Copulas find new role in derivatives pricing

Pricing model for exotic options revives poster child of 2008 credit crisis

The word ‘copula’ might still stir up bad memories for anyone in the markets at the time of the 2008 global financial crisis.

The Gaussian copula, which was widely used to price collateralised debt obligations, failed to capture correlation skew – the tendency of assets to move more in step in times of stress – with disastrous results.

Using copulas, one can bypass the unnecessary simulations

Ignacio Lujan

But not all copulas are created equal. Normal mean-variance mixture copulas can capture heavy tails and asymmetries in the dependence structure of random variables. In his latest paper, Ignacio Lujan, a quant analyst based in Madrid, describes how these types of copulas can be applied to improve the pricing of exotic options.

The tools do a good job of dealing with correlation skew, which is essential for pricing baskets and path dependent instruments, such as best-of and worst-of options. The main benefit is they dispense with the need to simulate multiple paths to maturity. 

“For European options, for which we only need the simulated price at maturity, it doesn’t make much sense to simulate the whole path to maturity,” says Lujan. “Using copulas, one can bypass the unnecessary simulations.”

The method described in the paper consists of two steps. First, copulas are used to generate the basket implied volatility surface and forward curve. With these inputs, the basket can be simulated as a one-dimensional process to reproduce the target joint distribution at each time step or observation date.

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Skewing the correlation in local and stochastic volatility frameworks via copulas 

A copula-based model to capture correlation skew in multi-asset derivatives is presented

This is a significant departure from current approaches to pricing exotic options, such as local correlation models, which capture the correlation skew but require a large number of simulations that are computationally intensive.

One of Lujan’s goals was to simplify the pricing of complex products, such as autocallables. “With this solution, I think the dimensionality of the problem is greatly reduced,” he says. Another objective was to control the path distribution and the joint distribution independently. This makes it possible to separate the effects of stochastic local volatility dynamics and correlation skew, which Lujan says has interesting implications for model risk management.

Lujan has also developed an ad hoc procedure for pricing products that either depend on a combination of baskets or have complex features that cannot be captured in a one-dimensional simulation. The approach involves calibrating a normal mean-variance copula to an index of the components of the baskets. This provides the dependence structure not only of the index but also any subset of assets. The dimensionality of the problem is reduced as there is no longer a need to simulate the index and its components.

Lujan concedes his approach has limited application in the front office. Trading desks need a clear P&L explained in terms of gamma and theta, which copulas cannot provide. “They are, however, well-suited for other tasks, such as indicative pricing, estimation of valuation adjustments, or the generation of stressed scenarios,” he says.

Another potential limitation is the size of the baskets the models can handle. Lujan tested them on baskets of up to 50 assets, suggesting the approach is suitable for small-to-medium-size stock indexes such as the Eurostoxx 50, but not for large indexes such as the S&P 500.

Given their history, banks will undoubtedly be cautious about using copulas for derivatives pricing. But that may be a case of throwing the baby out with the proverbial bathwater. Gaussian copulas were too simplistic to price credit derivatives. Lujan shows that a different copula, better equipped for complex and correlated structures, can be a viable alternative to established pricing methods.

“Copulas, in my opinion, are at a very early stage for pricing path-dependent options,” Lujan says.

The 1959 paper from Abe Sklar introducing copulas proved they can describe any dependency structure. Lujan’s paper shows this is true for basket options. It may be worth taking a look at whether this is also the case for other instruments.