Mathematical Finance Workshop

Christa Cuchiero: “Infinite-Dimensional Polynomial Processes”
Abstract: We consider a generic infinite dimensional framework for polynomial processes. We prove a Levy-Khintchine form of the involved operators and show how the moment formula looks like. We apply our results to forward variance models of polynomial type. The talk is based on joint work with Sara Svaluto-Ferro.

Paolo Guasoni: “Options Portfolio Selection”
Abstract: We develop a new method to optimize portfolios of options in a market where European calls and puts are available with many exercise prices for each of several potentially correlated underlying assets. We identify the combination of asset-specific option payoffs that maximizes the Sharpe ratio of the overall portfolio: such payoffs are the unique solution to a system of integral equations, which reduce to a linear matrix equation under suitable representations of the underlying probabilities. Even when implied volatilities are all higher than historical volatilities, it can be optimal to sell options on some assets while buying options on others, as hedging demand outweighs demand for asset-specific returns. Based on the joint work with Eberhard Mayerhofer.

Jan Kallsen: “Optimal Investment in High Dimensions under Small Proportional Transaction Costs”
Whereas nearly optimal portfolios can be determined quite easily for a single risky asset, much less can be said in situations of dozens of securities. In this talk we discuss the derivation and performance of several heuristics, and we compare them to an upper bound. The results are in particular applied to a DAX portfolio.

Martin Larsson: “Short- and Long-Term Relative Arbitrage in Stochastic Portfolio Theory”
Abstract: Stochastic Portfolio Theory is a mathematical framework for studying large equity markets, especially the performance of long-term investments. An important focus is universal features that only depend weakly on specific modeling assumptions. A basic result of this kind states that a mild nondegeneracy condition suffices to guarantee long-term relative arbitrage, that is, the possibility to outperform the market over sufficiently long time horizons. A longstanding open question has been whether short-term relative arbitrage is also implied. A qualitative answer, in the negative, was recently given by Fernholz, Karatzas & Ruf. In this work, we settle the question by characterizing and explicitly computing the critical time horizon beyond which relative arbitrage always exists. The key tool is a previously unknown connection between the existence of relative arbitrage and certain geometric PDE describing mean curvature flow.

Johannes Muhle-Karbe: “Equilibrium Asset Pricing with Transaction Costs”
Abstract: We study a risk-sharing equilibrium where heterogeneous agents trade subject to quadratic transaction costs. The corresponding equilibrium asset prices and trading strategies are characterized by a system of fully-coupled forward-backward stochastic differential equations. We show that a unique solution exists provided that the agents’ preferences are sufficiently similar. (Joint work in progress with Martin Herdegen and Dylan Possamai.)

Sergio Pulido: “Stochastic Volterra Equations”
Abstract: In this presentation I will give a brief summary of the theory of Stochastic Volterra Equations (SVEs). I will concentrate on the questions of existence and uniqueness of weak solutions. The motivation to study SVEs comes from the literature on rough volatility models. This is joint work with Eduardo Abi Jaber, Christa Cuchiero and Martin Larsson.

Scott Robertson: “Equilibria with Asymmetric Information”
Abstract: We study equilibria in multi-asset and multi-agent continuous-time economies with asymmetric information and bounded rational noise traders. We establish the existence of two equilibria. First, a full communication one where the informed agents’ signal is disclosed to the market, and static policies are optimal. Second, a partial communication one where the signal disclosed is affine in the informed and noise traders’ signals. Here, information asymmetry creates dynamic demands for portfolios which replicate linear and quadratic payoffs in the fundamental process. Results are valid for multiple dimensions; constant absolute risk averse investors; fundamental processes following a general diffusion; and non-linear terminal payoffs. Asset price dynamics and public information flows are endogenous and are established using multiple filtration enlargements, in conjunction with predictable representation theorems for random analytic maps. Rational expectations equilibria are special cases of the general results. Joint work with Jerome Detemple and Marcel Rindisbacher of Boston University.

Mihai Sîrbu: “Sensitivity Analysis of the Utility Maximization Problem With Respect to Model Perturbations”
Abstract: We consider the expected utility maximization problem and its response to small changes in the market price of risk in a continuous semimartingale setting. Assuming that the preferences of a rational economic agent are modeled by a general utility function, we obtain a second-order expansion of the value function, a first-order approximation of the terminal wealth, and construct trading strategies that match the indirect utility function up to the second order. The method, which is presented in the abstract version, relies on the simultaneous expansion with respect to both the state variable and the parameter, and convex duality in the direction of the state variable only (as there is no convexity with respect to the parameter). If a risk-tolerance wealth process exists, using it as a numeraire and under an appropriate change of measure, we reduce the approximation problem to a Kunita-Watanabe decomposition. Based on joint work with Oleksii Mostovyi.

Hao Xing: “Infinite Horizon Epstein-Zin Utility”
Abstract: Many macroeconomic and asset pricing models use Epstein-Zin utilities on an infinite horizon. However, the existence and uniqueness of such utilities were not well established. In particular, the transversality condition proposed in Duffie and Epstein (1992) identifies a trivial solution for the Esptein-Zin utility when its parameters are empirically relevant. Motivated by the power utility bounds in Seiferling and Seifried (2015), we establish a unique infinite horizon Epstein-Zin utility sandwiched between two power utilities. This is a joint work with Viet Dang.