5.2 Trinomial Processes

In contrast to binomial processes, a trinomial process allows a quantity to stay constant within a given period of time. In the latter case, the increments are described by:

$\displaystyle \P(Z_k = u) = p \, , \, \P(Z_k = - d) = q \, , \, \P(Z_k = 0) = r = 1 - p - q\ , $

and the process $ X_t$ is again given by:

$\displaystyle X_t = X_0 + \sum^t_{k=1} \, Z_k $

where $ X_0, Z_1, Z_2, \ldots $ are mutually independent. To solve the Black-Scholes equation, some algorithms use trinomial schemes with time and state dependent probabilities $ p$, $ q$ and $ r$. Figure 4.5 shows five simulated paths of a trinomial process with $ u=d=1$ and $ p=q=0.25.$

Fig.: Five paths of a trinomial process with $ p=q=0.25.$ ($ 2\sigma$)-intervals around the trend (which is zero) are given as well. 6396 SFETrinomp.xpl
\includegraphics[width=1\defpicwidth]{trinomialpr.ps}

The exact distribution of $ X_t$ cannot be derived from the binomial distribution but for the trinomial process a similar relations hold:

$\displaystyle \mathop{\text{\rm\sf E}}[X_t]$ $\displaystyle =$ $\displaystyle \mathop{\text{\rm\sf E}}[X_0] + t \cdot \mathop{\text{\rm\sf E}}[Z_1] = \mathop{\text{\rm\sf E}}[X_0] + t \cdot (pu - qd)$  
$\displaystyle \mathop{\text{\rm Var}}(X_t)$ $\displaystyle =$ $\displaystyle \mathop{\text{\rm Var}}(X_0) + t \cdot \mathop{\text{\rm Var}}(Z_1), \, \, \text{\rm where\ }$  
$\displaystyle \mathop{\text{\rm Var}}(Z_1)$ $\displaystyle =$ $\displaystyle p (1-p) u^2 + q (1-q) d^2 + 2pq\ ud\ .$  

For large $ t$, $ X_t$ is approximately    N$ (\mathop{\text{\rm\sf E}}[X_t], \, \mathop{\text{\rm Var}}
(X_t))$-distributed.