Graduate Seminar Financial and Actuarial Mathematics LMU and TUM

We propose a simple lifecycle strategy entailing contributions made during accumulation being invested entirely into a risky portfolio until pre-specified ‘switch age’ and then entirely into a risk-free portfolio after the switch age, followed by withdrawing during decumulation from both portfolios based on annuitization rates that vary with age according to remaining life expectancy. First, we show analytically that the strategy is optimal for range of investors with HARA risk preferences, and derive the dynamics of the investment strategy. Second, we show numerically that the proposed strategy delivers limited loss of utility versus an optimal solution for investors with CRRA preferences and low risk aversion, while significantly outperforming strategies commonly used in practice. The proposed strategy offers an attractive alternative for use in practical settings as it is simple to follow and removes the need for portfolio rebalancing.

We develop a simple and flexible technique to price executive stock options (ESOs) with vesting periods and liquidation penalties. The vesting period implies that the ESO is activated when a designed performance measure triggers a prespecified barrier. The performance measure is usually an accounting figure, such as the ROE or the EBITDA, normally correlated with the stock price. Once the option is activated, the holder has the right to buy the stock whenever she wants during the residual life of the option. The bivariate strutucture of the ESO, whose payoff depends jointly on the performance measure and the stock, makes usual lattice techniques difficult to apply. We first reduce the ESO to a compound forward-starting American call option on the stock. We then show how to evaluate the ESO option by means of an intuitive hybrid method that uses simulation to determine the bivariate distribution of the foward-starting date of the option and the corresponding price of the stock, and lattice techniques to retrieve the initial value of the activated call option. Liquidation penalties are common in ESOs, aiming at lowering the chances of selling the ESOs and the underlying company shares. We show that the presence of even mild liquidation penalties triggers the existence of optimal exercise opportunities for the ESOs that are absent when the option can be fully liquidated. Joint with M. De Donno and Alessandro Sbuelz

In environmental science applications, extreme events frequently exhibit a complex spatio-temporal structure, which is difficult to describe flexibly and estimate in a computationally efficient way using state-of-art parametric extreme-value models. In this talk, we propose a computationally-cheap non-parametric approach to investigate the probability distribution of temporal clusters of spatial extremes, and study within-cluster patterns with respect to various characteristics. These include risk functionals describing the overall event magnitude, spatial risk measures such as the size of the affected area, and measures representing the location of the extreme event. Under the framework of functional regular variation, we verify the existence of the corresponding limit distributions as the considered events become increasingly extreme. Furthermore, we develop non-parametric estimators for the limiting expressions of interest and show their asymptotic normality under appropriate mixing conditions. Uncertainty is assessed using a multiplier block bootstrap. The finite-sample behavior of our estimators and the bootstrap scheme is demonstrated in a spatio-temporal simulated example. Our methodology is then applied to study the spatio-temporal dependence structure of high-dimensional sea surface temperature data for the southern Red Sea. Our analysis reveals new insights into the temporal persistence, and the complex hydrodynamic patterns of extreme sea temperature events in this region. This is joint work with Raphael Huser.

We study the use of neural networks as nonparametric estimation tools for the hedging of options. To this end, we design a network, named HedgeNet, that directly outputs a hedging strategy given relevant features as input. This network is trained to minimise the hedging error instead of the pricing error. Applied to end-of-day and tick prices of S&P 500 and Euro Stoxx 50 options, the network is able to reduce the mean squared hedging error of the Black-Scholes benchmark significantly. We illustrate, however, that a similar benefit arises by a simple linear regression model that incorporates the leverage effect. (Joint work with Weiguan Wang)

Liquidity risk is typically added exogenously to a market price process. This is conceptually unsatisfying. We build a model, which integrates liquidity risk into the market price process. In particular, we add a liquidity (jump) component to the standard geometric Brownian motion and show that this approach models market prices better than without the liquidity component. Since long positions have to be liquidated at the bid price, we model bid and ask price individually. We verify our model with 50 million bond price data. We suggest that this model should underlie long positions in risk management approaches such as VaR (Value at Risk), ES (Expected Shortfall) and EVT (Extreme Value Theory). The talk is based on a joint work with Robert Engle and Anna van Elst.

The Dirichlet process represents the cornerstone of Bayesian Nonparametrics and is used as main ingredient in a wide variety of models. The many generalizations of the Dirichlet proposed in the literature aim at overcoming some of its limitations and at increasing the models' flexibility. In this talk we provide an overview of a large set of such generalizations by using completely random measures as a unifying concept. All the considered models can be seen as suitable transformations of completely random measures and this allows to highlight interesting distributional structures they share a posteriori in several statistical problems ranging from density estimation and clustering to survival analysis and species sampling. Furthermore, we discuss some natural approaches, which rely on additive, hierarchical and nested structures, to derive dependent versions of Bayesian nonparametric models derived from completely random measures.

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). *For participation on Zoom, please contact Felix Liebrich (liebrich@math.lmu.de)

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.

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.

https://www.fm.mathematik.uni-muenchen.de/teaching/teaching_winter_term_2022_2023/seminars/christmas_workshop/index.html

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 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.