Understanding option prices is important because your job, as an options trader, is to evaluate the price of an option and decide whether you should buy or sell it.

Too many people don’t really ‘get’ options prices, even though they trade them regularly. We should really understand expected return and theoretical value before we start thinking too much about the greeks. Here’s a quick intro to how and why option prices vary.

You already know some of the factors that must be included in an analysis of options prices. Here’s a short list of some of the most important factors:

- The current price of the stock
- The strike price of the option
- The time left until the option expires
- How quickly the price of the stock can change (the volatility)

Ideally, we’d know all of these inputs exactly, stick them into an equation, then get an answer for the true value of an option. We know the first three inputs exactly – they are all features of the option that we are considering trading. I’ve highlighted the last factor because it is the only thing we don’t know for sure when we trade an option. Volatility is the only thing we have to estimate.

For a discussion of volatility, let’s consider expected returns.

Let’s say you have a six-sided die. We all know that there is an equal probability of rolling any number between 1 and 6. Now let’s say I offer you the opportunity to roll the die and I will give you a dollar amount equal to whatever number come ups. If you roll a 1, you get $1. If you roll a 2, you get $2. And so on. How much would you pay to play this game with me?

Well, the first thing we need to think about is the **expected return** of the die. The average roll of the die will be (1+2+3+4+5+6)/6 = **3.5**. So, on an average roll, I will give you **$3.5**. For you to be profitable over the long term (assuming you love my game and want to keep playing it), the fee you should pay to play this game with me has to be **under $3.5**. The theoretical value of the bet is $3.5, so if you paid more than this, over the long run you would be a loser.

Now let’s do the same thing with a call option on a stock.

Let’s say there is a stock, that is currently trading at $40. At expiration, the stock could be at $20, $30, $40, $50, or $60. Each of these prices is equally likely (each one has a 20% chance). Now let’s say you buy a call with an exercise price of $40. How much should this cost?

Well, you will only make money if the stock ends up at $50, or $60. If it ends up at $20, $30, or $40, you get nothing. If it ends up at $50, you make $10, and if it ends up at $60, you make $20.

So, what is the AVERAGE return of the call option? We need to weigh the returns by their probabilities in order to find out, like this:

(20% x 0) + (20% x 0) + (20% x 0) + (20% x $10) + (20% x $20) = **$6**

The average return of the call is $6. This is its theoretical value, so you should always pay **under** $6 for this call, in order to keep playing this game and make a profit over the long run.

Now obviously we simplified things by saying there was an equal chance that the stock would end up at each of these prices. That is never really the case. A more realistic situation would be if the stock had a higher probability of doing nothing, and a lower probability of moving significantly up or down, like this:

Now at the end of the period, the **average stock price is the same **(it’s still $40, as it was in the first example). But because we’ve changed the probabilities, we’ve changed how much we should pay (on average) for this option. If we do the same calculation as before, but with the new probabilities, we find that the average return on the call option is now:

(10% x 0) + (20% x 0) + (40% x 0) + (20% x $10) + (10% x $20) = **$3**

Wow. So even though the stock’s **average ending price is still $40, the price we should pay for the call has HALVED to $3 from** **$6. **

The probability distribution (of where we expect the stock to finish at expiration) has a massive effect on how much we should pay for the option

So, the key input into option prices is the probability distribution. If we know the ‘real’ probability distribution, we will know the real price we should pay for the option. To consider the ‘real’ probability distribution of a stock, we can do a little thought experiment. Let’s say that each day, a stock has a 50/50 chance of going up or down by $1. The stock’s movement is essentially a ‘random walk’. We cannot predict where a stock will end up in 30 days time, but we can predict a rough probability distribution of where the stock *might* end up. This video shows what happens if you go through a path many times where there are multiple 50/50 decisions:

Now that you’ve seen the video, imagine what would happen if the nails were much, much fatter. The distribution of balls at the bottom would be much, much wider. This is equivalent to a much more **volatile** stock.

The purple line is how it would look with normal sized nails (equivalent to a stock with low volatility), while the blue line is how the distribution would look with wider nails (equivalent to a stock with high volatility). Clearly, a call option with a strike price around 5 would be more valuable for the high volatility stock (the blue line). So, a clear conclusion is that higher volatility stocks will have higher priced options.

It seems that the probability distribution of a stock is roughly normally distributed. Is this a reasonable conclusion?

Not really.

The problem is that the normal distribution is symmetrical, and as you can see in the distribution above, this means that there is the possibility of a negative stock price. No matter how high we set the middle of the distribution, the chart will always say there is the possibility of a negative stock price. Clearly, we can’t get a negative stock price, so this cannot be right. We can make the situation better by saying that the normal distribution is the distribution of the **percent return **of a stock, instead of the **stock price **itself. At every moment in time, the price of a stock can go up or down by a given percent, and it is **these percent changes that are actually normally distributed**, not the price itself.

If the **percent return of a stock is normally distributed**, then mathematically the **stock price of that stock must be** **lognormally distributed**. The key features of a lognormal distribution is that it does not fall below zero (which is important, as stocks also do not fall below zero), and that it has a longer tail on the positive side, like this:

So if we want to think about the probability of a stock ending up within a particular range, we should allow that range to be slightly higher on the upside. If we include interest rates into our calculation (which I deliberately ignored when I listed the four factors that affect option prices at the start of this article), then we should assume an even greater upside bias to the stock price.

Knowing the theory behind the probability distribution of stock prices, and how this affects option prices, simply helps you understand options theory better.

The two conclusions you should make coming away from this article are as follows:

Probability theory shows us that the distribution of a stock’s price at expiration explains why options have different prices. I.e. volatility is the key ingredient in option pricing.

And….

The probability distribution of a stock’s price is NOT normally distributed. A better approximation is a lognormal distribution, because stocks can (in theory) go up an unlimited amount but can only go down to zero.

Hi John. A good article. We discussed this briefly on my post before. I have a few questions. how do you calculate your risk probability? Also, I agree with your premise on the lognormal distribution (since the standard deviation gets bigger as the price goes higher and risk free rate of return). However, in my experience and my risk probability calculation shows that most trades that we do (1 day to 6 months duration), the distribution looks more like normal than lognormal. Just something to think about. Thanks again for a good article.

Thanks for your comment. I absolutely agree that you can approximate the price distribution as normal – I just don’t like the theoretical underpinning of immediately jumping to this assumption. It’s much more intuitive to work out that the percentage returns are normal, assume continuous compounding, and from there say that price distribution is lognormally distributed. Once we’ve done that, we can then look at the distribution and say that it is pretty much the same as a normal distribution. I just don’t like skipping the logical steps. There are clearly a few problems with assuming any kind of price distribution, but I think this is simple and reasonably accurate. The other reason I like to push the point of non-normal distribution is that lower priced stocks with higher vol will tend to be more “non-normal”, even though I don’t trade those kinds of stocks.

For risk probability, do you mean the chance of getting assigned on an option? I try to manage option structures within a neutral delta range, while avoiding expiration week gamma. For cash-secured puts and covered calls I generally approximate delta as chance of assignment. I can’t assume that I know more than the sum of the market, so I prefer to let the market tell me roughly what the risk is, assuming that there is a consistent and statistically exploitable volatility gap and that volatility reverts to the mean.

In my mind, the key risk is about cash management. I model back to 2008/2009 and see if I can afford it financially and psychologically.