Operational risk management and new computational needs in banks

Basel II banking regulation introduces new needs for computational schemes. They involve both optimal stochastic control, and large scale simulations of decision processes of preventing low-frequency high loss-impact events. This paper will first state the problem and present its parameters. It then spells out the equations that represent a rational risk management behavior and link together the variables: Levy processes are used to model operational risk losses, where calibration by historical loss databases is possible ; where it is not the case, qualitative variables such as quality of business environment and internal controls can provide both costs-side and profits-side impacts. Among other control variables are business growth rate, and efficiency of risk mitigation. The economic value of a policy is maximized by resolving the resulting Hamilton-Jacobi-Bellman type equation. Computational complexity arises from embedded interactions between 3 levels: * Programming global optimal dynamic expenditures budget in Basel II context, * Arbitraging between the cost of risk-reduction policies (as measured by organizational qualitative scorecards and insurance buying) and the impact of incurred losses themselves. This implies modeling the efficiency of the process through which forward-looking measures of threats minimization, can actually reduce stochastic losses, * And optimal allocation according to profitability across subsidiaries and business lines. The paper next reviews the different types of approaches that can be envisaged in deriving a sound budgetary policy solution for operational risk management, based on this HJB equation. It is argued that while this complex, high dimensional problem can be resolved by taking some usual simplifications (Galerkin approach, imposing Merton form solutions, viscosity approach, ad hoc utility functions that provide closed form solutions, etc.) , the main interest of this model lies in exploring the scenarios in an adaptive learning framework ( MDP, partially observed MDP, Q-learning, neuro-dynamic programming, greedy algorithm, etc.). This makes more sense from a management point of view, and solutions are more easily communicated to, and accepted by, the operational level staff in banks through the explicit scenarios that can be derived. This kind of approach combines different computational techniques such as POMDP, stochastic control theory and learning algorithms under uncertainty and incomplete information. The paper concludes by presenting the benefits of such a consistent computational approach to managing budgets, as opposed to a policy of operational risk management made up from disconnected expenditures. Such consistency satisfies the qualifying criteria for banks to apply for the AMA (Advanced Measurement Approach) that will allow large economies of regulatory capital charge under Basel II Accord.REGULAR - Operational risk management, HJB equation, Levy processes, budget optimization, capital allocation

On some expectation and derivative operators related to integral representations of random variables with respect to a PII process

Given a process with independent increments $X$ (not necessarily a martingale) and a large class of square integrable r.v. $H=f(X_T)$, $f$ being the Fourier transform of a finite measure $\mu$, we provide explicit Kunita-Watanabe and F\"ollmer-Schweizer decompositions. The representation is expressed by means of two significant maps: the expectation and derivative operators related to the characteristics of $X$. We also provide an explicit expression for the variance optimal error when hedging the claim $H$ with underlying process $X$. Those questions are motivated by finding the solution of the celebrated problem of global and local quadratic risk minimization in mathematical finance.Comment: 29 page

Variance optimal hedging for continuous time additive processes and applications

For a large class of vanilla contingent claims, we establish an explicit F\"ollmer-Schweizer decomposition when the underlying is an exponential of an additive process. This allows to provide an efficient algorithm for solving the mean variance hedging problem. Applications to models derived from the electricity market are performed

The obstacle problem for semilinear parabolic partial integro-differential equations

This paper presents a probabilistic interpretation for the weak Sobolev solution of the obstacle problem for semilinear parabolic partial integro-differential equations (PIDEs). The results of Leandre (1985) concerning the homeomorphic property for the solution of SDEs with jumps are used to construct random test functions for the variational equation for such PIDEs. This results in the natural connection with the associated Reflected Backward Stochastic Differential Equations with jumps (RBSDEs), namely Feynman Kac's formula for the solution of the PIDEs. Moreover it gives an application to the pricing and hedging of contingent claims with constraints in the wealth or portfolio processes in financial markets including jumps.Comment: 31 page