Question

n-3 n-0 u.So u.So u2.d.S So u.d2.S p u.d2s d2.So dB.So

We denote the collecction of sets determined by the first k tosses $$F_k$$

Suppose the imitial stock price is $$S_0$$ ,with up and down facter being $$u$$ and $$d$$.
Let the probability of each $$H$$ and $$T$$ be $$p$$ and $$q=1-p$$ and   $$F_t$$ the sigma-lgebra generated by the coin tosses up to (and inchudling) time $$t$$

1) Compute     $E(S_3|F_2)$

2) compute $Var(E(S_3|F_2))$

3)Compute $Var(S_3|F_2)$    such that $$$Var [X|G]= E[ (X-E(X|G) )^2|G]$$$

4) Compute $E(Var(S_3|F_2))$     

5)Compute $E(S_2|F_3)$

6) Compute $\sigma (S_3)$   the smallest sigma algebra contains S3

7) Compute $E(S_2|\sigma(S_3))$

Answer 1

2 wiIA probabili