A Stochastic Model of Optimal Debt Management and Bankruptcy

We consider a problem of optimal debt management which is modeled as a non-cooperative game between a borrower and a pool of risk-neutral lenders. Since the debtor may go bankrupt, lenders charge a higher interest rate to offset the possible loss of part of their investment. In this talk, I will present results on existence and properties of optimal strategies, both in a deterministic and in a stochastic framework.

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Optimal Consumption in the Stochastic Ramsey Problem without Boundedness Constraints

This paper investigates optimal consumption in the stochastic Ramsey problem with the Cobb-Douglas production function. Contrary to prior studies, we allow for general consumption processes, without any a priori boundedness constraint. A non-standard stochastic differential equation, with neither Lipschitz continuity nor linear growth, specifies the dynamics of the controlled state process. A mixture of probabilistic arguments are used to construct the state process, and establish its non-explosiveness and strict positivity. This leads to the optimality of a feedback consumption process, defined in terms of the value function and the state process. Based on additional viscosity solutions techniques, we characterize the value function as the unique classical solution to a nonlinear elliptic equation, among an appropriate class of functions. This characterization involves a condition on the limiting behavior of the value function at the origin, which is the key to dealing with unbounded consumptions. Finally, relaxing the boundedness constraint is shown to increase, strictly, the expected utility at all wealth levels.

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Epstein-Zin Utility Maximization over Random Horizon

This talk focuses on solving the consumption-investment problem for an agent with stochastic differential utility of Epstein-Zin type. In contrast to prior literature, our time horizon is random, taken from the class of stopping times in the augmented Brownian filtration. Parameter specification is empirically relevant, with both relative risk-aversion and the elasticity of intertemporal substitution (EIS) in excess of one. The theory of BSDE (backward stochastic differential equations) is used to establish existence and uniqueness of the continuation value process, while martingale method is employed to derive optimal strategies. Relevant examples in decision theory will be discussed. 

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American option pricing under stochastic volatility models via Picard iterations

This talk discusses the valuation of American options for a general one- factor stochastic volatility model. Using the local time-space calculus on surfaces we derive an early exercise premium representation for the option price, parametrized by the optimal exercise surface. The exercise surface is the unique solution to an integral equation of Volterra type. The paper proposes a new numerical scheme to solve the integral equation based on the Picard iterations method. The method is flexible and can handle a wide class of non-affine models. Performance is illustrated for the Black-Scholes, Heston and 3/2 models. The approach provides fast convergence, simple implementation and good runtime/RMSE tradeoff and can be extended to other multi-dimensional stopping problems 

Joint work with J. Detemple and L. Zhang

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Optimal Portfolio under Fractional Stochastic Environments

Rough stochastic volatility models have attracted a lot of attention recently, in particular for the linear option pricing problem. In this paper, starting with power utilities, we propose to use a martingale distortion representation of the optimal value function for the nonlinear asset allocation problem in a (non-Markovian) fractional stochastic environment (for all Hurst index H \in (0,1)). We rigorously establish a first-order approximation of the optimal value, where the return and volatility of the underlying asset are functions of a stationary slowly varying fractional Ornstein-Uhlenbeck process. We prove that this approximation can be also generated by a fixed zeroth order trading strategy providing an explicit strategy which is asymptotically optimal in all admissible controls. Similar results are also obtained under fast mean-reverting fractional stochastic environment. Furthermore, we extend the discussion to general utility functions, and obtain the asymptotic optimality of this fixed strategy in a specific family of admissible strategies.

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Nonzero-sum stochastic differential games with impulse controls

We consider a general class of nonzero-sum impulsive games with N players. By means of a suitable system of quasi-variational  inequalities, we provide a verification theorem for the equilibrium  strategies and the value functions of the game. In particular, we focus on the regularity conditions required by the theorem. We then  present some practical applications. Finally, we focus on some ongoing  extensions and generalizations.

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The Value of Scattered Information

We analyze a model in which the value of a security is comprised of multiple distinct parts and private information about these pieces is scattered among investors. We show that as information is scattered into smaller, distinctively informative pieces, endogenous information acquisition activity can increase, even if the acquisition cost does not decrease. Our paper generalizes Grossman-Stiglitz (1980) for an arbitrary number of distinct pieces of information and demonstrates that when information is scattered among investors, information free-riding can be alleviated. Our model generates new insights and testable predictions about financial information markets, segmentation of firm-specific information, and informed trading. 

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Machine Learning in Quantitative Finance

Machine learning has revolutionized fields such as image, text, and speech recognition. There is now growing interest in applying machine learning in financial applications. Recently, we have developed machine learning methods for modeling high-frequency financial data, solving high-dimensional partial differential equations, and estimating continuous-time models. In particular, we analyze stochastic gradient descent in continuous time, which is an efficient algorithm for estimating continuous-time models from a continuous stream of data. We prove convergence, convergence rate, and central limit theorem results for this algorithm. The analysis relies upon stochastic analysis and partial differential equation techniques.

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Portfolios, optimal transport and informations geometry

Can we outperform a market index in the presence of volatility? What is the optimal frequency to rebalance a portfolio? We show that these questions can be analyzed using modern ideas in probability and information geometry (geometry in information theory). We quantify market volatility by a logarithmic divergence which is a distance-like quantity analogous to the relative entropy, and in this context portfolio selection (such as the universal portfolio) has a lot in common with nonparametric statistics. Mathematically, the divergence is intimately related to exponentially concave functions and the solution of an optimal transport problem with a logarithmic cost. It induces a rich differential geometric structure with numerous applications. In particular, a dualistic Pythagorean theorem gives insight into the optimal frequency of discrete rebalancing.

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A Dynamic Equilibrium Model for Brokerage Fees

We  develop a dynamic equilibrium model for market liquidity. To wit, we solve for the equilibrium prices at which liquidity takers' demands are absorbed by liquidity providers, who can in turn gradually transfer these positions to a group of end users. We also find the optimal strategy of a liquidity taker in such a market and compute the equilibrium price dynamics. This is joint work in progress with Peter Bank and Johannes Muhle-Karbe.  

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