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They are different concepts, and the relation between them can be described as a conditional: "if EMH holds (all available information about future price movements is already priced into the market), then future price movements will follow a purely random walk as new and unpredictable information emerges"


Historically the RWT (Random Walk Theory) came first, as empirical observations by for example M.F.M. Osborne (1959) and others in the 1960s. The EMH came about as a result of theoretical work by Samuelson in 1965 ("Proof that properly discounted prices...") and E.Fama (1969) as a general empirical/theoretical hypothesis that guided the field for many ...


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 ...


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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