Mathematics in the financial world
Loan application
We have to buy the house. It costs a lot and the money we have is not enough. Where do we get what we lack? The lender has long been invented. Currently, banks are primarily responsible for this obligation.
We have gone to the bank and asked for the twelve million we need. Apart from guarantees, etc. (not for lack of importance, but because they have nothing to do with calculations), we have to define other things: how long will we return the money and what is the interest rate? Once known, the amount we will pay monthly to the bank is determined. Math is needed, but it is not difficult to count: How much money would they be if within the twelve million fifteen years today they increased by 0.50% in the month? For the M pesetas that we pay monthly, a similar calculation must be made, taking into account the months remaining to each payment to turn fifteen. To make them equal we get a value for M. Pay so many pesetas a month and ready. These calculations and formulas are taught at school. The truth is that although the justification of the formula is a little more difficult to count than money, it cannot be said that you have to use Mathematics at a high level. The formulas are well known and are in the hands of anyone.
Knowing the basic functioning of the argument, it is also easy to assess other possibilities offered by the bank: if after three years we decided to pay a million pesetas in a payment to relieve our debt with the bank, how much do we have to lower the monthly fee? Or how long do we have to continue paying if we want to take the opportunity to shorten the amortization period? How is the monthly fee changed if the interest rate is variable and its revision after one year?
The mathematics of the world of loans is simple. Besides that, it is old and even if time passes, formulas can be applied without changes and with "small" calculations. Paying off debts is more difficult than making these math accounts...
New products
On November 10 we have to make a payment of 5 million to make a business a reality. To do this we will use the money we include in K shares. If we sell as many shares as necessary and buy fixed interest letters with the money obtained, we are sure that on November 10 we will be able to make a payment of five million.
However, in view of the stock evolution of K, we would like not to sell now. The second option is being studied, the sale on November 10. If the price of the shares increased relative to the fixed interest letters, to get the five million we would have to sell less than today and we would go out winning, but the price can also go down and then... This second option risks and no one will tell us now what will happen to the actions within a few months.
What if someone buys us shares at least November 10 at the price set today? That is, we want to buy an opportunity: on November 10 we want to find someone willing to pay the P peseta for each K action, at least up to a number of shares, regardless of the price they have in stock that day. Then, when the day comes, we will do what suits us, selling in the market or going to sell to those who have signed the opportunity. This third option has the advantages of the second, but without risks. Is this possible?
In this example, unlike the loan application, we now have no lack of money, but since the day on which we will have to use the money is far away, we would like to devise a suitable strategy not to lose the benefits that the market can offer, keeping the guarantee we have now to deal with the payment.
Surprisingly, there are financial products like this and more and more. These optical stocks, so recently mentioned in the media, are nothing more than the right to acquire shares at a price prefixed at certain dates. European options are called when deciding whether or not to take place on the day mentioned above and whether Americans may be held at any time until the last day.
The role of mathematics
Let us now go to the other side, next to the product seller. If the customer makes a proposal like the previous ones, we assume the risk that it loses. In return we will ask you to pay the premium as requested by the insurers. How much to charge? The issue is measuring, valuing and pricing. In addition the price must be accepted by both parties.
Here is Mathematics. As in other sciences, it is necessary to develop a mathematical model. Determine what are the significant variables, establish relationships between them and write an equation; the way of modeling is as follows.
However, the model developed does not serve to predict reality. In physics, for example, knowing the equations of the description of the movement and the data of today, we can know where the Moon will be within eight months and what will be the course of the rocket that we emitted from Earth, and if we do well the calculations, we can get the rocket to be on the Moon on the preset day (as long as it is possible to overcome technical barriers).
In economics Mathematics does not. It is clear, therefore, that there is no forecasting model of market evolution within eight months. It is not possible to determine the attitude of investors. Have you ever imagined from television what would happen if they made the "scientific" forecast of the Stock Exchange as we are told by time? The mathematical equations would announce the decline of some stocks for tomorrow, investors selling them to the ramp and would convert what had to be a small decline in a huge amount, denying what was said by the equations.
In financial products the function of mathematics is not to predict, but to value. Take into account all the options (within the limits that you can have to say "all"), value the probability of each of them, decide the values of the parameters and finally extract a number of the necessary model that we must ask the client. When the buyer does not take advantage of the opportunity, the premium is won, but if not, we have to cover the difference between the market and the preset price. To avoid going to the disaster it is essential to know how to value it well.
Rating history
XVIII. At the end of the 20th century Daniel Bernoulli wrote an article on the measurement of risk, using the concept of variance used in Statistics. In 1900 the French Louis Bachelier presented in Paris his doctoral thesis entitled Thžorie de la SpŽculation. He led the prestigious Henri Poincarn. Apparently, the subject was not to Poincarés' liking and for many years Bachelier's work remained unknown. He proposed a Brownian movement for the evolution of asset prices, a concept that emerged in Biology in Bachelier's thesis before its use in Physics and Mathematics by Einstein.
The books published in 1947 and 1948 by the prestigious economist Paul Samuelson became a basic pillar for the economy. In Samuelson's view, without wasting time on words and mental exercises, economists had to seek light by the way of Mathematics, not because Mathematics would give the solution to all problems, but because it was the basis for understanding the situation. In the 1960s he introduced a change in the Bachelier model, making the Brownian movement not referring to the benefits of assets but to prices. Samuelson received the Nobel Prize in 1970; Professor Lindbeck began his presentation as follows: "One of the main characteristics of the economic evolution of recent decades is the increasing degree of formalization of analytical techniques, often with mathematical methods."
Years later there was a profound change, the formula of Black-Scholes, so famous for economists. The first financial derivatives market in Chicago began in 1973 (the aforementioned products are called derivatives because they depend on the evolution of others). The application of this formula and other resources offered by the Mathematical Economy in the market did not take long to enter into force, which has meant in recent years a spectacular growth of the derivatives market. They were born in Spain in 1990: In Barcelona of fixed income and in Madrid of variable income, because these derivatives are also sold and bought in the market, only look at the economic pages of the verification journals.
Three economists are linked to the new formulation: Fischer Black and Robert Merton from the United States and Myron Scholes from Canada. The last two received the Nobel Prize in 1997, unless Black died two years earlier. Robert Merton acquired a solid mathematical background and entered the California Institute of Technology in 1966 with the intention of doctorate in Applied Mathematics, but a year later he decided to move to Economics, was accepted at MIT and, being still very young, became a collaborator of Samuelson. There he met with Black and Scholes. Black began in physics, earned a doctorate in applied mathematics and finished in economics. Scholes' academic career was closer to economics, but when he began his thesis, he also had to work in computing at the University of Chicago.
All three worked on the problem of pricing derivatives. Using some basic ideas they were able to write an equation of partial derivatives, solving the equation and extracting the formula of Black-Scholes. The article entitled "The pricing of options and corporate liabilities" (1973) is one of the most cited in the scientific literature. Merton devised a new way to draw the formula in a work written in the same year and streamline some hypotheses. It was very important to try to introduce some ideas into the formulation, since they allow to obtain objective probabilities instead of the doubtful data of the future, which although they are not real, the final result is the same. It was also crucial to recognize that the market tends to balance and excludes by itself a situation known as arbitration (earning income).
Zu idazle
Zientzia aldizkaria
