Gaussian copulas are widely used in the industry to correlate two random variables when there is no prior knowledge about the co-dependence between them. The perturbed Gaussian copula approach allows introducing the skew information of both random variables into the co-dependence structure. The analytical expression of this copula is derived through an asymptotic expansion under the assumption of a common fast mean reverting stochastic volatility factor. This paper applies this new perturbed copula to the valuation of derivative products; in particular FX quanto options to a third currency. A calibration procedure to fit the skew of both underlying securities is presented. The action of the perturbed copula is interpreted compared to the Gaussian copula. A real worked example is carried out comparing both copulas and a local volatility model with constant correlation for varying maturities, correlations and skew configurations.
Copula models arise in the market place when only quoted information about the behaviour of single assets is available but nothing or very little is known about their joint relations. Products that depend on several assets allow to imply from the market some information about their joint behaviour. However, there is a lot of market information embedded in just a single price and therefore, it is necessary to make some assumptions about the joint relations in order to imply some meaningful parameters out of market prices. On the other hand, when products which depend on several assets are not available in the market or they are not liquid enough, it is necessary to make some assumptions about the joint relations of their underlying assets in order to describe them in a simple and intuitive way through some parameters whose values might be given as input. Then, the dependence and sensitivity of a given product to these "unobserved" parameters allow to know the risk associated with them and allows to take a conservative position for trading and managing them. Most of the assumptions of co-dependence among assets can be highly simplified through a copula model. One of the most popular contexts in which copula models have been used is credit (e.g. the valuation of Colateralized Debt Obligations). Another popular application appears in hybrid models which combine two different asset classes for which co-dependence information is not available. The particular application addressed in this research work is FX (foreign exchange) quanto options to a third currency different from the two currencies of the underlying FX pair of the option.
A copula model allows to obtain a joint probability distribution of two random variables when only their marginal distributions are known. As it is clearly explained in [5], the joint distribution of two random variables can be cleanly decomposed into two separate contributions: the distribution of the co-dependence (the copula function) and the marginal distribution of each random variable. This means that the copula function embeds completely the co-dependence information. The gaussian copula is the most popular, well-known and widely used copula model in the industry. It has become a reference proxy in the industry for it is analytical, easy to use and provides intuition. The main assumption of the gaussian copula is to consider that the random variables are normally distributed and their joint distribution is multi-variate normal. The copula function would only depend on the correlation matrix of the multivariate-normal distribution. This allows to embed the whole information of the co-dependence in just the correlation parameter. This hypothesis might not be reasonable when the distribution of the underlyings is skewed (the slope of the volatility surface with respect to strike is different from zero) or in the credit context, the probability of extreme events (the tail of the gaussian distribution) is too low.
The perturbed gaussian copula comes into play to improve the gaussian copula under the assumption that the two random variables have a common fast mean reverting stochastic volatility factor. This hypothesis allows to introduce the skew effect in the co-dependence of the two random variables. The joint distribution of the random variables is approximated through an asymptotic expansion to first order calculated using perturbation theory under the asumption that the common stochastic volatility factor is fast meanreverting. This allows to obtain an analytical expression for the joint and marginal distributions of both random variables and therefore the copula funtion (the joint density divided by the product of the two marginal densities). These “pertubed” marginal distributions will in general be different to the empirical ones obtained from the market. However, the closer these “pertubed” marginal distributions are to the empirical ones, the better their co-dependence will be modelled by the copula.
The general formulation of the perturbed copula has been simplified so that the whole information of the co-dependence of two random variables is condensed in just five parameters with a very intuitive interpretation. The skew information of each random variable is introduced through two parameters: one controls the volatility level and the other the slope of the volatility with respect to strike. Finally, the remaining parameter related to the co-dependence is the traditional correlation used for the gaussian copula. The current formulation of the perturbed copula only reflects properly the skew effect (the slope of the volatility with respect to the strike). In order to incorporate the smile effect (slope and convexity) it would be necessary to continue the asymptotic expansion up to order two. That’s why only two parameters (level and slope) are added to each of the random variables in addition to the tradicional correlation parameter.
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