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Intuition: Focusing on the unique tree with varying volatility and probabilities, we see that the "implied" volatility in the upper branch between $t_1$ and $t_2$ is $e^{\sigma(t_1,t_2)}=ln\left(\frac{120.27}{110.52}\right)\approx8.45\%$; we can therefore see that the unique implied tree starts off with a volatility of $10\%$ between $t_0$ and $t_1$ but then reduces the volatility in the upper branch to $8.45\%$, whilst "increasing" the implied volatility of the up-move from 0.625 to 0.682: in other words, by varying these two parameters (volatility and implied probability), the tree can solve for these two "unknowns" to satisfy the two equations for the two option prices (together with the "anchoring" condition that the tree recombines).

Intuition: Focusing on the unique tree with varying volatility and probabilities, we see that the "implied" volatility in the upper branch between $t_1$ and $t_2$ is $e^{\sigma(t_1,t_2)}=ln\left(\frac{120.27}{110.52}\right)\approx8.45\%$; we can therefore see that the unique implied tree starts off with a volatility of $10\%$ between $t_0$ and $t_1$ but then reduces the volatility in the upper branch to $8.45\%$, whilst "increasing" the implied probability of the up-move from 0.625 to 0.682: in other words, by varying these two parameters (volatility and implied probability), the tree can solve for these two "unknowns" to satisfy the two equations for the two option prices (together with the "anchoring" condition that the tree recombines).

Focusing on the unique tree with varying volatility and probabilities, we see that the "implied" volatility in the upper branch between $t_1$ and $t_2$ is $e^{\sigma(t_1,t_2)}=ln\left(\frac{120.27}{110.52}\right)\approx8.45\%$; we can therefore see that the unique implied tree starts off with a volatility of $10\%$ between $t_0$ and $t_1$ but then reduces the volatility in the upper branch to $8.45\%$, whilst "increasing" the implied volatility of the up-move from 0.625 to 0.682: in other words, by varying these two parameters (volatility and implied probability), the tree can solve for these two "unknowns" to satisfy the two equations for the two option prices (together with the "anchoring" condition that the tree recombines).

Intuition: Focusing on the unique tree with varying volatility and probabilities, we see that the "implied" volatility in the upper branch between $t_1$ and $t_2$ is $e^{\sigma(t_1,t_2)}=ln\left(\frac{120.27}{110.52}\right)\approx8.45\%$; we can therefore see that the unique implied tree starts off with a volatility of $10\%$ between $t_0$ and $t_1$ but then reduces the volatility in the upper branch to $8.45\%$, whilst "increasing" the implied volatility of the up-move from 0.625 to 0.682: in other words, by varying these two parameters (volatility and implied probability), the tree can solve for these two "unknowns" to satisfy the two equations for the two option prices (together with the "anchoring" condition that the tree recombines).

For our simple case here, we just consider two options from the paper (the initial price of the underlying is $S_0=100$):

Rates are 3% per year. In the B-S world, the two IVs give rise to two different Binomial trees (constructed as $S_{t+1}^{up}=S_{t}e^{\sigma_{bs}}$, $S_{t+1}^{down}=S_{t}e^{-\sigma_{bs}}$):

Focusing on the unique tree with varying volatility and probabilities, we see that the "implied" volatility in the upper branch between $t_1$ and $t_2$ is $e^{\sigma(t_1,t_2)}=ln\left(\frac{120.27}{110.52}\right)\approx8.45\%$; we can therefore see that the unique implied tree starts off with a volatility of $10\%$ between $t_0$ and $t_1$ but then reduces the volatility in the upper branch to $8.45\%$, whilst "increasing" the implied volatility of the up-move from 0.625 to 0.682: in other words, by varying these two parameters (volatility and implied probability), the tree can solve for these two "unknowns" to satisfy the two equations for the option prices (together with the "anchoring" condition that the tree recombines).

For our simple case here, we just consider two options from the paper and we demonstrate that one unique tree with varying volatility can generate the same two option prices as two different trees with a constant volatility.

We note that in our simple case, the initial price of the underlying is $S_0=100$. The two options we consider are:

Rates are 3% per year. 

In the B-S world, the two IVs give rise to two different Binomial trees with constant volatilities (the trees are constructed as $S_{t+1}^{up}=S_{t}e^{\sigma_{bs}}$, $S_{t+1}^{down}=S_{t}e^{-\sigma_{bs}}$):

Focusing on the unique tree with varying volatility and probabilities, we see that the "implied" volatility in the upper branch between $t_1$ and $t_2$ is $e^{\sigma(t_1,t_2)}=ln\left(\frac{120.27}{110.52}\right)\approx8.45\%$; we can therefore see that the unique implied tree starts off with a volatility of $10\%$ between $t_0$ and $t_1$ but then reduces the volatility in the upper branch to $8.45\%$, whilst "increasing" the implied volatility of the up-move from 0.625 to 0.682: in other words, by varying these two parameters (volatility and implied probability), the tree can solve for these two "unknowns" to satisfy the two equations for the two option prices (together with the "anchoring" condition that the tree recombines).