Stochastic Programming - Index of

146 STOCHASTIC PROGRAMMING
Figure 16
Network describing possible investments for a single scenario.
horizontally. For each stage we first have a column with one node for each
instrument. In the example these are four investment categories. The arcs
entering from the left brings the initial portfolio into the model, measured in
the amount of money held in each category. The arcs that run horizontally
between nodes of the same category represent investments held from one
period to the next. The node which is alone in a column represents trading.
Arcs that run to or from this cash node, represent the selling or buying of
instruments. A stage consists of one column of 4 nodes plus the single cash
node.
We mentioned that this is a generalized network. That means that the
amount of money that enters an arc is not the same as the amount that leaves
it. For example, if you put money in the bank, and the interest rate is 10%,
then the flow into the arc is multiplied by 1.1 to produce the flow out. This
parameter is called the multiplier of the arc. This way the investment generally
increases over time. For most categories this parameter is uncertain, normally
with a mean greater than one. This is how we represent uncertainty in the
model. For a given scenario, these multipliers are known.
Arcs going to the cash trading node from all nodes but the cash node to its
left, will have multiplyers less than 1 to represent variable transaction costs.
The arcs that leave the cash trading node have the same multipliers as the
horizontal arcs for the same investment categories, reduced (deterministically)
for variable transaction costs. Fixed transaction costs are not hard to model,
but they would produce very difficult models to solve.
We can also have arcs going backwards in time. They represent borrowings.
Since you must pay interest on borrowings (deterministic or stochastic), these
arcs have multipliers less than one, meaning that if you want 100 USD now,
you must pay back more than 100 USD in a later time period.