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1

We assume that $u=e^x$ and $d = e^{-x}$. Note that \begin{align*} u &\approx 1+ x +\frac{x^2}{2}, \textrm{ and}\\ d &\approx 1- x +\frac{x^2}{2}. \end{align*} Substituting these into your last equation, \begin{align*} u+d = \frac{\sigma^2 \Delta t}{1+\mu\Delta t} + 2, \end{align*} we obtain that \begin{align*} x^2 \approx \frac{\sigma^2 \Delta ...

2

As I mentioned above, I am not sure what the variable $r$ is. If we ignore that, or assume the questioner wanted to say its the risk free interest rate, then it has no effect on the number of paths. Then it is clear that after 50 steps going from \$1024 to \$2500 requires a net of 4 up movements with the given $x=y^{-1}=1.25$. Thus the number of steps ...

1

Two hints : The number of paths never going up to $3125$ when starting from $1024$ and stepping up by a multiplicative factor of $5/4$ and down by a multiplicative factor $4/5$ is the same as the number of paths starting from $0$ and and stepping up by an additive factor $+1$ and stepping down by an additive factor of $-1$ and never going up to $5$ Let ...

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