Some time ago, Stefano, a brilliant mathematician in our research group, came to my office with a smile painted on his face. He started with his classic “Look what I just found…”, and went on discussing a mostly complicate argument, the point of which was totally escaping me. This is a rather typical situation between us: I am more physically- than mathematically-minded, and he often looks for a little extra physical intuition from me, in order to better frame his original mathematical concoctions. This time he introduced us to a very special declination of stochastic differential equations that ends up into something called “infinite ergodicity”, applied to geometric Brownian motion. It was the occasion for me to learn quite a lot of new things, which ended up in a recent paper of our group (in which Stefano did practically all the work, with just a little disturbance by us coauthors). If you are not familiar with “geometric” Brownian motion, just think of it as a stochastic process, such as diffusion, with an underlying (deterministic) drift term that displaces the average value linearly in time. As it turns out, such a process has a number of interesting applications, but if you search the internet, you will invariably find just one and only use of the corresponding equations: predicting stock prices in the financial market. Hence, it’s been the occasion for me to learn something more about finance as well, which I am going to share with you today.

Let us imagine the following situation. You own a bakery downtown, and a law firm next door asks a quote for a wholesale supply of 10,000 sandwiches and croissants for a business reception. You provide the quote with a unit price for the food pieces, which is accepted. Only problem: the reception will be held exactly one year from today. Of course, you cannot sell them the croissants right now, but will have to make them just a few days before the reception. But what if, by that time, the price of flour and butter will have increased? For example, the flour costs today 200 dollars per ton. But it is known that in the past few years, this figure has been fluctuating up and down, by as much as 20, sometimes 30%, and you risk to lose a lot of money, with respect to the price fixed today. So, you go talk to your bank, which propose you a kind of insurance: if by next year the price has increased by, let’s say, more than 10%, the bank will sell you the flour at the maximum price of 220 dollars; if, on the other hand, the price decreases, you are free to not use the insurance, and can buy the flour at the best price you can find. Of course, such insurance has a price, and you will have to pay a monthly fee to the bank until next year, when your deal is closed. Then the question is: what is the amount of the fee the bank should ask (the “premium”), in order for you to pay a fair price, and them to not incur in a too high risk? Such a situation occurs all the time in finance, and is the problem of fixing the price of options in the stock market. (Care for the difference: options, not obligations.)

The baker’s example is what in finance is defined as a “call” option: it is a contract that allows you to buy a product at a fixed price, called the “strike price”, to a certain date in the future. At the opposite, there are so-called “put” options, in which you fix in advance the selling price of a product at a future date. In this case, you will sell at a fixed price, even if the price of that product fluctuates well below the price set in advance. It is still an option, since you remain free to sell at any price you want, if the price instead fluctuates to higher values. Such a principle holds in the stock market for any type of trade, from commodities to stocks to currencies. Therefore, the notion of option is a way to protect the investor (“hedging”) from the fluctuations and speculations on the prices, or of the exchange rates. However, note that this is different from a simple insurance against damage. In that case, the bank should make a relatively simple calculation. For example, let’s take the fire insurance: if the probability of a house catching fire is, e.g, 0.1% per year of exercise, and the average value of an insured property is 500,000 dollars, the bank would have to pay on average 500,000 dollars every 1,000 homes insured, each year; if they insure 1,000 homes, the “right” insurance premium per home will be then 500 dollars per year. This is because the damage event is a binary possibility: either it happens or it doesn’t.

With the stock market, instead, there is no such binary possibility: the price of an asset can continuously fluctuate up or down, in principle by any amount. In this case, the bank must be able to calculate any possible future scenario, only based on the history of fluctuations for that particular asset. Fluctuations in the stock market are stochastic, and unpredictable at any future time, with a behavior that shares all the typical characters of a unidimensional Brownian motion. As we know, at the microscopic scale Brownian motion can be described as a random walk, in any number of dimensions, while at the macroscopic scale the overall result of such a stochastic process is described by a parabolic (diffusion) equation, a differential equation of first order in time and second order in space. If for example, we look at the market fluctuations of the price of flour, a random walk with steps of about ±1 dollar/day does reproduce both qualitatively and quantitatively the typical fluctuation profile. This value, representing the typical width of the price jumps, is called the “volatility” of the price. However, a better reproduction of the fluctuation profile is obtained by considering the volatility not just as a fixed value, but as a function, notably a *stochastic* function. For example, a Gaussian distribution of random values, with the variance equal to the volatility, and a mean that grows linearly in time, representing the positive expectations of the investor. The mathematical model that is obtained in this case is called a “geometric” Brownian motion: it is the representation of a continuous (in time) stochastic process, in which the *logarithm* of the stochastic variable follows a random walk.

