I have written about it many times in the past but never as clearly as I want to lay it out here and it’s how to use a model in order to trade.

Many people on Twitter will often post about “entry models” or “smart money” models but can’t actually articulate what the model is.

I am not saying they don’t work, they might. But what I am saying is, they aren’t clearly articulating what they are actually doing with the vague representation of the “model” they are trading. That also goes for nearly every thread of FF.

To keep this post simple, we are going to look at the Black-Scholes model, specifically for pricing European Options.

For those that don’t know, a European option is simply a contract saying, after some amount of time, the holder of the contract has the right to purchase some number of shares or units of an asset for an agreed-upon price.

A simple example of this would be a 60-day EUR/USD call option with a strike price of 1.11. Whoever owns this contract has the right to buy EUR/USD at 1.11 after 60 days.

If 60 days pass and EUR/USD is above 1.11, the owner of the contract would be able to buy the shares at 1.11 and instantly show a profit from the difference between the current price of EUR/USD and their execution price of 1.11.

Now that we understand what the option is, and how it works, the most important question to ask is, what is an option like this actually worth?

This is where models come in, particularly the Black-Scholes model.

The purpose of the model is very simple. Given a set of inputs, what would be considered a “fair value” for an option with those parameters.

In this example, the parameters are:

- Probability Distribution for EUR/USD: Most economists typically start with a normal distribution.
- Risk-Free Rate: This is typically the interest rate difference between the 2 currencies.
- Current Price: We will use 1.09 in this post.
- Strike Price: As stated above, we will be using 1.11
- Duration: In this case, 60 days.
- Volatility: This is typically the annual volatility for the pair, but you can really use any period.

Now what the model does is, it takes probability distribution, and modifies it so that it accounts for the risk-free rate. It then calculates all of the possible results of the price of the EUR/USD using this assumed distribution over a 60-day period, using the given volatility.

For all of the possible outcomes where the price of EUR/USD ends higher than 1.11 after sixty days, you take the difference between the final price and the strike price as your payoff.

For all of the outcomes where EUR/USD ends below 1.11, you take 0 as your payoff.

You then average all of these possible payoffs together, discount it by the risk-free rate multiplied by the duration of the option, and just like that, you have a theoretical “fair value” for how much you should sell or buy this option for.

This is also the expected value of the trading activity that would manufacture this option (dynamic hedging).

In other words, if you bought or sold this option a thousand times for this fair price, and all of the model’s assumptions remained true, neither the buyer nor the seller would make any money because the price was “fair”.

I want you to re-read that last paragraph so it really sinks in. The main idea is, if your model’s assumptions (or inputs) are correct, neither you or your counterparty will make or lose any money transacting.

This is where risk and opportunity lie, in the assumption that your model’s inputs are correct…

Now in our model above, there is really only 1 main input that is a risk for someone using this model, and it’s volatility. We know the current price with a very high degree of certainty, the same goes for the strike price and the duration of the option. We also have a relatively high degree of certainty when it comes to how prices are distributed.

But one thing we do not have a high degree of certainty around is what kind of volatility EUR/USD will realize over the next 60 days. That is anyone’s guess…

Knowing this, think again about how we calculate the payoffs using the provided distribution and its assumed volatility.

If volatility is actually higher than what we used in this model, the model’s price for the option will be lower than what it is actually worth. This is because the probability of EUR/USD ending above the strike will be increased.

If the volatility is actually lower than what we used in the model, the mode’s price for the option will be higher than it’s actually worth. This is because the probability of EUR/USD ending above the strike will be decreased.

So even though a call option is inherently a long trade directionally, at the end of the day, it’s actually just a volatility bet.

If you are buying the call, you think the price you bought it for was essentially calculated using a volatility input that is lower than what you think the market will realize over the next 60 days. If you didn’t think this, you’d be explicitly buying the option for more than it’s worth.

If you are selling the call, you think the price you are selling it assumes more volatility in the market than what you think will actually take place. If you didn’t think this, you’d be explicitly selling the option for less than it’s worth.

In both cases, you are betting on the model being wrong, or the model having an error in regards to the volatility input. If you didn’t think the model had an error, there would be no reason to engage in the market at all because every price you transacted would be fair.

This is extremely important to understand. When you trade a model, you are betting on model errors as it relates to the inputs that are most likely to error…

The point of any model is to calculate what some kind of trading activity is worth (its expectancy), in order to price it.

You then need to understand which inputs into the model are most likely to be wrong, and in which direction they will be wrong. You then need to understand how these errors impact the expectancy of the model, and this is what you actually should be trading.

The opportunities are in the errors.

Everyone trading is doing this whether they formally recognize it or not. The problem is, they are betting on the wrong input for the model they don’t even know they are using and this is why they consistently lose.

Most retail traders are assuming the price will be more skewed in one direction or another if they use a certain entry signal, which is why they are either going long or short in a pair.

So in their model, whether they recognize it or not, they are betting that the expectancy of a fair market, which is essentially transaction costs, is wrong because the market is going to go up or down via skew. They think it’s higher than that using their entry signals.

But in reality, you can assume with a high degree of certainty that the skew of the pair’s distribution is essential right at or near 0 overtime.

So if you are betting that the skew of the distribution will be well above or below 0 consistently over time using your entry signal, you will never make any money because that input is one of the least likely to error in the model they don’t even realize they are using…

There is also a bit of a fallacy in this line of thought because, if there was some signal that if used would consistently result in the skew being significantly positive or negative, you wouldn’t need the signal because the “edge” would be built into the distribution itself.

Thinking about trading MA crosses on SPY over the last 100 years… It’s not the MA crosses creating positive expectancy for “trend-traders”, it’s the distribution itself. This is why timing SPY almost always will perform worse than buying and holding it. By timing it, you are actually handicapping the one thing that is driving your expectancy, the distribution itself.