arXiv subjects

Harry Zheng

Publications and source records attributed to Harry Zheng.

At least 37 records · Page 2Linked to original sources

arXiv2021

Duality for optimal consumption with randomly terminating income

We establish a rigorous duality theory, under No Unbounded Profit with Bounded Risk, for an infinite horizon problem of optimal consumption in the presence of an income stream that can terminate randomly at an exponentially distributed time, independent of the asset prices. We thus close a duality gap encountered by Vellekoop and Davis in a version of this problem in a Black-Scholes market. Many of the classical tenets of duality theory hold, with the notable exception that marginal utility at zero initial wealth is finite. We use as dual variables a class of supermartingale deflators such that deflated wealth plus cumulative deflated consumption in excess of income is a supermartingale. We show that the space of discounted local martingale deflators is dense in our dual domain, so that the dual problem can also be expressed as an infimum over the discounted local martingale deflators. We characterise the optimal wealth process, showing that optimal deflated wealth is a potential decaying to zero, while deflated wealth plus cumulative deflated consumption over income is a uniformly integrable martingale at the optimum. We apply the analysis to the Vellekoop and Davis example and give a numerical solution.

q-fin.MF↗

arXiv2020

Optimal Liquidation in a Mean-reverting Portfolio

In this work we study a finite horizon optimal liquidation problem with multiplicative price impact in algorithmic trading, using market orders. We analyze the case when an agent is trading on a market with two financial assets, whose difference of log-prices is modelled with a mean-reverting process. The agent's task is to liquidate an initial position of shares of one of the two financial assets, without having the possibility of trading the other stock. The criterion to be optimized consists in maximising the expected final value of the agent, with a running inventory penalty. The main result of this paper consists in finding a classical solution of the Hamilton-Jacobi-Bellman (HJB) equation associated to this problem, which is proved to not coincide with the value function. However, we find the value function as a solution to the forward-backward stochastic differential equation (FBSDE) associated to the problem. We provide numerical tests showing that the HJB and FBSDE solutions are close to each other and analysing performance of the described model. We also prove a verification theorem and a comparison principle for the viscosity solution to the HJB equation.

math.OC↗

arXiv2018

Global Closed-form Approximation of Free Boundary for Optimal Investment Stopping Problems

In this paper we study a utility maximization problem with both optimal control and optimal stopping in a finite time horizon. The value function can be characterized by a variational equation that involves a free boundary problem of a fully nonlinear partial differential equation. Using the dual control method, we derive the asymptotic properties of the dual value function and the associated dual free boundary for a class of utility functions, including power and non-HARA utilities. We construct a global closed-form approximation to the dual free boundary, which greatly reduces the computational cost. Using the duality relation, we find the approximate formulas for the optimal value function, trading strategy, and exercise boundary for the optimal investment stopping problem. Numerical examples show the approximation is robust, accurate and fast.

q-fin.MF↗

arXiv2018

Dynamic Portfolio Optimization with Looping Contagion Risk

In this paper we consider a utility maximization problem with defaultable stocks and looping contagion risk. We assume that the default intensity of one company depends on the stock prices of itself and other companies, and the default of the company induces immediate drops in the stock prices of the surviving companies. We prove that the value function is the unique viscosity solution of the HJB equation. We also perform some numerical tests to compare and analyse the statistical distributions of the terminal wealth of log utility and power utility based on two strategies, one using the full information of intensity process and the other a proxy constant intensity process.

q-fin.MF↗

arXiv2018

Turnpike Property and Convergence Rate for an Investment and Consumption Model

We discuss the turnpike property for optimal investment and consumption problems. We find there exists a threshold value that determines the turnpike property for investment policy. The threshold value only depends on the Sharpe ratio, the riskless interest rate and the discount rate. We show that if utilities behave asymptotically like power utilities and satisfy some simple relations with the threshold value, then the turnpike property for investment holds. There is in general no turnpike property for consumption policy. We also provide the rate of convergence and illustrate the main results with examples of power and non-HARA utilities and numerical tests.

q-fin.PM↗

arXiv2017

Dual control Monte Carlo method for tight bounds of value function under Heston stochastic volatility model

