Processus de Lévy en Finance - Laboratoire de Probabilités et ...

166 CHAPTER 5. APPLICATIONS OF LEVY COPULAS
Lévy copula of a Lévy process is not linked to any given time scale; it describes the dependence
structure of the entire process.
One example of a financial modelling problem where Lévy copulas naturally appear is the
pricing of multi-asset options on underlyings, described by exponential Lévy models. In this
setting the parameters of the marginal Lévy processes can be calibrated from the prices of European
options, quoted at the market, and the dependence structure will typically be estimated
from the historical time series of returns. Since the sampling rate of returns is different from
the maturity of traded options as well as from the maturity of the basket option that one wants
to price, ordinary copulas are not well suited for this problem. Pricing of basket options in a
Lévy copula model is discussed in more detail in Section 5.3.
Lévy copulas can also be useful outside the realm of financial modelling. Other contexts
where modelling dependence in jumps is required are portfolios of insurance claims and models
of operational risk.
Consider an insurance company with two subsidiaries, in France and in Germany. The
aggregate loss process of the French subsidiary is modelled by the subordinator {X t } t≥0 and
the loss process of the German one is {Y t } t≥0 . The nature of processes X and Y may be
different because the subsidiaries may not be working in the same sector and many risks that
cause losses are local. However, common risks like floods and pan-European windstorms will
lead to a certain degree of dependence between the claims. In this setting it is convenient
to model the dependence between X and Y using a Lévy copula on [0, ∞] 2 . In this modelling
approach, the two-dimensional Lévy measure of (X, Y ) is known and the overall loss distribution
and ruin probability can be computed.
Another example where jump processes naturally appear is given by models of operational
risk. The 2001 Basel agreement defines the operational risk as “the risk of direct and indirect
loss resulting from inadequate or failed internal processes, people and systems or from external
events” and allows banks to use internal loss data to compute regulatory capital requirements.
Taking into account the dependence between different business lines in this computation, due to
a diversification effect, may lead to substantial reduction of regulatory capital [14]. Aggregate
loss processes from different business lines can be dynamically modelled by subordinators and
the dependence between them can be accounted for using a Lévy copula on [0, ∞] 2 .