Stochastic Calculus For Finance: Conditional Expectation of Binomial Tree Model

Question

I'm reading through Stochastic Calculus For Finance Volume 1, page 32.

The conditional expectation of $S_2$ knowing $1=H$ is $p*S_2(HH) + q*S_2(HT) = 0.5*16 + 0.5*4 = 10$

But how do I calculate the conditional expectation of $S_3$ knowing $1=H$? The given answer is $12.50$ but I couldn't reason through it.

Hope that someone could help.

Thanks!

Answer 1

Hint: from $S_1(H)$ you get to

1. $S_3(HHH)$ with probability $p^2$
2. $S_3(HHT)$ with probability $pq$
3. $S_3(HTH)$ with probability $qp$
4. $S_3(HTT)$ with probability $q^2$

Answer 2

$1 = H$ so at epoch 1 you start with a value of $8$. If the second and third coin tosses are bot heads, then you double that twice to get $32$. This happens with probability $\frac{1}{4}$ If they are 1 H and one T you double it and half it,k in either order, (total probability $\frac{1}{2}$) you get $8$, and if they are both T (probability $\frac{1}{4}$) you get $8/2/2 = 2$. The expectation is $$ \frac{32 + 2\cdot 8 + 2}{4} = 12.5 $$