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3 step binomial tree process with $S_0=4,u=2,d=0.5,r=0.25.$ Determine the probability p and q such that the stock price process is a martingale (i.e. $E[S3]=S_0)$

I know P = 1/3 and Q = 2/3 but having trouble to get to $E[S_2]$ and $E[S_3]$ to prove it's the same as $S_0$

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Under Martingale framework you can admit ,without loss of generality, to be under an arbitrage free market. By the way the martingale process is the discounted spot, you then need to use $$\exp^{-3*0.25} E[S_3]=S_0 $$. Finally, remember that under Up event $$S_{t+1} = S_t * u$$. You'll be able to solve your tree recursively.

I may have made a mistake but still hope to be useful, good luck.

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