Applied Stock maket Stochastics

A stochastic variable is pretty much a non-deterministic variable, in other words a random variable. I've been thinking about stock market in terms of statistics to better quantify risks and expected values because it really boils down to understanding the risk-reward profile and creating a portfolio with the desired characteristics given the available financial vehicles.

The challenge is then quantifying the risks and rewards of which is a black art. I doubt that anyone has a perfect system of setting these values a priori because risk and reward are particularly fuzzy numbers until after the fact.

Given the current fragility of the world markets, one occurring theme that I've been contemplating is the mechanics of hedging. There are of course several ways of not losing money before a market downturn may include:

1. Selling off equities
2. Buying bonds
3. Shorting stocks, selling calls or buying puts
4. Buying short ETFs

The list is not extensive but those are some of the options that one would have during market uncertainty. The next question becomes, what do you think will happen in the next time frame (which can include spans of days, months or years). The idea will be to track current and future issues and setup a portfolio that tracks your sentiment based on the data collected and your interpretation of them and make bets on sentiments that you feel the most confident about.

Suppose that you don't believe that the stock market is quite volatile but not in a position to make huge gains. If you have securities already and already made significant gains but don't want to incur capital gains taxes, selling your securities might not be the best course of action. In this case it might be wise to lock in gains by hedging against future drops.

This begs 2 questions:

1. What is the probability of the stock dropping n%?
2. Give the probability of a drop, how much should you pay for insurance?

Now obviously without part 1 of the answer, answering part 2 is going to be hard. So I've started first looking at part 1 of the problem. What is the probability of the stock market moving in some direction in some time frame?

The most obvious method that came to mind was to first get the probability distribution function of a stock market for the range of all monthly percent changes (I could have done daily, but I already crunched the monthly data), The graph looks like the one below:

 

The above plot contains all monthly data from 1928. The plot looks fairly Gaussian yielding a mean of 0.764% and a standard deviation of 4.445.

The next step that one takes with this data is determining where things may go in the next few months. The easiest method to calculating what may happen in the next 6 months is to create a random Gaussian variable with the mean and variance of the Dow Jones index and multiply them together to determine the probability of what may happen in the next 6 months. The result is the following graph:

Given the cdf of the next 6 month, the next question to answer is what is the probability of seeing the Dow increasing or decreasing by some factor? This requires some integration but I've done some sample calculations and those results are provided below:

Based on the information below, we have a general idea on the probability of seeing growth in the Dow based on all historical data within the next 6 months.  What would be more important is to find means and standard deviations that represents the market right now; which could be done through calculating the distributions of %changes in price over a more recent time fame, or more importantly, values that they might expect in some future time frame based on current research.

The idea will be to create a portfolio that represents a personal bias on individual stocks and the stock market in general to create an adequately hedged portfolio against catastrophic crashes while exposing one's to certain amount of gains to mitigate risks while trying to aim for the most optimal profits. Modeling the motions of the stock market in general are important tools to help create a portfolio that matches one's risk-reward profile.

This is still a work in progress with many imperfections but it is a start to creating portfolios with explicit characteristics.