Was hoping somebody could help me with the following question.
Prove that under the risk-neutral probability $\tilde{\mathsf P}$ the stock and the bank account have the same average rate of growth. In other words, if $S_0$ and $S_N$ are the initial and final stock prices, and $B_0$ and $B_N$ the initial and final bank prices, show that:
$$ \tilde{\mathsf E}\left[\frac{S_N}{S_0}\right]=\tilde{\mathsf E}\left[\frac{B_N}{B_0}\right]=c $$ and find the constant c.
I have the following:
I know that the risk neutral (or non risk neutral) expectation of the bank account will simply be $B_N/B_0$, as the expectation of any bank related investment will simply be the same as whatever is in the bracket (there is no uncertainty in the bank).
Also, I know $B_N=B_0(1+r)^N$ ($B_N$ is equal to initial investment multiplied by interest rate to the power $N$), So I can simplify $$ \tilde{\mathsf E}\left[\frac{B_N}{B_0}\right]= B_N/B_0 = \frac{B_0(1+r)^N}{B_0}=(1+r)^N. $$
My problem is trying to show that this is the case for the stock $\tilde{\mathsf E}\left[\frac{S_N}{S_0}\right] = (1+r)^N.$ As the stock is a martingale, I know I can say that:
$$ S_0/(1+r)^0 = \text{(by multi step ahead property)} = \tilde{\mathsf E}\left[\frac{S_N}{S_0}\right]. $$But I cannot work out what to do after this. I have found a way online that says this implies: $\tilde{\mathsf E}\left[\frac{S_N}{S_0}\right]=(1+r)^N$, but I cannot see how the previous statement implies this.
Would greatly appreciate any help.
it is fully correct
or not? :) $\endgroup$