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Question: Conditional expectation and filtration

Question:Conditional expectation and filtration

Zeineb 520 Maple
probability
March 15 2019
0

Hi all

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

Suppose the imitial stock price is $S_0$ ,with up and down facter being $u$ and $d$.

Up : S1(H)=u S0 and S1(T)=d S0

S_{N+1}= alpha S_N

where alpha =u or 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:

After three coin tosses.

Can we propose a code computing the element of the filtration F1 and F3 and sigma(S3) (the sigma algebra generated by S3).

For example by hand we have F1={ emptyset, Omega, AH, AT}

Where AH={ w: w1=H}

AT={w: w1=T}

Can we compute

 

$E[ S_2|F_3] \text { and } E[ S_2|\sigma(S_3) ] $

 

$$E[ \frac{S_2}{S_1} | F_1] \text { and } E[ \frac{S_2}{S_1} | \sigma(S_1) ] $$

 

 

restart;
with(Finance);
S := [7.9, 7.5, 7.1, 6.5, 5., 3.7, 3.3, 2.95, 2.8];
         [7.9, 7.5, 7.1, 6.5, 5., 3.7, 3.3, 2.95, 2.8]
T := BinomialTree(3, S, .3);
TreePlot(T, thickness = 2, axes = BOXED, gridlines = true);

 

 

many thanks
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