Calculate risk-neutral probability Consider a two-step binomial model in which at each step the share price either doubles, with probability $p ∈ (0, 1)$, or halves, with probability $1 − p ∈ (0, 1)$. Initially the price is $S_0 = 4$. Assume each step takes one unit of time and that over one unit of time the risk-free rate is $r = \log(5/4)$.
Calculate the risk-neutral probability of an up-move.
Well, I am in the very beginning of my course and the notion of "risk-neutral probability" was not fully developed yet. I tried to do some calculations, but it seems to me that I need p to calculate such probability.
Can someone please help me by clarifying the concept and showing beginning of the right path?

Assuming that the interest rate is compounded continuously, the cost of the investment must equal the (present-time value) expected value of the payoff of that investment. If this weren't the case, then we would have arbitrage (non risk-neutral).
Let $P$ be the payoff of the investment of purchasing $1$ unit of the stock price at initial time $0$. The cost of this investment is then just $S_0$.
Then $$E[PV[P]] = S_0$$
So $$e^{-r}E[P]=S_0$$
And $E[P]=2S_0*P(S_1=2S_0) + \tfrac12S_0*P(S_1=\tfrac12S_0)=2S_0P+\tfrac12S_0(1-P)$. Thus, $$E[P]=S_0e^r$$
$$2S_0P+\tfrac12S_0(1-P)=S_0e^r$$
Do some algebra, with using $r=log(\tfrac54)$, to get $$P=\tfrac12$$
Which makes sense intuitively.
