Application of Copula in the pricing of credit derivatives

Abstract

This is an excerpt from an article in a popular American Magazine Wired Recipe for Disaster: the Formula That Killed Wall Street A year ago, it was hardly unthinkable that a math wizard like David X. Li might someday earn a Nobel Prize. After all, financial economists—even Wall Street quants—have received the Nobel in economics before, and Li's work on measuring risk has had more impact, more quickly, than previous Nobel Prize-winning contributions to the field. Today, though, as dazed bankers, politicians, regulators, and investors survey the wreckage of the biggest financial meltdown since the Great Depression, Li is probably thankful he still has a job in finance at all. Not that his achievement should be dismissed. He took a notoriously tough nut—determining correlation, or how seemingly disparate events are related—and cracked it wide open with a simple and elegant mathematical formula, one that would become ubiquitous in finance worldwide. For five years, Li's formula, known as a Gaussian copula function, looked like an unambiguously positive breakthrough, a piece of financial technology that allowed hugely complex risks to be modeled with more ease and accuracy than ever before. With his brilliant spark of mathematical legerdemain, Li made it possible for traders to sell vast quantities of new securities, expanding financial markets to unimaginable levels. His method was adopted by everybody from bond investors and Wall Street banks to ratings agencies and regulators. And it became so deeply entrenched—and was making people so much money—that warnings about its limitations were largely ignored. Then the model fell apart. Cracks started appearing early on, when financial markets began behaving in ways that users of Li's formula hadn't expected. The cracks became full-fledged canyons in 2008—when ruptures in the financial system's foundation swallowed up trillions of dollars and put the survival of the global banking system in serious peril. The rest as they say is history. There were many questions that arose from the financial crisis of 2008 which shook industries, economies and nations. Some of them were fundamental, some technical. The collapse of a global housing bubble, which peaked in the U.S. in 2006, caused the values of securities tied to real estate pricing to plummet thereafter coupled with a new era of financial engineering were cited as the main causes for the crisis. The term financial engineering refers to the ongoing development of financial products designed to achieve particular client objectives, such as offsetting a particular risk exposure (such as the default of a borrower) or to assist with obtaining financing. Examples pertinent to this crisis included: the adjustable-rate mortgage; the bundling of subprime mortgages into mortgage-backed securities (MBS) or collateralized debt obligations (CDO) for sale to investors, a type of securitization; and a form of credit insurance called credit default swaps (CDS). The usage of these products expanded dramatically in the years leading up to the crisis. As described in the section on subprime lending, the CDS and portfolio of CDS called synthetic CDO enabled a theoretically infinite amount to be wagered on the finite value of housing loans outstanding, provided that buyers and sellers of the derivatives could be found. For example, selling a CDS to insure a CDO ended up giving the seller the same risk as if they owned the CDO, when those CDO's became worthless. Certain financial innovation may also have the effect of circumventing regulations, such as offbalance sheet financing that affects the leverage or capital cushion reported by major banks. For example, Martin Wolf wrote in June 2009: "...an enormous part of what banks did in the early part of this decade – the off-balance-sheet vehicles, the derivatives and the 'shadow banking system' itself – was to find a way round regulation. The entire CDO- CDS would not have been possible without the path breaking work of Mr. Li. His paper "On Default Correlation: A Copula Function Approach"(2000) was the first appearance of the Gaussian copula applied to CDOs, which quickly became a tool for financial institutions to correlate associations between multiple securities. This allowed for CDOs to be supposedly accurately priced for a wide range of investments that were previously too complex to price, such as mortgages. However not much unlike the immensely popular Black Scholes Option Pricing Model, the Gaussian copula function had some inherent deficiencies which were not completely understood or accounted for.