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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 ...

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The equality $1 + R = q_u · u + q_d · d$ is not particularly significant or difficult to prove. In fact, any number $b$ can be written as a linear combination of 2 other distinct arbitrary numbers $a,c$: $b=qa+(1−q)c$. (Easy: just set $q=\frac{b−c}{a−c}$). But in addition iff $a\le b\le c$ then it is a convex linear combination i.e. $0\le q \le 1$ and $0\... 1 Is there a faster way to calculate the option price? With a recombining binomial tree, the terminal asset price has a binomial distribution -- as you might have expected. For a tree with$n$steps, the probability of reaching price$S_{n,k}$where$k\$ is the number of up moves is $$P_{n,k} = \frac{n!}{k!(n-k)!}q^k(1-q)^{n-k}$$ The option price is the ...

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Usually, you would use the volatility from a fitted volatility curve or surface. Those are based on implied volatilities. You can use historical volatility, but then your valuation is likely to be off because the volatility curve/surface is not constant and at historical vol. You should use a yield curve to present value nodes. This is unlikely to make a big ...

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