Oberseminar Finanz- und Versicherungsmathematik

We consider a class of optimal liquidation problems where the agent's transactions create both temporary and transient price impact driven by a Volterra-type propagator. We formulate these problems as minimization of a revenue-risk functionals, where the agent also exploits available information on a progressively measurable price predicting signal. By using an infinite dimensional stochastic control approach, we characterize the value function in terms of a solution to a free-boundary L^2-valued backward stochastic differential equation and an operator-valued Riccati equation. We then derive explicitly the optimal trading strategy by solving these equations. Our results also cover the case of singular price impact kernels, such as the power-law kernel.

After a brief introduction to the Model Risk Assessment literature, this talk will present two recent results in this field.
The first part of this talk focuses on risk aggregation problems under partial dependence uncertainty. The main point of our analysis is to show that the knowledge of a dependence measure such as Pearson correlation, Spearman's rho or the average correlation, has typically no effect on the worst-case scenario of the aggregated (Range)Value-at-Risk, with respect to the case of full dependence uncertainty.
The second part of the talk deals with the robust assessment of a life insurance contract when there is ambiguity regarding the residual lifetime distribution function of the policyholder. Specifically, we show that if the ambiguity set is described using an L^2 distance constraint from a benchmark distribution function, then the net premium bounds can be reformulated as a convex linear program that enjoys many desirable properties.

We investigate whether risk-taking for resurrection type of risk preference (non-constant risk aversion) can increase the probability of achieving inflation-indexed pension benefits at retirement, especially when the starting position is underfunded. By maximizing the expected utility of the ratio of final wealth to a close approximation of this inflation-indexed target fund, we find that this non-constant risk aversion type of utility gives a high degree of certainty about achieving a certain percentage of this desired target fund. The CRRA utility is too risk-averse to overcome under-funding.

We develop a dual-control method for approximating investment strategies in multidimensional financial markets with convex trading constraints. The method relies on a projection of the optimal solution to an (unconstrained) auxiliary problem to obtain a feasible and near-optimal solution to the original problem. We obtain lower and upper bounds on the optimal value function using convex duality methods. The gap between the bounds indicates the precision of the near-optimal solution. We illustrate the effectiveness of our method in a market with different trading constraints such as borrowing, short-sale constraints and non-traded assets. We also show that our method works well for state-dependent utility functions.

We formalize a consumption-investment-insurance problem with the distinction of a state-dependent relative risk aversion. The state-dependence refers to the state of the finite state Markov chain that also formalizes insurable risks such as health and lifetime uncertainty. We derive and analyze the implicit solution to the problem, compare it with special cases in the literature, and illustrate the range of results in a disability model where the relative risk aversion is preserved, decreases, or increases upon disability. We also discuss whether the approach is appropriate to deal with uncertainty in relative risk aversion and consider some alternative ideas.

In this paper we study an optimisation problem for a couple including two breadwinners with uncertain lifetimes. Both breadwinners need to choose the optimal strategies for consumption, investment, housing and life insurance purchasing during to maximise the utility. In this paper, the prices of housing assets and investment risky assets are assumed to be correlated. These two breadwinners are considered to have dependent mortality rates to include the breaking heat effect. The method of copula functions is used to construct the joint survival functions of two breadwinners. The analytical solutions of optimal strategies can be achieved, and numerical results are demonstrated.

We study the range of prices at which a rational agent should contemplate transacting a financial contract outside a given securities market. Trading is subject to nonproportional transaction costs and portfolio constraints and full replication by way of market instruments is not always possible. Rationality is defined in terms of consistency with market prices and acceptable risk thresholds. We obtain a direct and a dual description of market-consistent prices with acceptable risk. The dual characterization requires an appropriate extension of the classical Fundamental Theorem of Asset Pricing where the role of arbitrage opportunities is played by good deals, i.e., costless investment opportunities with acceptable risk-reward tradeoff. In particular, we highlight the importance of scalable good deals, i.e., investment opportunities that are good deals regardless of their volume. The talk is based on joint work with Maria Arduca (LUISS Rome).