A simple exercise shows how profitable trading of options can be easily achieved without any hedged position and refusing the actual assumption that options prices are independent of underlying-asset expectations.

Digital options allow to have only two events of profit and loss thus avoiding multiple integrations in the equations.

Moreover, I give here an example of how a stochastic process can not be outlined without the use of integral dependencies.

This paper has not been written for publication so please excuse any typing error.

I will suppose to take positions on digital call options which only allow the owner to receive a fixed quantity (per option) on the underlying passing a fixed threshold.

The same exercise can, of course, be done with more structured options but the main aim here is to show that the actual theory is misleading. At the same time, more complex options immediately require harder mathematics than those presented here.

Suppose that you have a portfolio whose value at the beginning of trading is x₀ and whose evolution is characterized by a probability density

at each trading step n, due to the uncertainty of your trades. This means that if you want to know with how many chances y the value of your portfolio is comprised between x₁ and x₂ at trading step n you simply perform the integral

being, of course, x₁ and x₂ comprised between zero and infinity.

Obviously you have

otherwise it would be foolish to invest.

Now, if at trading step n you have an amount of money equal to x, at trading step n+1 you may have either

with p chances or

with 1-p chances. Therefore your expectation value is

At the same time, you don’t have an exact value for x_n but a probability density because we are dealing with a stochastic process. This means that also for x_n+1 you will obtain at most a probability density by averaging on all possible values of x_n which is

This is a linear Boltzmann equation depending on time (trading steps) differences. Solving is quite complex but those who are interested may read on my BEFASP article about the use of the Mellin transform to achieve this.

Of course, the solution allows the reader to inspect the process much better and study in detail eventual asymptotic behavior of portfolio or even stability problems.

At this stage, however, it is only my intention to show if and how money grows during time.

Fortunately, the terms under the sign of integration can be worked out with no difficulty to those who know the properties of Delta functions.

In this way we obtain

The given formula already allows us to calculate the average value of our portfolio as a function of the previous one.

We have that

which with some minor algebra becomes

From this expression it is obvious that our strategy will be successful if, and only if,

because this implies the capital to grow.

Notice that, in sharp contrast with actual options customs, we have:

1. No hedging.
1. The expectations on the underlying influencing our strategy.
1. The expectations on the underlying influencing our estimations of the options price.

Point 3 may seem difficult at first but comes out from the fact that we will decide acceptable options prices from Eq. 2 in function of their return; once we have evaluated p.

It is custom of traditional options theory to object that a stochastic approach to trading may bring to ruin with consistent probability.

However, if you take Eq. 1, you see that there is no transition to the state with zero money unless you invest all your money in the trade. In fact the second term in the RHS tends to

It is usually written on books that: "Bulls and Bears agree on options prices" but what is not written is that bulls and bears don’t use Boltzmann equations.

Good trading,