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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 sell-hold-hedge 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:

1. Exercising the option: In this case, the expected value equals the strike price. 
2. 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 risk-less 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.