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one of the most fundamental results states that the binomial model converges towards the Black Scholes model if the step size $\Delta t$ converges to zero. The Black Scholes model is an option pricing model where the underlying is given by $$S_T = S_0 \cdot \exp \Bigl(\sigma W_T - \frac 12 \sigma^2 T \Bigr).$$ By choosing  u = \exp(\sigma \sqrt{\...
Risk neutral valuation tells you to discount the expected payoff in the risk free world. Therefore, it should be $\sum_{i=0}^{N-1}R^{i+1}(1-p)^i p$, where $N$ is the number of periods.