With such a model we could realize a computer simulation of the process, for example by a Monte Carlo method. As a result, after many millions of different realizations of time trajectories of the price, we would be able to give the bank an estimate for the average value of the price P at any time t in the future. However, to perform millions of Monte Carlo simulations can be both cumbersome and annoying, and eventually it is a numerical procedure, always affected by some error. In 1973, the finance world was taken by surprise, with the publication of a closed-form, analytical equation that allowed the markets to calculate the optimum premium of an option, without recourse to simulations: the celebrated **Black-Scholes formula**. As the story goes, the economists Fischer Black, then at the University of Chicago, and Myron Scholes, then at MIT, had formulated their model earlier around 1970, but they did not publish it until 1973, when also Robert Merton (who also worked with Scholes) came up with a very similar theoretical result. As the story goes again, in 1994 Merton and Scholes founded the LTCM hedge fund that, after a very successful start, went on to lose 4,6 billion dollars in a few weeks, after the 1998 financial crises. Just before such an unfavourable end, which proved the volatility of financial markets beyond any predictive model and involved the two partners in endless legal suits, the two were awarded the Nobel prize in economic sciences (not Black, who had died in the meantime).

A stochastic simulation is based on a model that we could define as “wait-and-see”. By following the average prediction of the simulation, there will be situations in which the bank gains some money with respect to the average, and situations in which they lose money. Such a degree of uncertainty, however, is not what banks generally like. The Black-Scholes model asks the bank to follow a more proactive strategy. The principle is that the seller (the bank) must *reinvest* a fraction of the premium periodically collected from the buyer, in the same asset that is under the option. As time goes, the fraction of the premium that is reinvested is adjusted according to the past fluctuations. The solution of the Black-Scholes model gives the optimal strategy as far as adjusting the fraction reinvested, such that at the expiration of the option the amount paid is constant. Technically, this is called the “delta-neutral trading” of the option. If P_{t} is the price in time, the stochastic differential equation is: dP_{t}=μP_{t} dt + σP_{t} dW_{t}, with the constants μ being the percent drift, and σ the volatility; dW_{t} is a noise term that provides the stochastic fluctuation. This is a stochastic differential equation, that cannot be solved as a normal differential equation in time, and requires a special calculus (the “Itô stochastic integral”) . If you are interested in the details of how this equation is solved, you can get a nice and bookkeeping introduction at this web page. The final “exact” result for the premium to pay for the option at time T (finite), given a strike (or fixed) price K of the option, is a Gaussian integral chiefly depending on the logarithm of the ratio P_{0}/K.

A dangerous outcome of this logic is that any merchandise that displays fluctuations in price can be subject to this same mathematical treatment. The option itself can become an asset to trade, hence one can set an “option on the option”: this is the strategy behind the so-called *derivatives*, which are among the most speculative products on the financial market. Imagine that you presume, on the basis of some analysis, that the price of flour will increase in the next months up to 250$/ton. A traditional investor should buy large amounts of flour, and resell them when the price will be higher. But a higher bet could be to buy instead “call” options on flour: you go to the baker at the beginning of our story, and find he is afraid that, despite the bank option at 220$, he could still lose money. So, you propose to buy back his options at a slightly better price than the strike price, for example 230$. This will make him happy, and give you a margin when the price will rise, for example to 250$: you will then exert the option and immediately resell the flour at market price, gaining the difference. Of course, it is a risky speculation with a tight margin, in which you hope for the worst to happen in order to gain. It is a bit like getting an insurance on top of somebody’s insurance. You know that your friend John is a very bad driver, so you make a bet on him having a car accident (for which he is normally insured). In this case, you could even “push” him to have an accident: he will be covered by his insurance, and you will get paid by yours. You don’t run any risk personally, but you are hoping that your neighbor gets damaged. This is exactly what happened in 2008 with the subprime crisis, in which speculators kept betting on the downfall of high-risk assets.

What is interesting to note, for us physicists, is that this “discovery” is nothing more than a “rediscovery” of a work due to Richard Feynman and Marek Kac, about a quarter of a century earlier. In 1947, the two scientists were both at Cornell but ignored each other’s presence and work. One day, Kac went to listen to Feynman giving a lecture on his newly introduced path-integral approach to quantum mechanics, and realized that they were working pretty much on the same problem. In fact, a notion similar to the path integral had been introduced at least ten years earlier by Norbert Wiener, to deal with (guess what…?) Brownian motion. Feynman and Kac established a formal connection between general parabolic differential equations (such as the diffusion or the heat equations) and stochastic processes. Basically, the solution to any such PDE can be reformulated as a conditional expectation value for a stochastic process X_{t}, which follows exactly the same stochastic equation written above for P_{t}. This established a subtle connection between quantum mechanics and stochastic processes, which would deserve another Sunday letter… However, other interesting applications of the geometric Brownian model are found in many other scientific fields, such as epidemiology. In this case, the stochastic variable P_{t} represents the infected population, the deterministic drift μ is the transmission rate of the virus, the volatility σ is the recovery rate from the disease, and the Brownian process dW_{t} is the randomness in the infection process, such as the binary encounter between an infected and a healthy individual. The problem is to decide whether or not, and at what time, to implement costly (both economically and socially) control strategies, as opposed to “wait-and-see”, hoping to learn enough on the pathogen in order to prevent the spreading. This is the dilemma, for example, for the amazing olive trees in the Puglia region in Italy: attacked by the terrible *Xylella* pest, against which there is no chemical or biological treatment, one has to decide whether to cut them, up to where, and when. Black-Scholes may help, but probably the trees have a different opinion.