Stochastic Ruminations: Decision Making Under Uncertainty
I've always admired those entrepreneurs who could seemingly
analyze mountains of data and determine the correct course of
action with little more than a few figures on a napkin. Is this intuition, luck, or
superior reasoning? While we may not be able to attain their
innate skills, we can apply mathematical modeling techniques to
assist us in decision making under uncertainty.
Let's consider a relatively simple decision model for
evaluating a sellholdhedge decision on an equity. You purchased
the equity at $40/share some time ago and it has been trading at
$45/share until recently when it dipped to $42/share. You could:
 sell
 hold
 hedge by purchasing an option to sell
Here is the basic decision tree:
The box represents the decision and the circles represent the
uncertainty. In a classic Bayesian decision tree, the lines
emanating from the circles represent the discrete outcomes. This
example contains multiple discrete outcomes for each uncertainty
because equities are usually priced in increments of eighths or
sixteenths. However, it is easier to consider the outcomes as
continuous with a bounded lower level ($0) and an unbounded upper
limit. We will also include a further simplifying assumption that
the probability density function (pdf) that describes the
uncertainty is symmetrical around the current price. This means
that it is equally likely for the price to go up by a specific
amount as it is to go down by that same amount.
We calculate the expected value for each possible decision. For
the Sell decision, the expected value equals the current price
because there is no uncertainty. The expected value for the Hold
decision is based on the pdf. For a symmetrical pdf, the expected
value equals the median and in this case is also the current
price. The expected value is the same for Selling and Holding, but
there is no risk in Selling. If the pdf were not symmetrical, then
a higher or lower expected value may result and the Hold decision
may appear more or less favorable than the Sell decision.
To calculate the expected value for the Hedge decision, we need
to know the shape of the pdf. There are studies that can be
referenced for selecting an appropriate distribution, but we will
choose the normal pdf for simplicity in our example. In the normal
distribution, larger changes in price are less likely than smaller
changes in price. The expected value for the Hedge decision is
composed of two components:
 Exercising the option: In this case, the expected value
equals the strike price.
 Not exercising the option: In this case, the expected value
equals the integral from the Strike Price to infinity of the
product of the pdf and x (using differential dx).
The composite expected value is found by multiplying the
expected value for each component by its probability (i.e. the
expected value for the first component is the product of the
strike price and the probability that the equity trades at or
below the strike price).
Skipping past the calculus steps, we find that the expected
value for the Hedge decision is dependent on the strike price and
the standard deviation of the equity price. We graph this result
for various values of the standard deviation:
If the asking price for the option is $0.50/share with a strike
price of $40/share, then the expected value of the Hedge decision
can be determined from the graph to be between $0.09/share and
$1.05/share over the current price depending on the variability
(standard deviation). The net profit for the Hedge decision must
account for the larger cost (i.e. the $0.50/share cost of the
option).
We haven't yet factored in:
 your preference/tolerance of risk  selling is riskless
while holding and hedging have varying amounts of risk.
 analyst recommendations  you can use reports from respected
analysts and industry studies in order to choose a more appropriate and realistic
pdf.
 time sensitivity  a probabilistic analysis of uncertainty
is usually only valid under independence (e.g. subsequent
roles of the dice are not influenced by prior roles). In this
situation, stock prices may very well be influenced by prior
pricing events.
We'll explore these factors in future issues.
