Calibrating an Implied Binomial Tree to American Put Prices

# Building implied binomial tree with American input options


# Building implied binomial tree with American input options












i want to build an implied volatility binomial tree with American input options, so the setup is the following:

1) We know the market Price P of the American Put $P_{am}(t_i,K)$, where $t_i$ is the expiration date and K the strike price.

2) $S_{i,j}$ is the value of the stock at node (i,j), where i is the time and j the displacement on the tree

3) $v^{put}_{i-1,j}$ is the value of $P_{am}(t_i,S_{i-1,j})$ at node (i-1,j), so we set the the strike $K=S_{i-1,j}$

4)$\lambda_{i-1,m}$ the Arrow-Debreu price of node (i-1,m)

5) r the risk free rate

6) $D_i$ is the dividend payment wit ex-dividend date $t_i$

Lets assume we are at some arbitrary point (i-1,j) in the tree and want to determine the value $v^{put}_{i-1,j}$ (we take the put, because we are below the center). Then: 1) All nodes at time $t_{i-1}$ that are above the strike $K=S_{i-1,j}$ have value zero, because the payout of the put option is zero at all those nodes

2)According to a formula, which i don't want to derive here, it holds

$v^{put}_{i-1,k}=max[e^{-r\Delta t} K - S_{i-1,k} +e^{-r\Delta t}D_i,K-S_{i-1,k}] $ for $k<j$ (intrinsic vs time value)

Then my source follows, that the only node in the option value tree for $P_{am}(t_i,K)$ we do not know is (i-1,j).

It offers the solution of the false position method to determine a root for $P(v):=\pi(v)-P$ with $\pi(v)$ defined as the tree value and P the market price of the American option. It takes two guesses $v_0$ and $v_1$ with $P(v_0)\leq P \leq P(v_1)$ and gives the following formula for the next guess:

\begin{align*} v_2=\frac{v_1 m + P(v_1)}{m}, \enspace \enspace m=\frac{P(v_1)-P(v_0)}{v_1-v_0}. \end{align*}

Now i wonder, how do i calculate $P(v_1)$, because i am not sure what the function $\pi(v)$ exactly is. In the European case, this tree value $\sum^j_{k=0}\lambda_{i-1,k}v^{put}_{i-1,k} = \sum^{j-1}_{k=0}\lambda_{i-1,k}(e^{-r\Delta t}K -S_{i-1,k}+e^{-r\Delta t}D_i) + \lambda_{i-1,j}v^{put}_{i-1,j}$ is set equal to the market price $P^{eu}(t_i,K)$ and solved for $v^{put}_{i-1,j}$.

Why can't we do that in the American case, as we know all values $v_{i-1,k}$ for $k<j$ ?? It must be somehow related to the non linearity of the max operator, but i don't see how...

Maybe the notation is confusing or i didn't explain the problematic good enough, still in any case i would also be very grateful for any literature or example of building such a tree for American input options with dividends, because i could only find proper examples for the case of Europeans.

Thanks.