Mathematics and Financial Derivatives

Warren Buffet, considered by many to be the most successful investor of the 20th century, has called derivatives financial weapons of mass destruction (see link).

As a matter of fact, looking at the causes of the greatest financial losses of all time due to incorrect investments, we find financial derivatives in many of the top positions (see link).

At the top of this special chart there is a loss of nine billion dollars, suffered by the JPMorgan Chase investment bank in 2012.

Why am I speaking about this in a blog dedicated to mathematics and physics?

Let’s start (as is often appropriate) from the beginning.

From Thales to Wall Street

It seems that the oldest financial derivative was bought in the sixth century BC by the Greek mathematician and philosopher Thales.

According to some anecdotes, it seems that Thales had a certain flair for business and, foreseeing an exceptional olive harvesting season, agreed upon what in modern terms would be called a derivative to earn from it.

This type of agreement was quite common among ancient merchants to fix, in advance, the price of goods that would be available at a later date.

A more systematic use of this kind of contracts was developed in the Dojima Rice Exchange in Osaka, Japan during the 18th century, and then in the 19th century in the United States.

The derivative market remained quite limited up to about the end of the 1990’s when an expansion caused a boom in the recruitment of mathematicians and physicists by banks, insurance companies and consultancy firms working in the financial sector.

Why do banks need mathematicians and, above all… what the hell are these derivatives?

Definition and examples

You can give different definitions of what a derivative is, a rather complete one is the following:

A derivative is a contract that provides for one or more future exchanges of money or other assets, where the exchanged amounts depend on the values taken in the future by certain variables called underlying.

Using a more mathematical language a derivative can be described by a structure like this:

• a vector of times $(t_1, \ldots, t_n)$ representing payment dates;
• a vector of variables $\vec{x}=(x_1, \ldots, x_k)$ representing the underlying assets;
• n functions $f_1(\vec{x}_{t_1}), \ldots, f_n(\vec{x}_{t_n})$ describing the payments due on dates $t_1, \ldots, t_n$ depending on the values of the underlying assets x. In financial jargon, these functions are called payoffs.

In many practical cases, this structure is simplified because there is just one payment, just one underlying, or the payment functions are all the same.

After all this theory it is time to look at some real examples. The following are derivatives that have crude oil price as the underlying, which is today about 60 dollars/barrel (for updated prices look at link).

Derivative 1: On 31 December 2018 if the price of crude is greater than or equal to 60 dollars/barrel, then those who had previously bought the derivative will receive 30,000 $, otherwise they won’t receive anything.

Note that in general a derivative offers you the chance to make a profit, so if you want to invest in it, you have to buy it for a certain price. In this case the graph of the payoff function is a step function.

Derivative 2: define the variable $x$ as “the price of a crude oil barrel at 31 December, 2018”. Those who had previously bought the derivative will receive $30,000\cdot \max [x-60; 0]$ dollars on the last day of 2018. Written another way, they will receive

$$f(x) = \begin{cases}30,000\cdot ( x – 60 ) & \text{if } x \geq 60\\ 0 & \text{if } x < 60 \end{cases}$$

The graph of the payoff function is a countinuous line made up of two straight lines.

All clear but… what are they for?

Derivatives can be used for two purposes:

• To protect against a risk. In this case you have to buy a derivative that pays you when the event from which you want to protect yourself against occurs.
• To speculate. In this case you buy a derivative and try to sell it later at a higher price.

Taking the above examples, Derivative 2 is suitable for hedging against the risk of an increase in oil price because it pays you proportionally to how much the price exceeds the level of 60 dollars/barrel.

Let’s assume that I have an oil-related activity and that when the price rises too much, my business is at risk. I could buy this derivative as a sort of “insurance”. If the price rises my business will suffer a loss but I will also receive some money from the derivative that will mitigate that loss.

Derivative 1, on the other hand, is considered speculative because its payoff is discontinuous and piecewise constant . As a result:

• Small changes in oil price (around the value of 60 dollars/barrel) can imply very different payments;
• When the derivative pays you some money, it always gives you the same quantity even if the price reaches extremely high values.

These characteristics make it unsuitable to be used as an insurance against the rise in oil price.

Now you can understand how derivatives can be useful for hedging against some risk, but also dangerous if their own risks are not properly assessed.

The Price Is Right!

The standard recipe for giving a price to a derivative comprises:

1. Set up a stochastic model that describes the evolution of the underlying assets.
2. Find the probability distribution of the underlying assets at payment dates.
3. From the distribution of the underlying assets, through the payment functions, find the distribution of payments.
4. Calculate the mean value for each payment distribution.
5. The price is given by the sum of the mean values found in step 4.

The price is the average profit I can get from the derivative.

This calculation can be more or less difficult depending on the complexity of the model used for the underlying, and how complicated the payment functions are.

In the simplest cases the price can be found through closed-form formulae, but when this is not possible, the price is usually calculated using Monte Carlo simulations (for an introduction to Monte Carlo methods see my previous articles Monte Carlo Methods and Monte Carlo Methods in Action).

Since you need to use stochastic processes to price a derivative, now you can understand why companies dealing with derivatives hire so many mathematicians and physicists.

What are stochastic processes? Maybe I will write something about them in a future post, but for the moment you just need to know that they are used to describe phenomena that evolve in a non-deterministic way (for example the price of a share or the chaotic motion of a molecule colliding against other molecules).

See you soon!

P.S.

Those who already work in the field have surely noticed that I made some simplifications. In particular, I have omitted:

• The fact that generally the underlying asset values used to calculate the payoff are not taken on payment dates but on previous dates;
• The fact that the expected value of future payments should to be discounted with a rate curve to find a corresponding present-day value.