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.

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 \(X_0\).

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 \(\Delta x_{i+1} = V_i \sqrt{3}\).

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 : \[ \begin{aligned} p_u &= \frac{1}{6}+ \frac{\eta_{j,k}^2}{6 V_{i}^2} + \frac{\eta_{j,k}}{2 V_i \sqrt{3}}, \\ p_m &= \frac{1}{2} - \frac{\eta_{j,k}^2}{3 V_i^2}, \\ p_u &= \frac{1}{6}+ \frac{\eta_{j,k}^2}{6 V_{i}^2} - \frac{\eta_{j,k}}{2 V_i \sqrt{3}}. \end{aligned} \]

A short rate \(r(t)\) is an interest rate at which an entity can borrow money for an infinitesimally short period at time \(t\).

Vasicek (1977) assumed that the short rate \(r_t\) follows an Ornstein-Uhlenbeck process with constant coefficients under the risk-neutral measure : \[ dr_t = k(\theta - r_t ) dt + \sigma dW_t \] where \(r_0=r(0)\), \(k\), \(\theta\) and \(\sigma\) are positive constants. The conditional mean and the variance of \(r_t\) are given by \[ \mathbb{E} (r_t | \mathcal{F}_s ) = r_s e^{-k(t-s)} + \theta (1 - e^{-k(t-s)} ),\] \[ \mathbb{V} (r_t | \mathcal{F}_s ) = \frac{\sigma^2}{2k} ( 1 - e^{-2k(t-s)} ).\]

We build the basic tree for the Vasicek model with : \[\Delta r = V \sqrt{3}\] and \[V=\sqrt{ \frac{\sigma^2}{2k}(1-e^{-2k\Delta t}) }.\]

We notice that \(r_t\) follows a normal distribution with mean and variance given by

\[\mathbb{E} ( r_t )= r_{0}e^{-kt} + \theta (1 - e^{-kt} ),\] \[\mathbb{V} ( r_t ) = \sqrt{ \frac{\sigma^2}{2k}(1-e^{-2kt}) }.\]

We will fix two boundaries \(0<a<b<1\) and at each time step we will compute the two quantiles \(Q_{min}^i\) and \(Q_{max}^i\) such that

\[\mathbb{P}(r_{t_i}<Q_{min}^i)=a \quad \text{and} \quad \mathbb{P}(r_{t_i}<Q_{max}^i)=b.\]

The equivalence of the conditional mean and variance, and the additional conditions bring us to

\[ \begin{aligned} p_u &= \frac{1}{6}+ \frac{\eta_{j,k}^2}{6 V^2} + \frac{\eta_{j,k}}{2 V \sqrt{3}},\\ p_m &= \frac{1}{2} - \frac{\eta_{j,k}^2}{3 V^2},\\ p_u &= \frac{1}{6}+ \frac{\eta_{j,k}^2}{6 V^2} - \frac{\eta_{j,k}}{2 V \sqrt{3}}, \end{aligned} \] with \[M_{i,j}= r_{i,j}e^{-k\Delta t_i} + \theta (1 - e^{-k\Delta t} ),\] \[V^2 = \frac{\sigma^2}{2k}(1-e^{-2k\Delta t})\] and \(r_{i,j} = j\Delta r \), where \(\Delta r := V \sqrt{3}\) and \(\eta_{j,k} = M_{i,j} - r_{i+1,k}\).

We consider the process \[ y_t = e^{k t}(r_t - \theta) \]

where \(r_t \) follows a Vasicek model.

By Ito's formula, we get \[ dy_t = \sigma e^{k t }dW_t. \]

The conditional mean and variance are then

\[\mathbb{E} ( y_t ) | \mathcal{F}_s ) = y_{s},\] \[\mathbb{V} ( y_t ) | \mathcal{F}_s ) = \frac{\sigma^2}{2k}(e^{2kt}-e^{2ks}).\]

In the graphs, we represent both the trees for \(y_t\) and for \(r_t\).

We build the basic tree for the process \(y_t\) with \[\Delta y_{i} = V_i \sqrt{3}\] and \[V_i=\sqrt{ \frac{\sigma^2}{2k}(e^{2 k t_{i}}-e^{2 k t_{i-1}}) }.\] Once we build the tree for \(y_t\), we multiply the nodes \(y_{i,j}\) by \(e^{-k i \Delta t}\) and we add \(\theta\) in order to obtain \(r_t\).

We notice that \(y_t\) follows a normal distribution with \[\mathbb{E} ( y_t )= y_{0},\] \[\mathbb{V} ( y_t ) = \sqrt{ \frac{\sigma^2}{2k}( e^{2 k t} - 1 ) }.\]

As we did for the standard construction, at each time step we constrain the process \(y_t\) to stay between two choosen quantiles.

Once we build the tree for \(y_t\), we make the computations to transform it in the tree for \(r_t\).

The equivalence of the conditional mean and variance and the other conditions that we saw in the general framework, applied to the process \(y_t\), bring us to \[ \begin{aligned} p_u &= \frac{1}{6}+ \frac{\eta_{j,k}^2}{6 V_{i}^2} + \frac{\eta_{j,k}}{2 V_i \sqrt{3}},\\ p_m &= \frac{1}{2} - \frac{\eta_{j,k}^2}{3 V_i^2},\\ p_u &= \frac{1}{6}+ \frac{\eta_{j,k}^2}{6 V_{i}^2} - \frac{\eta_{j,k}}{2 V_i \sqrt{3}}, \end{aligned} \] with \[M_{i,j}= y_{i,j},\] \[V_{i}^2 = \frac{\sigma^2}{2k}(e^{2 k t_{i}}-e^{2 k t_{i-1}}),\] \[y_{i,j} = j\Delta y_i, \] where \(\Delta y_i := V_i \sqrt{3}\) and \(\eta_{j,k} = M_{i,j} - y_{i+1,k}\).

Once the tree build for \(y_t\), we can build the one for \(r_t\).

Realized by Thibaut Montes

Developed by Daphné Giorgi and Altaïr Pelissier