Vasicek and Extended Vasicek Model

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?

1. In general, for $$t, $$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$$.
• This is exactly because of the discounting property described above. You can use the ZCB prices to discount other cash flows. Basically, if you have a coupon bearing bond that pays 5% annually on a \$100 principal for 3 years, you get \$5 a year for 3 years. This is equivalent to holding 5 ZCBs with maturity 1, 5 ZCBs with maturity 2, and 5 ZCBs with maturity 3. Hence, the coupon bearing bond price is simply the sum of its component ZCB prices. – AdB May 26 at 7:38