The Mathematics of Financial Modelingand Investment Management

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676 The Mathematics of Financial Modeling and Investment Management

tually on a smaller set of variables) at each subsequent stage. Here we

provide a brief description of multistage stochastic optimization.^19

The key idea of stochastic programming is that at every stage a deci-

sion is made based on conditional probabilities. Scenarios form an informa-

tion structure so that, at each stage, scenarios are partitioned. Conditional

probabilities are evaluated on scenarios that belong to each partition. For

this reason, stochastic optimization is a process that runs backwards. Opti-

mization starts from the last period, where variables are certain, and then

conditional probabilities are evaluated on each partition.

To apply optimization procedures, an equivalent deterministic prob-

lem needs to be formulated. The deterministic equivalent depends on the

problem’s objective. Taking expectations naturally leads to deterministic

equivalents. A deterministic equivalent of a stochastic optimization

problem might involve maximizing or minimizing the conditional

expectation of some quantity at each stage.

We will illustrate stochastic optimization in the case of CFM as a

two-stage stochastic optimization problem. The first decision is made

under conditions of uncertainty, while the second decision at step 1 is

made with certain final values. This problem could be equivalently for-

mulated in a m-period setting, admitting perfect foresight after the first

period. This two-stage setting can then be extended to a true multistage

setting. At the first stage there will be a new set of variables. In this case,

the new variables will be the portfolio’s weights at stage 1. Call S the set

of scenarios. Scenarios are generated from an interest rate model. A

probability ps, s ∈ S is associated with each scenario s. The quantity to

optimize will be the expected value of final cash. The two-stage stochas-

tic optimization problem can be formulated as follows:

Maximize ∑ pshs , subject to the constraints

s ∈ S

∑αiKi, 0 + b 0 + B = r 0

i ∈ U

s s s s s s s s

∑αiKit, + (^1 + ρt )rt – 1 + bt = Lt + (^1 + βt )bt – 1 + rt

i ∈ U

s s

∑αiPi = ∑γ iPi

i ∈ U i ∈ U

(^19) For a full account of stochastic programming in finance, Zenios, Practical Finan-
cial Optimization.