Joint Seminar day Leeds-Liverpool

Part of the Probability, Stochastic Modelling and Financial Mathematics seminar. The event will take place at the University of Liverpool, Department of Mathematical Sciences.

This is a 1 day event jointly organized by Julia Eisenberg (University of Liverpool) and Katia Colaneri (University of Leeds) and held at University of Liverpool (Department of Mathematical Sciences - Institute for Financial and Actuarial  Mathematics). The speakers for this event are:  

• Jan Palczewski (University of Leeds)

• Tiziano De Angelis (University of Leeds)

• Katia Colaneri (University of Leeds)

• Carmen Boado Penas (University of Liverpool)

• Apostolos Papaioannou (University of Liverpool)

• Paul Eisenberg (University of Liverpool)

Programme

Abstracts

Carmen Boado Penas

Title: Automatic Balancing Mechanisms for Mixed Pension Systems under Different Investment Strategies

Abstract: State pension systems are usually pay-as-you-go financed, i.e. current contributions cover pension expenditure. However, some countries combine funding and pay-as-you-go (PAYG) elements within the first pillar. Our aim is twofold. First, using nonlinear optimisation, it seeks to assess the impact of a compulsory funded defined contribution pension scheme that complements the traditional PAYG on the level of pension benefits. Future expected returns for both the funded part and the buffer fund of the PAYG are simulated through the non-overlapping block bootstrap technique. Second, in the case of a partial financial sustainability, we design different optimal strategies, that involve variables such as the contribution rate, age of retirement and indexation of pensions, to restore the long-term financial equilibrium of the system.

Katia Colaneri

Title: Optimal Converge Trading with Unobservable Pricing Errors

Abstract: We study a dynamic portfolio optimization problem related to convergence trading, which is an investment strategy that exploits temporary mispricing by simultaneously buying relatively under-priced assets and selling short relatively overpriced ones with the expectation that their prices converge in the future. We build on the model of Liu and Timmerman (2013) and extend it by incorporating unobservable Markov-modulated pricing errors into the price dynamics of two co-integrated assets. We characterize the optimal portfolio strategies in full and partial information settings both under the assumption of unrestricted and beta-neutral strategies. By using the innovations approach, we provide the filtering equation that is essential for solving the optimization problem under partial information. Finally, in order to illustrate the model capabilities, we provide an example with a two-state Markov chain.

Tiziano De Angelis

Title: Optimal dividends with partial information and stopping of a degenerate reflecting diffusion

Abstract: Motivated by the optimal dividend problem, we study a problem of singular stochastic control with partial information on the drift of the underlying process and with absorption. In the Markovian formulation, we have a 2-dimensional degenerate diffusion, whose first component is singularly controlled and it is absorbed as it hits zero. The free boundary problem (FBP) associated to the value function of the control problem is challenging from the analytical point of view due to the interplay of degeneracy and absorption. We find a probabilistic way to show that the value function of the dividend problem is a smooth solution of the FBP and to construct an optimal dividend strategy. Our approach establishes a new link between multidimensional singular stochastic control problems with absorption and problems of optimal stopping with `creation'. One key feature of the stopping problem is that creation occurs at a state-dependent rate of the `local-time' of an auxiliary 2-dimensional reflecting diffusion.

Paul Eisenberg

Title: Occupation Estimates

Abstract: Consider a stochastic optimal control problem where the optimal control can't be found in closed form. Say, we want to evaluate the goodness of an arbitrary strategy compared to the unknown optimal control. This would require the comparison of the process with the chosen control to the unknown optimally controlled process, or in essence, we deal with at least one process where we have insufficient information on the coefficients.
In this talk, we take a generic diffusion X where we have partial knowledge of its coefficients and try to find estimates for its density at fixed times and estimates on its occupation density or expected occupation density.
These type of estimates allow us in some control problems to obtain goodness estimates for given choices of controls compared to the unknown optimal control.

Jan Palczewski

Title: Value of stopping games with asymmetric information                                                                                                                                   

Abstract: We study the value of a zero-sum stopping game in which the terminal payoff function depends on the underlying process and on an additional randomness which is known to one player but unknown to the other. Such asymmetry of information arises naturally in insider trading when one of the counterparties knows an announcement before it is publicly released, e.g., central bank’s interest rates decision or company earnings/business plans. In the context of game options this splits the pricing problem into the phase before announcement (asymmetric information) and after announcement (full information); the value of the latter exists and forms the terminal
payoff of the asymmetric phase. The above game does not have a value if both players use pure stopping times as the informed player’s actions would reveal too much of his excess knowledge. The informed player manages the trade-off between releasing information and stopping optimally employing randomised stopping times. We reformulate the stopping game as a zero-sum game between a stopper (the uninformed player) and a singular controller (the informed player). We prove existence of the value of the latter game for a larger class of underlying strong Markov processes including multi-variate diffusions and Feller processes. The main tools are approximations by smooth singular controls and by discrete-time games.

Apostolos Papaioannou/ Lewis Ramsden

Title: On Risk Models with Dependent Delayed Capital Injections

Abstract: We propose a generalisation to the Cramer-Lundberg risk model, by allowing for a delayed receipt of the required capital injections whenever the surplus of insurance firm is negative. Delayed capital injections often appear in practice due to the time taken for administrative and processing purposes of the funds from a third party or the shareholders of an insurance firm.
The delay time of the capital injection depends on a critical value of the deficit in the following way: If the deficit of the firm is less than the fixed critical value, then it can be covered by available funds and therefore the required capital injection is received instantaneously.  On the other hand, if the deficit of the firm exceeds the fixed critical value, then the funds are provided by an alternative source and the required capital injection is received after some time delay. In this modified model, we derive a Fredholm integral equation of the second kind for the ultimate ruin probability and obtain an explicit expression in terms of ruin quantities for the Cramer-Lundberg risk model. In addition, we show that other risk quantities, namely the expected discounted accumulated capital injections and the expected discounted overall time in red, up to the time of ruin, satisfy a similar integral equation, which can also be solved explicitly. Finally, we extend the capital injection delayed risk model, such that the delay of the capital injections depends explicitly on the amount of the deficit. In this generalised risk model, we derive another Fredholm integral equation for the ultimate ruin probability, which is solved in terms of a Neumann series.