The aim of this paper is to study the fast computation of the lower and upper bounds on the value function for utility maximization under the Heston stochastic volatility model with general utility functions. It is well known there is a closed form solution of the HJB equation for power utility due to its homothetic property. It is not possible to get closed form solution for general utilities and there is little literature on the numerical scheme to solve the HJB equation for the Heston model. In this paper we propose an efficient dual control Monte Carlo method for computing tight lower and upper bounds of the value function. We identify a particular form of the dual control which leads to the closed form upper bound for a class of utility functions, including power, non-HARA and Yarri utilities. Finally, we perform some numerical tests to see the efficiency, accuracy, and robustness of the method. The numerical results support strongly our proposed scheme.

q-fin.CP↗

arXiv2017

Constrained Quadratic Risk Minimization via Forward and Backward Stochastic Differential Equations

In this paper we study a continuous-time stochastic linear quadratic control problem arising from mathematical finance. We model the asset dynamics with random market coefficients and portfolio strategies with convex constraints. Following the convex duality approach, we show that the necessary and sufficient optimality conditions for both the primal and dual problems can be written in terms of processes satisfying a system of FBSDEs together with other conditions. We characterise explicitly the optimal wealth and portfolio processes as functions of adjoint processes from the dual FBSDEs in a dynamic fashion and vice versa. We apply the results to solve quadratic risk minimization problems with cone-constraints and derive the explicit representations of solutions to the extended stochastic Riccati equations for such problems.

q-fin.PM↗

arXiv2016

Dynamic Convex Duality in Constrained Utility Maximization

In this paper, we study a constrained utility maximization problem following the convex duality approach. After formulating the primal and dual problems, we construct the necessary and sufficient conditions for both the primal and dual problems in terms of FBSDEs plus additional conditions. Such formulation then allows us to explicitly characterize the primal optimal control as a function of the adjoint process coming from the dual FBSDEs in a dynamic fashion and vice versa. Moreover, we also find that the optimal primal wealth process coincides with the adjoint process of the dual problem and vice versa. Finally we solve three constrained utility maximization problems, which contrasts the simplicity of the duality approach we propose and the technical complexity of solving the primal problems directly.

q-fin.MF↗

arXiv2016

Weak Convergence of Path-Dependent SDEs in Basket CDS Pricing with Contagion Risk

We investigate the computational aspects of the basket CDS pricing with counterparty risk under a credit contagion model of multinames. This model enables us to capture the systematic volatility increases in the market triggered by a particular bankruptcy. The drawback of this problem is its analytical complication due to its path-dependent functional, which bears a potential failure in its convergence of numerical approximation under standing assumptions. In this paper we find sufficient conditions for the desired convergence of the functionals associated with a class of path-dependent stochastic differential equations. The main ingredient is to identify the weak convergence of the approximated solution to the underlying path-dependent stochastic differential equation.

q-fin.MF↗

arXiv2016

On Correlated Defaults and Incomplete Information

In this paper, we study a continuous time structural asset value model for two correlated firms using a two-dimensional Brownian motion. We consider the situation of incomplete information, where the information set available to the market participants includes the default time of each firm and the periodic asset value reports. In this situation, the default time of each firm becomes a totally inaccessible stopping time to the market participants. The original structural model is first transformed to a reduced-form model. Then the conditional distribution of the default time together with the asset value of each name are derived. We prove the existence of the intensity processes of default times and also give the explicit form of the intensity processes. Numerical studies on the intensities of the two correlated names are conducted for some special cases. We also indicate the possible future research extension into three names case by considering a special correlation structure.

q-fin.MF↗

arXiv2014

Optimal Liquidation in a Finite Time Regime Switching Model with Permanent and Temporary Pricing Impact

In this paper we discuss the optimal liquidation over a finite time horizon until the exit time. The drift and diffusion terms of the asset price are general functions depending on all variables including control and market regime. There is also a local nonlinear transaction cost associated to the liquidation. The model deals with both the permanent impact and the temporary impact in a regime switching framework. The problem can be solved with the dynamic programming principle. The optimal value function is the unique continuous viscosity solution to the HJB equation and can be computed with the finite difference method.

q-fin.PM↗

arXiv2014

Turnpike Property and Convergence Rate for an Investment Model with General Utility Functions

