“Sometimes there’s a strange dichotomy between applied mathematics and pure mathematics,” said Professor Raul Tempone, a founding KAUST faculty member and principal investigator of the University's Stochastic Numerics Research Group. “People think that applied math is simply the application of theory.” The argument is that applied math is just classified by its ends in the sense that one is trying to solve a real problem.
Tempone argues that this doesn’t mean that new theories don’t need to be created. It also does not mean that the kinds of problems are less challenging than the ones faced in pure math. “Sometimes it’s more difficult to formulate and it’s harder to solve from the theoretical point of view. Applied math does require theory, and quite deep theory, actually,” he said.
A prime example of how the field of applied mathematics has necessitated the development of novel theories to ensure its continued efficiency is in the world of finance, and specifically for European options pricing. An option is a derivative which gives its holder the right to buy or sell a particular financial security in the future for a given price. In the case of European options, this right can only be exercised at maturity, the time at which the option contract expires.
To illustrate, Fabián Crocce, a postdoctoral fellow from the group, explains: “Options are kind of an insurance contract, and our goal is to put the right price to that contract to compute the premium. If someone needs to buy certain amount of oil in one year, it’s important to ensure some protection from price fluctuations on that commodity.”
Basically, the buyer needs assurances that he or she will have enough money to acquire the oil needed regardless of sudden market fluctuations. The buyer may purchase a call option to set a guaranteed upper limit for the price. “It’s similar to how insurance companies compute the price of your car insurance,” Crocce explained. We’re essentially paying to cover ourselves from the randomness in the middle.
“There’s a fair price and the goal is to compute the fair price. This is what option pricing is about,” Ph.D. student Juho Häppölä, also from Tempone’s group, further explained.
Using Fourier methods for pricing options
In a research article due to appear in The Journal of Computational Finance entitled “Error analysis in Fourier methods for option pricing,” Tempone, Crocce, Häppölä and collaborator Jonas Kiessling outline their method for pricing European options by tuning the parameters in a systematic way to achieve the required accuracy with minimal computational work.
“The idea is that this could be implemented in a computer to run automatically and not just through trial and error, but by inputting parameters and yielding accurate prices for the options contracts,” Tempone explained.
Financial derivative securities are used, among other things, to buy or sell financial securities in advance to ensure against adverse market movements by arranging today the price of an asset in the future. The price of derivatives is often evaluated using a simplified assumption of the world — a model. The role of the model is to distill the immense complexity of the markets into a handful of parameters. Like all financial contracts, given a model, options have a value implied by that model. Ideally, that model price is very close to the price at which one is actually able to buy or to sell that option.
A pioneering model from the 1970s for option pricing was the Black-Scholes model. This model not only laid the groundwork for much of what we know as quantitative finance today, but gave rise to a definitive pricing formula that was easy to evaluate. However, demonstrating how applied mathematics also spurns new theories, newer methods have since been developed to extend upon the work of Black and Scholes. These extensions allow for modelling many phenomena that are absent in the original work by Black and Scholes, such as real-time pricing and taking into account important factors like stochastic volatility and jumps. Many of these recent mathematical techniques make use of Fourier transforms. These transforms are named after French mathematician Joseph Fourier (1768-1830), who developed them for the purpose of studying heat flow in matter. Since the 1960s, this methodology has seen a renaissance reaching far beyond its original intended purpose, including option pricing.
The most realistic financial models must indeed take into account jumps related to large market movements. Pricing options in the presence of jumps requires delving into partial integro-differential equations (PIDEs).
“This brings a lot of interesting mathematics and a lot of twists,” said Tempone.
In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function. Those equations represent a pivotal part of mathematical modeling in many fields of science and engineering, as well as when taking into account real world problems.
How academia interacts with the trading floor
Tempone’s group that makes up the core of the KAUST Strategic Research Initiative for Uncertainty Quantification represents a good mix of people with both theoretical and financial market experiences. Ph.D. student Häppölä completed a master's degree in theoretical physics and worked in commodity markets for a large energy company.
“I got bitten pretty 'big time' by the financial bug there, but I still felt like I wanted to come back to academia to pursue a Ph.D.,” he said. Häppölä also has a sense of adventure, so when he learned about the opportunity to come to KAUST from a friend who was already at the University, he jumped at the opportunity.
Postdoctoral fellow Crocce completed his Ph.D. in mathematics in his native Uruguay. He comes from a pure math background and enjoys delving into theoretical matters. He also has a background in mathematical finance.
“It’s a nice group because all the backgrounds from industry to the theoretical side are represented,” he said. “Pure math is beautiful and I would like to go back at some point, but it’s also good to see how things are actually applied and to get your hands dirty.”
“I’m very happy to have this vibrant team and the other lines we pursue in the group. This is essentially computational finance at the core,” said Tempone.
By Meres J. Weche, KAUST News.