# Distribution and parameters for the amount at time T of Bond

An investor follows the following investment strategy from time t to time T: buys a 10-year zero coupon bond, holds it for a time-length dt, sells it and buys a new 10-year ZCB with the proceeds. The process is repeated continuously. Derive the distribution and parameters for the amount at time T if he starts with a value of 1

Im not really sure where you will start on this question

• It seems like there is a missing "dt" in the stochastic differential equation. Jun 18 '20 at 18:04

Here is a simpler version which you can generalise to your problem. As you outlined in the question, the price of a T maturity ZC-bond is related to the short rate as follows:

$$P\left(t,T\right)=e^{a\left(t,T\right)- b\left(t,T\right)\, r_t}$$

Which is essentially an expression of this form (assume given t and T):

$$P\left(r\right)=Ae^{-rB}$$

You know r under the Vasicek follows a normal distribution, so you are essentially after the determination of the density of the exponential of a normal, you know the result will be log normal, but in terms of the steps to get there, you can use the density transformation formula for a monotonic function of a random variable:

$$f_p(p)=f_r (P^{-1}(p)) \left| \frac{dr}{dp} \right|$$

You can calculate the inverse and the derivative of r with respect to P easily:

$$P(r)=A e^{-r B}=p \quad \Rightarrow r=-\frac{1}{B}ln \frac{p}{A} =P^{-1}(p), \mbox{ and } \frac{dr}{dp}= -\frac{1}{Bp}$$

And you can then plug these in the density transformation formula to get:

$$f_p(p)= f_r \left( -\frac{1}{B}ln \frac{p}{A} \right) \frac{1}{Bp}$$

You jsut need to plug in the density of r, which is $$f_r$$, and you have the density of P.

• Thank you, I still don't understand how this will be made into continuously Jun 19 '20 at 10:47
• Additonal hint: every day you have the P&L of a fixed T year zero coupon bond and Vasicek parameters driving distribution of r are fixed Jun 19 '20 at 11:10