In this paper we aim to address two questions faced by a long-term investor with a power-type utility at high levels of wealth: one is whether the turnpike property still holds for a general utility that is not necessarily differentiable or strictly concave, the other is whether the error and the convergence rate of the turnpike property can be estimated. We give positive answers to both questions. To achieve these results, we first show that there is a classical solution to the HJB equation and give a representation of the solution in terms of the dual function of the solution to the dual HJB equation. We demonstrate the usefulness of that representation with some nontrivial examples that would be difficult to solve with the trial and error method. We then combine the dual method and the partial differential equation method to give a direct proof to the turnpike property and to estimate the error and the convergence rate of the optimal policy when the utility function is continuously differentiable and strictly concave. We finally relax the conditions of the utility function and provide some sufficient conditions that guarantee the turnpike property and the convergence rate in terms of both primal and dual utility functions.

econ.GN↗

arXiv2014

A robust algorithm and convergence analysis for static replications of nonlinear payoffs

In this paper we propose a new robust algorithm to find the optimal static replicating portfolios for general nonlinear payoff functions and give the estimate of the rate of convergence that is absent in the literature. We choose the static replication by minimizing the error bound between the nonlinear payoff function and the linear spline approximation and derive the equidistribution equation for selecting the optimal strike prices. The numerical tests for variance swaps and swaptions and also for the static quadratic replication and the model with counterparty risk show that the proposed algorithm is simple, fast and accurate. The paper has generalized and improved the results of the static replication and approximation in the literature.

q-fin.CP↗

arXiv2014

Intensity Process for a Pure Jump Lévy Structural Model with Incomplete Information

In this paper we discuss a credit risk model with a pure jump Lévy process for the asset value and an unobservable random barrier. The default time is the first time when the asset value falls below the barrier. Using the indistinguishability of the intensity process and the likelihood process, we prove the existence of the intensity process of the default time and find its explicit representation in terms of the distance between the asset value and its running minimal value. We apply the result to find the instantaneous credit spread process and illustrate it with a numerical example.

q-fin.MF↗

arXiv2013

Lower Bound Approximation to Basket Option Values for Local Volatility Jump-Diffusion Models

In this paper we derive an easily computed approximation to European basket call prices for a local volatility jump-diffusion model. We apply the asymptotic expansion method to find the approximate value of the lower bound of European basket call prices. If the local volatility function is time independent then there is a closed-form expression for the approximation. Numerical tests show that the suggested approximation is fast and accurate in comparison with the Monte Carlo and other approximation methods in the literature.

q-fin.PR↗

arXiv2013

Weak Necessary and Sufficient Stochastic Maximum Principle for Markovian Regime-Switching Diffusion Models

In this paper we prove a weak necessary and sufficient maximum principle for Markovian regime switching stochastic optimal control problems. Instead of insisting on the maximum condition of the Hamiltonian, we show that 0 belongs to the sum of Clarke's generalized gradient of the Hamiltonian and Clarke's normal cone of the control constraint set at the optimal control. Under a joint concavity condition on the Hamiltonian and a convexity condition on the terminal objective function, the necessary condition becomes sufficient. We give four examples to demonstrate the weak stochastic maximum principle.

math.OC↗

arXiv2013

On Modeling Economic Default Time: A Reduced-Form Model Approach

In the aftermath of the global financial crisis, much attention has been paid to investigating the appropriateness of the current practice of default risk modeling in banking, finance and insurance industries. A recent empirical study by Guo et al.(2008) shows that the time difference between the economic and recorded default dates has a significant impact on recovery rate estimates. Guo et al.(2011) develop a theoretical structural firm asset value model for a firm default process that embeds the distinction of these two default times. To be more consistent with the practice, in this paper, we assume the market participants cannot observe the firm asset value directly and developed a reduced-form model to characterize the economic and recorded default times. We derive the probability distribution of these two default times. The numerical study on the difference between these two shows that our proposed model can both capture the features and fit the empirical data.

q-fin.CP↗

arXiv2013

On Infectious Model for Dependent Defaults

In this paper, we propose a two-sector Markovian infectious model, which is an extension of Greenwood's model. The central idea of this model is that the causality of defaults of two sectors is in both direction, which enrich dependence dynamics. The Bayesian Information Criterion is adopted to compare the proposed model with the two-sector model in credit literature using the real data. We find that the newly proposed model is statistically better than the model in past literature. We also introduce two measures: CRES and CRVaR to give risk evaluation of our model.

q-fin.RM↗