or T.
- In these formulas we will treat the parameters a, b and s as constants in time.
- The standard deviation, s.
To calculate the price of a zero coupon bond that pays € 1 at maturity (T) in any period of time (t) we only have to give values to the parameters a, b and s and simulate the short-term interest rates (r (t )).
Representation of the Vasicek Model: Zero Coupon Bond Price
P (t, T) represents the price of the bond from time t to T.
So… Are bond prices always going to be this way?
Not at all, as we said at the beginning, interest rates depend on a Brownian geometric process, and therefore, implies the presence of a random component, N (0,1). So each time we calculate the above formulas, the short-term rates will change and so will the long-term interest rates, bond prices, and their representation.
We will use the following formulas to find r (T) and R (T).
Vasicek Model Formula: Short-Term Interest Rates
Short-term interest rate formula (r T ):
Long-term interest rate formula (R T ):