The world of derivatives is based on three main concepts: hedging, arbitrage and pricing. Most investment banks, hedge funds and other financial institutions base their behavior in derivatives’ markets on the assumption that a position in derivatives can be replicated and therefore balanced by a position in other equities.

As a consequence, these financial institutions pull-out derivatives prices from the prices of equivalent positions and given an imbalance between theoretical prices and market ones they operate an arbitrage. In such a strategy they claim to lock-in sure profits.

It is the aim of this article to redo the path from position-structuring to pricing and show how the arguments mentioned above are absolutely false.

This is evident once the problem is treated with mathematical rigor and without the need for any assumptions on the behavior of markets.

Moreover, I show that the impossibility of hedging even holds under more extensive position-structuring capabilities who do not actually take place on markets.

A complete deduction is done for simple options but the method can be easily generalized to more complex derivatives.

Some non-mathematical conclusions try to give an explanation for such a mistake and are aimed to prevent investors from taking unneeded and uncontrollable risks.

The core concept in derivatives-pricing is that of replication. It is assumed in the industry that such financial instruments need not to be priced by forecasting the behavior of their underlying assets. On the contrary, the price of the derivative is that of the equivalent position, being this equivalence intended in the sense of the investment outcome.

This equivalence, however, has never been proven in mathematics in an extensive way. In just one case, the existence of replicated position is a matter of fact and this is the binomial random-walk. In all other cases, it has been assumed in the financial industry that replication exists as a postulate.

It is therefore natural for us, to begin our inspection from this point.

We will pose the question in a quantitative way to mathematics under no particular assumption on the behavior of the underlying assets.

Let us consider the position of an investor who owns a certain stock and who is willing to hedge his position by the use of options and fixed-income instruments.

We will assume that options exist on all strikes from zero to infinity; this will give the investor more hedging power than he has in the true world as we all know strike prices are discretized; we will show at end how this enforces our conclusions in a tighter way.

In order to avoid any problems between American and European options behavior, we will assume that we are only one trading step away from expiration whose duration is T.

The investor has a stock valued x at initial time. After the trading interval, the stock will probably no longer be valued x but will be valued at a different uncertain price y. We of course neither know y in the deterministic sense (an exact value) nor in the stochastic sense, which means a probability density

giving information on the distribution of the final prices.

Both the above-mentioned kinds of knowledge for y must not be assumed as on our hypothesis of no superior forecasting ability. It may also be shown that they can even be expressed in a single sentence when deterministic events are given stochastic formalism by the use of the Dirac’s Delta function.

Should the actual theory hold true, we will be able to land to a price for the option which does not-at-all depend on y and, therefore, F(y) (y) will have to disappear from the final expression.