Discrete-Time Finance

\[ \def\BB#1{{\mathbb{#1}}} \def\BF#1{{\mathbf{#1}}} \]

yedlu, Winter 2024

Modeling evolutions of a variable is at the center of economic disciplines. We will first try to see tools we can use in discrete times and then push them forward in continuous times.

Linear Difference Equations

Consider the following evolution:

\[\begin{aligned} y_t &= a + b y_{t-1} \end{aligned}\]

with \(y_0\) being a pre-determined constant.

Then, we can iterate this over discrete times:

\[\begin{aligned} y_1 &= a + b y_0 \\ y_2 &= a + b y_1 = a + ba + b y_0 \\ y_3 &= a + b y_2 = a + ba + b^2 a + b^3 y_0 \\ & ... \\ y_t &= a (\sum_{i=0}^{t-1} b^i) + b^t y_0 \end{aligned}\]

We can further derive the geometric sum:

\[\begin{aligned} y_t &= a (\sum_{i=0}^{t-1} b^i) + b^t y_0 \\ &= a (\frac{1 - b^t}{1 - b}) + b^t y_0 \end{aligned}\]

Convergence

This condition

\[\begin{aligned} \lim_{t \to \infty} \frac{1 - b^t}{1 - b} = c, \quad c \in \BB{R} \end{aligned}\]

would be true if

\[\begin{aligned} \lim_{t \to \infty} b^t &= 0 \\ \implies \mid b \mid &< 1 \end{aligned}\]

Types of Convergence

In financial models we would typically assume the convergence is monotonic:
\(b \in (0,1)\).
Nevertheless, alternating convergence might be useful in some cases:
\(b \in (-1,0)\).

Solution

Assuming convergence, the steady-state of \(y_t\) would then be:

\[\begin{aligned} \overline{y} &= \lim_{t \to \infty} = a (\frac{1 - b^t}{1 - b}) + b^t y_0 \\ &= \frac{a}{1-b} \end{aligned}\]

Another way to find the steady state is to assume existance of \(\overline{y}\):

\[\begin{aligned} \overline{y} &= a + b \overline{y} \\ \implies \overline{y} &= \frac{a}{1-b} \end{aligned}\]

Exponential Growth

The general solution
\[\begin{aligned} y_t &= \overline{y} + b^t (y_0 - \overline{y}) \end{aligned}\] tells us a story. The discrete-time process begins at \(y_0\), converges to \(\overline{y}\) in the end exponentially (\(b^t\)).

Transversality: Dividend Discount Revisited

We know that the price of a stock can be denoted by the dividend discount model:

\[\begin{aligned} p_t &= \frac{d_{t+1}}{1 + r} + \frac{d_{t+2}}{(1 + r)^2} + ... + \frac{d_{t+n}}{(1 + r)^n} + \frac{p_{t+n}}{(1 + r)^n} \end{aligned}\]

If we were to allow the time period goes to infinity:
\[\begin{aligned} p_t &= \sum_{i = 1}^{n} \frac{d_{t+i}}{(1 + r)^i} + \lim_{n \to \infty} \frac{p_{t+n}}{(1 + r)^n} \end{aligned}\]

We can then see that the following assumption makes the price of the stock to be unique (transversality condition):
\[\begin{aligned} \lim_{n \to \infty} \frac{p_{t+n}}{(1 + r)^n} &= 0 \end{aligned}\]

If this term explodes, we would not be able to obtain a unique price for this stock.

Tip

This would be an interesting concept in Asset Pricing called market completeness.