Graduate Seminar Financial and Actuarial Mathematics LMU and TUM

We propose a continuous-time stochastic model to analyze the dynamics of impermanent loss in liquidity pools in decentralized finance (DeFi) protocols. We replicate the impermanent loss using option portfolios for the individual tokens. We estimate the risk-neutral joint distribution of the tokens by minimizing the Hansen–Jagannathan bound, which we then use for the valuation of options on relative prices and for the calculation of implied correlations. In our analyses, we investigate implied volatilities and implied correlations as possible drivers of the impermanent loss and show that they explain the cross-sectional returns of liquidity pools. We test our hypothesis on options data from a major centralized derivative exchange.

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The goal of this work is to disentangle the roles of volume and participation rate in the price response of the market to a sequence of orders. To do so, we use an approach where price dynamics are derived from the order flow via no arbitrage constraints. We also introduce in the model sophisticated market participants having superior abilities to analyse market dynamics. Our results lead to two square root laws of market impact, with respect to executed volume and with respect to participation rate. This is joint work with Bruno Durin and Grégoire Szymanski.

Motivated by the tradeoff between exploitation and exploration in reinforcement learning, we study a continuous-time entropy-regularized mean variance portfolio selection problem in the presence of jumps. A first key step is to derive a suitable formulation of the continuous-time problem. In the existing literature for the diffusion case (e.g., Wang, Zariphopoulou and Zhou, Mach. Learn. Res. 2020), the conditional mean and the conditional covariance of the controlled dynamics are heuristically derived by a law of large numbers argument. In order to capture the influence of jumps, we first explicitly model distributional controls on discrete-time partitions and identify a family of discrete-time integrators which incorporate the additional exploration noise. Refining the time grid, we prove convergence in distribution of the discrete-time integrators to a multi-dimensional Levy process. This limit theorem gives rise to a natural continuous-time formulation of the exploratory control problem with entropy regularization. We solve this problem by adapting the classical Hamilton-Jacobi-Bellman approach. It turns out that the optimal feedback control distribution is Gaussian and that the optimal portfolio wealth process follows a linear stochastic differential equation, whose coefficients can be explicitly expressed in terms of the solution of a nonlinear partial integro-differential equation. We also provide a detailed comparison to the results derived by Wang and Zhou (Math. Finance, 2020) for the exploratory portfolio selection problem in the Black-Scholes model. The talk is based on joint work with Thuan Nguyen (Saarbrücken).

https://www.fm.math.lmu.de/en/teaching/winter-term-2023-24/seminars/lmu-christmas-workshop-in-stochastics-and-finance.html

We study two aspects of making optimal investment decisions for pension investors in the savings phase. First, we explore the impact of an investor’s perception towards inflation risk on their investment strategy. We find that mis-specifying inflation risk reduces the expected utility of the risk averse investors, and more risk averse investors face larger reductions. For investors who adopt terminal wealth constraints (e.g. minimum guarantee), ignoring inflation results in real wealth not adhering to the real constraints. The conclusion is that investors ignore inflation at their peril. Secondly, we compare the retirement outcomes derived from the risk averse and loss averse utility functions. We use a numerical dynamic programming approach and a model that includes ongoing pension contributions to savings, prohibits short-selling and borrowings, and, when applicable, includes wealth constraints. We find that the loss averse utility function, without wealth constraints, naturally results in a more favourable retirement income distribution that peaks at the investor's chosen income goal with some level of robustness. We conclude that the investor can benefit from adopting a loss aversion-derived optimal investment strategy to target a sufficient level of income at retirement.

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.