Applied Stock Market Stochastics 2 (applications to options)

I wrote earlier in Applied Stock Market Stochastics about learning using the mean and the standard deviation of stock market data to calculate the probability distributions of where a stock might go in the future. I've decided to take what I learned from the previous post and take it a step further in application to Option derivatives.

Understanding options is a little math heavy; option pricing comes from the black-scholes equation pricing equation by assuming that the price of an equity moves in a Geometric Brownian motion like way with a constant drift and volatility. We can model this using the mean and standard deviation by throwing the parameters into a Gaussian function. The black-scholes equation is basically a closed form solution to the differential equation representing this condition.

I got tired of simply just looking at the equation so I decided that it would be more interesting to create a statistical model to determine the motion of the price of a stock. In my last post in Applied Stock Market Stochastics, I analyzed the historical prices of the Dow Jones index as a distribution of percentage change on a month by month basis. The result showed that the motion of the stock market moves around in a Gaussian like manner.

I will assume that you have a basic understanding of put and call options. If you don't know about them, the short story is that the purchase of one of these contracts is a bet that the price of some equity will be above or below some price. If it is above or below that price, your profit is proportional to that difference (and if you need to learn more, hit the links or Google for more details).

With that in mind, the question that I want to solve is the likelyhood of making a profit or loss and by how much. If we understand the probabilities of making a profit or loss we can then calculate for the "expected value" of the trade-- which is the average loss or profit that we expect to get if we made the same bet over and over again. If the expected value is positive, we expect to make money and vice versa.

Now, I am not aiming for the level of sophisticated analysis hedge funds do out there and I won't pretend that I know that much either, but I do have a great fascination of learning about things though a first principles approach; and I think this is a great first principles approach to tackling this problem!

So let's first model the motion of an equity instead of a stock market index. In this case, I've downloaded Google's historical prices for the last 3 months (between June and September) and calculated the mean and standard deviation of it's % chance in price on a daily basis. The numbers I got from this calculation were 0.0373% and 2.409 for the mean and standard deviation in percents. For actual use in the models, I divided these numbers by 100 and ran a simulation using 100,000 simulations to simulate the resulting price of the Google stock by October 20, 2011 which is the expiry date for a set of call and put options. The resulting plot looks like the one below:

The change factor is the multiplier which you would multiply the stock price to determine it's final value. The plot here, is also normalized (meaning calculating the area of this graph will equal 1 or 100%).

Now for illustrative purposes, I took an option at random, which was the $555 strike call option dated for Oct 20th 2011, which was priced at about $12.00 when I checked at Google Finance and plotted out the returns as a function of price:

$555 strike, call option Profit/Loss profile @ a price of $12.00

So now we have now have 2 pieces of information: the probability distribution of Google's stock changing by some factor by the option expiry date and the Profit/Loss profile of the stock option. Now the most important question to answer is, given the current price of Google's stock at $537 (when I checked today), what is the weighted profit/loss as a function of price? The answer to that is to take the probability distribution of final prices and multiply it by the return distribution of the option to get the following plot:

What this graph tells us is what you expect to earn/lose over all the possible outcomes. Summing the values together will yield the expected value of this trade, or the amount of money you expect to make when making this bet over and over again. I've done that calculation and the number resulted in -$3.023/per share (and 1 contract is a multiple of 100 shares) meaning that you expect to lose $302.3 on average when making this trade.

But!! There is an upside to this however, in that you can instead sell a call contract instead of buying one and expect to make $302.3 instead and turning this into a profit.

Caveats

Obviously, if this were the case then everyone would already be doing this to make money in the stock market (and to some extent, there are people and organizations out there that are already doing this). So there are 2 important assumptions that you needs to be aware of, and those are:

1. that the motion of price can be modeled by a random Gaussian variable and
2. the mean and standard deviations are accurate and do not change

But there will be times when Google's stock price won't behave in the random manner that we modeled it as and we don't event know that numbers that were put into the model were right in the first place! This is the risk we take when using this calculation (and any other model for weighted profits and losses!).

So what is this good for?

This is a problem that plagues historical analysis when trying to make predictions of the future. But the most important take away from this is developing an understanding of the likely and unlikely future outcomes to help in making better decisions; in the long run, this is the most important part.

Now, even with these points of uncertainty, this could be used as a model outcomes based on your research and how you feel about it. The great power bestowed from this is being able to quantify your opinions of the future and risks involved by turning them into an expected value. This helps considerably in decision making instead of simply just going off on "a hunch."

There is obviously more work in modeling random variables and outcomes but I hope that you find this a useful tool in thinking about the future movement of a stock. Ideas and comments are most welcome.