We consider a diffusion process \(X_t\) which evolves according to \[ dX_t = \mu(t,X_t)dt + \sigma(t,X_t)dW_t \] where \(\mu\) and \(\sigma\) are smooth scalar real functions and \(W_t\) is a scalar standard Brownian motion.
We observe the process on the time interval \([0,T]\) and we make the time discretization \(t_0 = 0, t_1 = \frac T N, \ldots, t_N = T\).
We will see three different trinomial trees constructions.
Basic trinomial tree
At each time step \(i\), starting form \(x_{i,j}\) the process can go up, straight or down as follows :
\[
\begin{cases}
x_{i+1,j+1}= x_{i,j} + (j+1)\Delta x_{i+1}\\
x_{i+1,j}= x_{i,j} + j \Delta x_{i+1} \\
x_{i+1,j-1}= x_{i,j} + (j-1)\Delta x_{i+1}
\end{cases}
\]
with \(\Delta x_{i+1} = V_i \sqrt{3}\) and \(x_{0,0}=X_0\).
The result of this construction is a symmetric tree around \(X_0\).
Trinomial tree with boundaries conditions
We suppose that the conditional law of the process at time \(i\) depends only on \(i\). Then if we set two levels \(0<a<b<1\), at each time step \(i\) we can compute the inferior and the superior quantile \(Q_{min}^i\) and \(Q_{max}^i\), such that \[\mathbb{P}(X_{t_i}<Q_{min}^i)=a \quad \text{and} \quad \mathbb{P}(X_{t_i}<Q_{max}^i)=b.\]
Starting from \(x_{i,j}\), we proceed at the tree construction depending on whether we passed or not the quantiles boundaries at the next step:
- If \(j\Delta x_{i+1} > Q_{max}^{i+1}\), the process can go straight or twice down as follows :
\[
\begin{cases}
x_{i+1,j}= x_{i,j} + j\Delta x_{i+1}\\
x_{i+1,j-1}= x_{i,j} + (j-1) \Delta x_{i+1} \\
x_{i+1,j-2}= x_{i,j} + (j-2)\Delta x_{i+1}
\end{cases}
\]
- If \(j\Delta x_{i+1} < Q_{min}^{i+1}\), the process can go straight or twice up as follows :
\[
\begin{cases}
x_{i+1,j}= x_{i,j} + j\Delta x_{i+1}\\
x_{i+1,j+1}= x_{i,j} + (j+1) \Delta x_{i+1} \\
x_{i+1,j+2}= x_{i,j} + (j+2)\Delta x_{i+1}
\end{cases}
\]
- Otherwise, the tree follows a basic evolution
\[
\begin{cases}
x_{i+1,j+1}= x_{i,j} + (j+1)\Delta x_{i+1}\\
x_{i+1,j}= x_{i,j} + j \Delta x_{i+1} \\
x_{i+1,j-1}= x_{i,j} + (j-1)\Delta x_{i+1}
\end{cases}
\]
with \(\Delta x_{i+1} = V_i \sqrt{3}\).
Trinomial tree with Hull-White method
We state that starting from \(x_{i,j}\) we can reach the positions
\[
\begin{cases}
x_{i+1,k+1}=(k+1)\Delta x_{i+1} \quad \text{with probability} \quad p_u, \\
x_{i+1,k}=k\Delta x_{i+1} \quad \text{with probability} \quad p_m, \\
x_{i+1,k-1}=(k-1)\Delta x_{i+1} \quad \text{with probability} \quad p_d.
\end{cases}
\]
We compute the conditional mean and the conditional variance of the process,
\[\mathbb{E} ( X(t_{i+1} ) | X(t_{i}) = x_{i,j} ) = M_{i,j},\] \[\mathbb{V} ( X(t_{i+1} ) | X(t_{i}) = x_{i,j} ) = V_{i,j}^2.\]
We search for \(p_u\), \(p_m\) and \(p_d\) such that the conditional mean and the conditional variance match those in the tree.
We notice that
\[
\begin{aligned}
x_{i+1,k+1} &= x_{i+1,k} + \Delta x_{i+1},\\
x_{i+1,k-1} &= x_{i+1,k} - \Delta x_{i+1}.
\end{aligned}
\]
Then we can write
\[
\begin{aligned}
p_u( x_{i+1,k} + \Delta x_{i+1} ) + p_m x_{i+1,k} +p_d (x_{i+1,k} - \Delta x_{i+1}) &= M_{i,j}, \\
p_u( x_{i+1,k} + \Delta x_{i+1} )^2 + p_m x_{i+1,k}^2 +p_d (x_{i+1,k} - \Delta x_{i+1})^2 &= V_{i,j}^2 + M_{i,j}^2,
\end{aligned}
\]
which brings us to
\[
\begin{aligned}
x_{i+1,k} + (p_u + p_d) \Delta x_{i+1} &= M_{i,j},\\
x_{i+1,k}^2 + 2x_{i+1,k} \Delta x_{i+1} (p_u-p_d) + \Delta x_{i+1}^2 (p_u + p_d ) &= V_{i,j}^2 + M_{i,j}^2.
\end{aligned}
\]
Setting \(\eta_{j,k} = M_{i,j} - x_{i+1,k}\), we can write
\[
\begin{aligned}
(p_u + p_d) \Delta x_{i+1} &= \eta_{j,k}, \\
(p_u + p_d ) \Delta x_{i+1}^2 &= V_{i,j}^2 + \eta_{j,k}^2
\end{aligned}
\]
and remembering that \(p_m = 1- p_u -p_d\), we obtain
\[
\begin{aligned}
p_u &= \frac{V_{i,j}^2}{2 \Delta x_{i+1}^2}+ \frac{\eta_{j,k}^2}{2 \Delta x_{i+1}^2} + \frac{\eta_{j,k}}{2 \Delta x_{i+1}},\\
p_m &= 1 - \frac{V_{i,j}^2}{\Delta x_{i+1}^2} - \frac{\eta_{j,k}^2}{\Delta x_{i+1}^2},\\
p_d &= \frac{V_{i,j}^2}{2 \Delta x_{i+1}^2}+ \frac{\eta_{j,k}^2}{2 \Delta x_{i+1}^2} - \frac{\eta_{j,k}}{2 \Delta x_{i+1}}.
\end{aligned}
\]
We take advantage of the available degrees of freedom, in order to obtain quantities that are always positive. For that, we assume that \(V_{i,j}\) is independent of \(j\). From now on we write \(V_i\) instead of \(V_{i,j}\). We set \(\Delta x_{i+1} = V_i \sqrt{3}\), we choose the level \(k\) and \(\eta_{j,k}\) in such a way that \(x_{i+1,k}\) is as close as possible to \(M_{i,j}\) : \[k = round\left( \frac{M_{i,j}}{\Delta x_{i+1}}\right),\] where \(round(x)\) is the closest integer to the real number \(x\).
Finally we get : \[
\begin{aligned}
p_u &= \frac{1}{6}+ \frac{\eta_{j,k}^2}{6 V_{i}^2} + \frac{\eta_{j,k}}{2 V_i \sqrt{3}}, \\
p_m &= \frac{1}{2} - \frac{\eta_{j,k}^2}{3 V_i^2}, \\
p_u &= \frac{1}{6}+ \frac{\eta_{j,k}^2}{6 V_{i}^2} - \frac{\eta_{j,k}}{2 V_i \sqrt{3}}.
\end{aligned}
\]