Copula (probability Theory) - Applications - Quantitative Finance

Quantitative Finance

The applications of copulas in quantitative finance are numerous, both in the real-world probability of risk/portfolio management and in the risk-neutral probability of derivatives pricing.

In risk/portfolio management, copulas are used to perform stress-tests and robustness checks: panic copulas are glued with market estimates of the marginal distributions to analyze the effects of panic regimes on the portfolio profit and loss distribution. Panic copulas are created by Monte Carlo simulation, mixed with a re-weighting of the probability of each scenario.

As far as derivatives pricing is concerned, dependence modelling with copula functions is widely used in applications of financial risk assessment and actuarial analysis – for example in the pricing of collateralized debt obligations (CDOs). Some believe the methodology of applying the Gaussian copula to credit derivatives to be one of the reasons behind the global financial crisis of 2008–2009. Despite this perception, there are documented attempts of the financial industry, occurring before the crisis, to address the limitations of the Gaussian copula and of copula functions more generally, specifically the lack of dependence dynamics and the poor representation of extreme events. There have been attempts to propose models rectifying some of the copula limitations.

While the application of copulas in credit has gone through popularity as well as misfortune during the global financial crisis of 2008–2009, it is arguably an industry standard model for pricing CDOs. Less arguably, copulas have also been applied to other asset classes as a flexible tool in analyzing multi-asset derivative products. The first such application outside credit was to use a copula to construct an implied basket volatility surface, taking into account the volatility smile of basket components. Copulas have since gained popularity in pricing and risk management of options on multi-assets in the presence of volatility smile/skew, in equity, foreign exchange and fixed income derivative business. Some typical example applications of copulas are listed below:

• Analyzing and pricing volatility smile/skew of exotic baskets, e.g. best/worst of;

• Analyzing and pricing volatility smile/skew of less liquid FX cross, which is effectively a basket: C = S₁/S₂ or C = S₁*S₂;

• Analyzing and pricing spread options, in particular in fixed income constant maturity swap spread options.