5.5 Binomial Models with State Dependent Increments

Binomial processes and more general random walks model the stock price at best locally. They proceed from the assumption that the distribution of the increments $Z_t = X_t - X_{t-1}$ are the same for each value $X_t$, regardless of whether the stock price is substantially greater or smaller than $X_0$. Absolute increments $X_t - X_{t-1} = (R_t - 1)\ X_{t-1}$ of a geometric random walk depend on the level of $X_{t-1}.$ Thus, geometric random walks are processes which do not have independent absolute increments. Unfortunately, modelling the stock price dynamics globally the latter processes are too simple to explain the impact of the current price level on the future stock price evolution. A class of processes which take this effect into account are binomial processes with state dependent (and possibly time dependent) increments:

$$X_t = X_{t-1} + Z_t \, \, , \quad t = 1, 2, \ldots$$ (5.6)

$$\P(Z_t = u) = p(X_{t-1}, t) \, , \, \, \, \P(Z_t = - d) = 1 - p (X_{t-1}, t) \, .$$

Since the distribution of $Z_t$ depends on the state $X_{t-1}$ and possibly on time $t$, increments are neither independent nor identically distributed. The deterministic functions $p(x, t)$ associate a probability to each possible value of the process at time $t$ and to each $t$. Stochastic processes $\{ X_t; \, \, t \ge 0 \}$ which are constructed as in (4.6) are still markovian but without having independent increments.

Accordingly, geometric binomial processes with state dependent relative increments can be defined (for $u > 1,\ d < 1$):

$$X_t = R_t \cdot X_{t-1}$$ (5.7)

$$\P(R_t = u) = p (X_{t-1}, t) \, , \, \, \, \P(R_t = d) = 1-p (X_{t-1}, t) \, .$$

Processes as defined in (4.6) and (4.7) are mainly of theoretic interest, since without further assumptions it is rather difficult to estimate the probabilities $p(x, t)$ from observed stock prices. But generalized binomial models (as well as the trinomial models) can be used to solve differential equations numerically, as the Black-Scholes equation for American options for example.