Risk Latte - Our Time in Finance

Our Time in Finance

Rahul Bhattacharya
January 1, 2012

Another New Year is upon us. Time is of essence in life and the financial markets; and it flies. Time is imaginary in Minkowski space and Einstein thought "time is suspect" so he made it a relative measure. Every day "time" beats down on us in the financial markets; "time to maturity" of options, "discrete time", "continuous time", "delta t", daily and weekly stock charts, and on and on. And in April 2007, StephaneMattatia and his team at SocieteGenerale made the wonderful world of Black-Scholes stand on its head with a new definition of time to maturity. Time is up when volatility budget is expended.

In the movie, In Time, starring Justin Timberlake and Amanda Seyfried, the characters don’t biologically age beyond 25 but have to earn "time" to live. "Living Time" is money. One needs to earn "time" like we earn "money" to live. In this hypothetical, techno-scientific society, you buy goods and services with "time" and not money and when you no more "time" to buy food or shelter you die.

Time puts a boundary on our lives and the lives of everything around us. It constrains our imagination. Life is time to maturity. Life insurance policies have a time to maturity, financial derivative contracts have a time to maturity, and bonds have maturity. An average human being lives for 75 years or so.On a bigger scale there is time to maturity as well, when the game gets over. The Sun will live for another 5 to 7 billion years or so. The earth will be swallowed by the expanding Sun in another 7.6 billion years and life of all living organisms on earth will certainly end in another billion years. Of course, it is not just the gigantic time horizons that define our lives and the Universe. A year, a month, a day, an hour or even a minute could be as pivotal to our scheme of things. The stock market in most cities opens at 9:30 am and closes at around 4:00 pm;a single day, characterizing a day trade, a daily volatility or a daily close.

Or, our agony and misery could be estimated minute by minute. A trader’s nail-biting stare at his screen to see if his level was hit or if the order got filled. Inside a classroom, time is continuous and simply, the familiar notation of differential calculus. This is the abstract world where we cannot live. In real life, time is discrete and becomes , delta t, measurable by the minute, hour, days, weeks and months. Take an equation that explains the random movement of your favourite stock and you’ll find the continuous time expression of there. Open a spreadsheet to simulate that equation to see how it actually unfolds before your eyes – an undulating, jagged curve of peaks and troughs – and you are faced with how to capture and slice time. Should you make delta t one hour, one day or one week? It matters what unit you choose because at the smallest level the dynamics of the process can become very interesting. Add all those units of time and you’ll get the time to maturity. Add 252 days of trading and you’d get one year movement of the stock price.

Keep slicing time into smaller units of minutes, seconds, microseconds, and yet it remains finite. Unlike the world of differential calculus, an infinitesimally small unit of time in the real world is still finite. The Higgs Boson, that elusive particle thought to give mass to all elements in the Universe and termed the Gold Particle, has a life span of between 600 microseconds and one hundredth of a billionth of a billionth of a second. That figure is as incomprehensively small as the figure of 14 billion years is incomprehensively large. Yet, both bounds are finite.

Time makes our life finite and deterministic. This determinism is perhaps the greatest philosophical truth in this world. No physicist or philosopher has ever questioned this truth. But in the world of finance, a few brave bankers and their mentors in the academia have dared to imagine a world where time is not deterministic but stochastic or random.

Quants and bankers have taken philosophy to a level where no physicist has ever dared to go. Only a fool or a very brave man would do such a thing. In search of an elusive truth, and greater profits, quants – the practicing bankers and the academicians in the Universities – have ventured into a world beyond that of the physicists. Physics has randomness in the form of Brownian motion and diffusion to describe the motion of molecules, gases and heat. Finance has taken the same Brownian motion and has made the medium in which this Brownian motion occurs random. In finance, the coefficient of diffusion is also random. To a physicist this would be incomprehensibleand totally bizarre but to a banker this is just stochastic volatility.

And so we land in the realm of random time.

How weird is the notion of random time? Quants talk about random time and random clocks. In fact, there are financial products based on the notion of a random time. These masters of an alternative universe view a stochastic – by the way, "stochastic" is just another name for "random" – volatility process as a Brownian motion subordinated to a random clock. In a random clock, the clock time is nothing but the trading time which is identified with the volume of trades or the frequency of trading. Time happens when trading happens. I have often wondered if finance is truly that esoteric or it is just that the ex-physicists who have invaded the field in the last 30 years have made it so.

In finance, time can be random in many ways. And time to maturity can take on other guises. Time to default is the time when a firm defaults, goes bankrupt. This time to default is randomized by characterizing it as, what mathematicians call, a "Poisson process". It arrives in front of us all of sudden from nowhere with a certain intensity – literally speaking, bankruptcy is an intense process – stays for a while and then flies away somewhere. A finite and discrete process that is expressed as a very simple equation and yet which has the power to devastate our lives.

In 2007, StephaneMattatia, head of the hedge fund engineering team at SocieteGenerale bank in Paris made the Black-Scholes world stand on its head by proposing a financial product which made use of a different kind of a random time. The product introduced the notion that time to maturity, the day when a financial product matures, is random or uncertain. Up until now, we have always taken the notion of time to maturity, the capital that we use in our formulas and equations, of a financial product as sacrosanct. A three year bond will mature in three years and after the maturity date the product will cease to exist. A three month call option on an equity index will mature in three months and after the maturity date the option will cease to exist. The Black-Scholes world stipulates a fixed time to maturity for financial derivatives and for the past 40 years that is how it has been. Numerous innovations and changes have been made to the original Black-Scholes option pricing theory over the years but no one has thought of making the time to maturity of an option random.

How can we invest in and trade financial products if we do not know when they will mature? The notion that a financial product can mature anytime, in three days, three months, six months or one year may not instill confidence in an investor. Mathematically as well, a random or uncertain time to maturity is a bizarre notion.

However, in justifying the rationale behind this product, called the Timer Call option, one’s arguments can be precisely the opposite. Randomness in time to maturity is just an illusion. Time to maturity is always fixed in advance because that is how we desire it. There is nothing fixed or deterministic about time. It is just the way we are conditioned to think.

Time appears immovable and fixed because its importance is paramount in our minds. All our investment decisions are guided by a simple fact: what is our investment horizon? In other words, when do we want our money back? This "when" is inextricably tied to the notion of time? This "when" is always "a month", "3 months", "6 months", "one year", "five years" or some other unit of time. Why can’t we define our investment horizon in some other unit or use some other parameters to fix the time to maturity. We could as well say that our "time to maturity" would be when the product returns 15% on an annualized basis or when we have made 20% more than what we’veoriginally invested. If this happens in three months then three months would be my equivalent time to maturity in the units of time or if this happens in one year then one year would become the equivalent time to maturity in the units of time.

Therefore, one can argue, rather than looking at an imperfect measure of the life of a financial product – that is, time to maturity, which constrains our perspective – why not focus on something that is directly related to what we desire. A variance budget – or, a volatility budget – that is what investors’ desire and should focus on when they buy options and derivatives. Time to maturity is when that volatility budget is consumed. This makes lot of sense. We buy financial derivatives to buy volatility or protect ourselves against volatility. So why not make that goal as the defining feature of financial contracts? The life of a financial product should not be measured in the units of absolute time but rather in the units of what we desire. Time to maturity, or simply time, needs to be redefined. And this makes it random

The year 2012 will be gone soon and before we realize this decade will be over. The incessant march of time shall continue and we shall wait for our own time to maturity.

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