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Buzz Words of Quantitative Finance

As an avid reader of quantitative finance publications, I've been very impressed by the creativity and thoughtfulness financial professionals have displayed in naming trivial concepts.

With terms like supermartingale, antithetic sampling, gamma patterns, and Parisian options, it is tempting to read a romance novel instead.

Truth be told, quantitative finance is not rocket science; although it is commonly practiced by individuals of the same background. So if you are a mathematics PhD who has been intimidated by the carefully crafted language barrier, this article is for you. If you are simply trying to impress your friends with some nifty finance jargon, even better.

The latest significant innovation in quantitative finance was accomplished in the mid 1970s when Fischer Black and Myron Scholes derived a remarkable, extraordinarily elegant, closed-form solution to a problem nobody cares about: finding the fair price of a derivative security, which depends on an underlying asset that follows the path of a geometric Brownian motion with constant volatility in a world with no transaction costs and infinite liquidity. Even if you are unfamiliar with some of these terms, an easy mnemonic to remember the validity of these underlying assumptions is to remember the initials of the authors' last names.

The Swedish in-crowd clearly must have been very impressed by the fact that this formula is not applicable to even a single real-life security because they immediately decided to reward this humorous discovery with a Nobel prize. Immediately in this context means 20 years later but that is considered preferred processing in communist
Sweden.

Practitioners of modern finance have become significantly more sophisticated. Instead of relying on the outdated pen-and-paper methods of the 1970s, modern quants are leveraging highly sophisticated, cutting edge computer technology, capable of running many millions of computations per second. These computers are dedicated to state of the art Monte Carlo simulations.

As the name implies, Monte Carlo simulation basically means "We are either not skilled enough or too lazy to solve these equations or there is simply no closed-form solution -- so why not just flip a coin instead?". And, "Heck, now that we have all of these expensive computers collecting dust and playing Minesweeper, why not just let _them_ flip the coins while we pick our noses or grab a sandwich?". There are even reported cases where these activities were combined.

Of course, you cannot admit this at a finance convention. Instead, argue that Monte Carlo simulation is a sophisticated method for numerically solving high-dimensional problems with probabilistic methods, typically enhanced with powerful variance reduction techniques, such as low-discrepancy random number generators, moment matching, antithetic sampling and stratified sampling, for faster convergence. Don't worry about the fact that combining some of these methods will lead to unexpected (i.e. disastrous) results because as soon as you mention these magic key words, you have the potential to become a spiritual leader in the finance community.

You may have noticed a pattern by now: 1. quants love to appear sophisticated, 2. as indicated in the opening paragraph, a great amount of energy is invested in selecting complicated terms for trivial phenomena, and 3. quants like to pick their noses and let computers do all of the work for them.

Now that we posess the necessary background, let's return to some of the words we have encountered along the way.

A martingale is a zero-drift stochastic (i.e. random) process. You may have seen a co-worker drifting around aimlessly with no discernible direction. As soon as you think you discovered a tendency for them to move towards the coffee maker, they turn around and head for the water cooler or the printer. Your best guess of where they will end up next is their current location. That's a martingale.

What about that guy who keeps arguing with his boss, shows up later to work everyday, and has an increasing number of mysterious illnesses that prevent him from coming to work entirely? You know he is on his way out but sometimes he astonishes everyone with some excellent work and while his departure is painted on the wall, he follows a path of many ups and downs before he eventually gets there. Quants call this a supermartingale because it is super fun to watch. If he were, instead, headed for the CEO post it would be a submartingale because it is suboptimal for your own career to have such an ambitious coworker.

I know you are yearning to learn about antithetic sampling. Intuitively, it must be something that allows quants to do less work. Imagine a scenario where the resident quant has already installed millions of dollars worth of processing power and is running Monte Carlo simulations. The computer is happily flipping coins in the form of pseudo-random picks from a normal distribution. Unfortunately, with all of the CPU cycles used up with flipping coins, the quant's machine can no longer be used to play a graphics intensive first-person shooter game on the same server.

Obviously, this is a mission-critical emergency situation. In response, the quant has a brilliant idea: "Why not just let the computer flip half as many coins and construct the other half of the coin flips from the outcomes we already know?". Luckily, the normal distribution is symmetrical. So he decides to let the computer flip coins from a normal distribution and then use the mirror image of the first flip as his second flip. In this manner, valuable CPU cycles are freed up for the purpose of shooting aliens.

Unfortunately, there is a more subtle concern: how can this move be justified to his supervisors? The answer clearly involves inventing some benefits of using the new method; but to really make the work credible, a fancy term needs to be coined for the procedure. The first part is easy: by using mirror images of previous random samples, the mean of the samples will always be zero. While this is only useful in fairly low-dimensional problems, his bosses don't know that. However, by calling this simple-minded procedure by the cryptic term "antithetic sampling" rather than a more user friendly English word such as "opposite sampling", he is sure to be rewarded with an ungodly bonus.

Gamma patterns are almost too embarrassing to write about. The uninitiated usually wear suits when discussing something of such great significance and class as Gamma patterns. The term Gamma patterns originates with a fellow named Erich Gamma and his "Gang of Four". They simply cataloged well known approaches to patching together software and called them patterns. For example, instead of saying "Hey, pal, we only need one About screen in our swaption pricing engine", he suggests using the term Singleton pattern, as in "Hey, pal, let's use a Singleton pattern when developing the About screen in our swaption pricing engine -- you know, Singleton is that one pattern Gamma and his Gang of Four wrote about, where there is only a single instance of something, in our case the About screen." See, much more succinct and classy.

Parisian options are essentially a cross between Asian options and barrier options. Asian options were originally developed as a racial slur, much like its European and American counterparts.

What the author sadly didn't realize was that Paris is a city and not a country, much less a continent. This is the sort of branding mistake that would be ill-advised for a quant because it is precisely the type of mistake an unsophisticated supervisor may notice. I am not aware of the consequences this had for the inventor of the Parisian option but it is safe to say that he probably moved back in with his parents.

European options are very simple. They can only be exercised on a single future date and their payoff only depends on the price of the underlying security on that date. In contrast, American options can be exercised at any time before or at the specified expiration date and their payoff depends on the price of the underlying security on the date it is exercised. Evidently, American options try to outdo the Europeans.

Asian options deploy fancy gadgets that will only be released years later in Europe and the Americas. In particular, their payoff doesn't just depend on the price of the underlying security on the exercise date. Instead, Asian options are path dependent and their payoff depends on how the underlying instrument reached its current price.

The easiest way to memorize these features is by considering a bike race, for example the Tour de France (which conveniently concludes in Paris). European riders might take some short cuts and avoid steep mountains but in the end, they arrive at the Champs-Elysee and get their payoff (a medal and a drug test). American riders may follow the same strategy but are prone to declare victory in the middle of the race (mission accomplished). This is called an early-exercise feature. In contrast, Asian racers are very traditional and know that if they don't follow the exact path specified, they will not receive a good payoff. That's karma.

In summary, quantitative finance is an area where it is more important to use the right terminology than to have any earth-shattering insights. Acquire the right vocabulary and you too can start a hedge fund or entertain people at the annual Christmas dinner of a large bank.