Vasicek and Extended Vasicek Model

Question

I want to ask about basic reasoning in Vasicek and Extended Vasicek model.

1. Why $P(T,T) = 1$ for non arbitrage model? Can we place $P(T,T) = 10$ or other numbers? Is it correlated with The Law of Single Price?

2. How can you have an idea to write $P(t,T,r) = A e^{-Br}$ or $e^{A - Br}$. Why not other function?

3. What is actually market price of risk?

Answer 1

1. In general, for $t<T$, $P(t, T)$ is the price at time $t$ of a $T$-maturity zero coupon bond with a principal of $1$. It is commonly called the discount factor between time $t$ and $T$, since it is the value at time $t$ of receiving $1$ unit at time $T$. Using this idea, for $t=T$, it is easy to see why $P(T, T)=1$. What is the value at time $T$ of receiving $1$ unit at time $T$? It is simply $1$! Of course, you can change your notation and let $P(t, T)$ denote a zero coupon bond with a principal of e.g. $10$, thereby changing your numeraire, which would result in the no-arbitrage condition $P(T, T) = 10$.

2. This was simply presented as a "guess" in the initial papers. This guess in turn seems to work out very nicely (in this model and other affine term structure models).

3. The market price of risk is extensively covered many places on this site and elsewhere.