Which of those processes indicate an arbitrage free family of bonds in terms of r? $T^*=1$. The process $r_t$ is given by:
a) $r_t = 1$ for all $t \in [0,1]$
b) $r_t = B_t$ for all $t \in [0,1]$
c) $r_t = B_1$ for all $t \in [0,1]$
d) $r_t = t + B_t$ for all $t \in [0,1]$
I know that I should check if:
- $B(T,T)=1$
- there exists a probility meassure such that: $\frac{B(t,T)}{B_t}$ is a martingale
What's is more:
$$\frac{B(t,T)}{B_t} = \mathbb{E}(\frac{1}{B_T}|\mathcal{F}_t)$$ and $\mathbb{E}(X|\mathcal{F})$ is a martingale for X - measurable, integrable random variable
To be honest I am not quite sure if $B_t$ donotes Brownian motion or a banking account process.
Ad a) yes
$B_t = \mathbb{E}(\exp(\int \limits_0^t r_u du)) = e^t$
$\frac{B(t,T)}{B_t} = \mathbb{E}(\frac{1}{B_T}|\mathcal{F}_t)= e^{-T}$ - a martingale
$B(1,1) = 1$
Please check, if my solution to point a) is correct and help me with others.