Umberto Cherubini: Compatibility problems

Since the beginning of this century I have been focusing my research on an issue that is known as compatibility. In statistics compatibility means that joint probability distributions must be consistent with marginal distributions. To gauge the relevance of this concept in financial applications consider the following instances.

·Pricing a multivariate derivative consistently with univariate derivatives: basket equity derivatives consistent with plain vanilla options, basket credit derivatives consistent with single name CDS
·Assigning VaR or Expected Shortfall limits to a business line, consistently with limits assigned to the different desks, or to the different risk factors.

Copula functions provide the basic answer to the compatibility problem: every joint distribution can be written as a function of marginal distributions. We are then free to calibrate the distribution of each financial asset, or each risk factor as a first step, and their co-movement as a second step. 

Compatibility problems span the specification of the dependence structure of prices both in the cross-section and time series dimensions. To fix ideas, and to introduce the topic from the basic to more involved features, consider the three (compatible) pricing problems below. 

·Problem 1.
Altiplano. Consider a product paying a digital amount if all the prices of n assets are above a given threshold at a given future date T. Consider n digital products paying a digital amount if the corresponding asset is above a given threshold at a given future date T. Copula functions are the ideal tools to ensure compatible prices of the multivariate digital options. 

·Problem 2
No touch options. Assume an option paying a digital pay-off if an asset is above a given barrier at time T (digital call). Assume an option paying a digital pay-off if the same asset remains above the same barrier by time T – 1, on a discrete grid of dates (time T – 1 no-touch option). Copula functions must ensure compatibility of the price of the time T no-touch option with the two prices above.

·Problem 3
Barrier altiplano. Consider a product paying a digital amount if all the prices of n assets are above a given threshold by a given future date T versus a product paying the same amount if they are above the same threshold at a given future date T. You may easily guess which product is cheaper. How much cheaper is a compatibility problem?

Problem 1 is discussed in my research on copula, surveyed in chapter 8 of my book Copula Methods in Finance. Problems 2 and 3 are my current research agenda and will be the subject of my presentation at the Conference: “The World Congress on Computational Finance: The First Decade”.    

Umberto