Mini course by Gilles Pagès (LPSM): “Functional convex order and applications to Finance”
Convex order between two integrable vectors U and V having values in R^d is defined by IE f(U) <= IE f(V) for every convex function f: R^d –> R (with some variants like monotonic convex order in dimension d=1 where f is also supposed monotonic). After recalling the deep connections of such an ordering with martingality through Kellerer and Strassen theorems and “p.c.o.c.” (processus croissants pour l’ordre convexe aka “peacocks” following the terminology introduced by Yor et al.), we will expose some first applications to Finance (sensitivity to volatility of options written on convex payoffs, risk measure, etc). Then we will come to the core of the course which is to introduce functional convex order i.e. to extend, when dealing with stochastic processes (X_t), the above definition to (convex) functionals F((X_t)_t) of the whole trajectories. The comparison is also based on functional “hyper-parameters” (typically the diffusion coefficient \sigma in the case of martingale or scaled Brownian diffusions). We will investigate various classes of stochastic processes, first belonging to Brownian diffusions and then to other classes of processes like diffusions with jumps, diffusion of McKean Vlasov type but also non-Markovian processes, such as the solutions of Volterra equations with (possibly) singular kernels like those appearing in rough volatility modeling in Finance. On our way we will revisit, unify and extend or discuss former results from the literature like the old Hajek’s theorems for monotone marginal convex ordering of one dimensional diffusions, or more recent contributions by Rüschendorf and coauthors, Hobson, Schied and Stadje among others. A typical result for scalar martingale Brownian diffusions with respective non-negative diffusions coefficients sigma_1 and sigma_2 if \sigma or \theta_2 is convex & \sigma <= \theta then IE F((X^{\sigma}_t)_t) <= IE F((X^{theta}_t)_t) for every lower semi-continuous convex functional F. Such results can make possible to compare/bound European option prices in different models in Finance in the spirit of the seminal paper by El Karoui-Jeanblanc-Shreve. Similar results hold for non linear problems like American options by adapting our approach to optimal stopping theory, stochastic control (and swing option pricing on energy) or to mean-field games when dealing with McKean-Vlasov equations. In the first two cases we rely on a Backward Dynamic Programming Principle. We will highlight how convex ordering is closely related to a problem of (at least) equivalent importance for applications: the propagation of convexity which can be summed up as the fact that, if a functional F is convex, x–> E F((X_t)_t) is also convex (under appropriate assumtptons). We will systematically establish both our comparison and propagation results starting from a discrete time approximation procedure of Euler scheme type, generally simulable and concluding by appropriate strong or weak functional limiting theorems à la Jacod-Shiryaev. Among other virtues, this approach makes it possible in Finance to ensure that the prices of derivative products computed by simulation cannot give rise to convexity arbitrages since our approximations share the same convexity properties. If time is not too short we will make a focus on one dimensional Brownian diffusions where the convexity assumption on the (one of the) diffusion coefficient can be relaxed and the class of admissible functionals extended to directionally convex functionals. A bibliography will be joined to the slides.