Imperial College London

ProfessorHarryZheng

Faculty of Natural SciencesDepartment of Mathematics

Professor of Mathematics
 
 
 
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Contact

 

+44 (0)20 7594 8539h.zheng Website

 
 
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Location

 

6M16Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Publication Type
Year
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58 results found

Gu J-W, Steffensen M, Zheng H, 2020, A note on P- vs. Q-expected loss portfolio constraints, Quantitative Finance, ISSN: 1469-7688

We consider portfolio optimization problems with expected loss constraints un-der the physical measure P and the risk neutral measure Q, respectively. Using Merton’s portfolio as a benchmark portfolio, the optimal terminal wealth of the Q-risk constraint problem can be easily replicated with the standard delta hedg-ing strategy. Motivated by this, we consider the Q-strategy fulfilling the P-risk constraint and compare its solution with the true optimal solution of theP-riskconstraint problem. We show the existence and uniqueness of the optimal solution to theQ-strategy fulfilling theP-risk constraint, and provide a tractable evalua-tion method. The Q-strategy fulfilling the P-risk constraint is not only easier toimplement with standard forwards and puts on a benchmark portfolio than the P-risk constraint problem, but also easier to solve than either of theQ- or P-riskconstraint problem. The numerical test shows that the difference of the values ofthe two strategies (the Q-strategy fulfilling the P-risk constraint and the optimal strategy solving the P-risk constraint problem) is reasonably small.

Journal article

Jang HJ, Jia L, Zheng H, 2020, Why should we invest in CoCos than stocks? An optimal growth portfolio approach, The European Journal of Finance, ISSN: 1351-847X

We investigate an optimal growth portfolio problem with contingent convertible bonds (CoCos). As the conversion risk in CoCos is closely associated with the issuer's capital structure and the stock price at conversion, we model both equity and credit risk to frame this optimisation problem. This study aims to answer two questions that (i) how investors should optimally allocate their financial wealth between a CoCo and a risk-free bond; and (ii) which approach – investing in a CoCo or in a stock issued by the same bank – could result in higher expected returns. First, we derive the dynamic of a coupon-paying CoCo price under a reduced-form approach. We then decompose the problem into pre- and post-conversion regimes to obtain closed-form optimal strategies. A comparative simulation leads us to conclude that, under various market conditions, investing in a CoCo with a risk-free bond provides a higher expected growth than investing in stock.

Journal article

Zheng H, gu J, Si S, 2020, Constrained utility deviation-risk optimization and time-consistent HJB equation, SIAM Journal on Control and Optimization, Vol: 58, Pages: 866-894, ISSN: 0363-0129

In this paper we propose a unified utility deviation-risk model which covers both utilitymaximization and mean-variance analysis as special cases. We derive the time-consistentHamilton-Jacobi-Bellman (HJB) equation for the equilibrium value function and significantlyreduce the number of state variables, which makes the HJB equation derived in this papermuch easier to solve than the extended HJB equation in the literature. We illustrate theusefulness of the time-consistent HJB equation with several examples which recover theknown results in the literature and go beyond, including mean-variance model with stochasticvolatility dependent risk aversion, utility deviation-risk model with state dependent riskaversion and control constraint, and constrained portfolio selection model. The numericaland statistical tests show that the utility and deviation-risk have significant impact on theequilibrium control strategy and the distribution of the terminal wealth.

Journal article

Dong Y, Zheng H, 2020, Optimal investment with S-shaped utility and trading and value at risk constraints: An application to defined contribution pension plan, European Journal of Operational Research, Vol: 281, Pages: 341-356, ISSN: 0377-2217

In this paper we investigate an optimal investment problem under loss aversion (S-shaped utility) and with trading and Value-at-Risk (VaR) constraints faced by a defined contribution (DC) pension fund manager. We apply the concavification and dual control method to solve the problem and derive the closed-form representation of the optimal terminal wealth in terms of a controlled dual state variable. We propose a simple and effective algorithm for computing the initial dual state value, the Lagrange multiplier and the optimal terminal wealth. Theoretical and numerical results show that the VaR constraint can significantly impact the distribution of the optimal terminal wealth and may greatly reduce the risk of losses in bad economic states due to loss aversion.

Journal article

Ma J, Li W, Zheng H, 2020, Dual control Monte-Carlo method for tight bounds of value function under Heston stochastic volatility model, European Journal of Operational Research, Vol: 280, Pages: 428-440, ISSN: 0377-2217

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 to 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 Yaari 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.

Journal article

Ching W, Gu J, Zheng H, 2019, On correlated defaults and incomplete information, Journal of Industrial and Management Optimization, ISSN: 1547-5816

In this paper, we study a continuous time structural asset value model for two correlatedfirms using a two-dimensional Brownian motion. We consider the situation of incompleteinformation, where the information set available to the market participants includes the defaulttime of each firm and the periodic asset value reports. In this situation, the default timeof each firm becomes a totally inaccessible stopping time to the market participants. Theoriginal structural model is first transformed to a reduced-form model. Then the conditionaldistribution of the default time together with the asset value of each name are derived. Weprove the existence of the intensity processes of default times and also give the explicit formof the intensity processes. Numerical studies on the intensities of the two correlated names areconducted for some special cases.

Journal article

ma J, Xing J, Zheng H, 2019, Global closed-form approximation of free boundary for optimal investment stopping problems, SIAM Journal on Control and Optimization, Vol: 57, Pages: 2092-2121, ISSN: 0363-0129

In this paper we study a utility maximization problem with both optimal control and opti-mal stopping in a finite time horizon. The value function can be characterized by a variationalequation 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 functionand 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 approx-imate formulas for the optimal value function, trading strategy, and exercise boundary for theoptimal investment stopping problem. Numerical examples show the approximation is robust,accurate and fast.

Journal article

Dong Y, Zheng H, 2019, Optimal investment of DC pension plan under short-selling constraints and portfolio insurance, Insurance: Mathematics and Economics, Vol: 85, Pages: 47-59, ISSN: 0167-6687

In this paper we investigate an optimal investment problem under short-selling and portfolio insurance constraints faced by a defined contribution pension fund manager who is loss averse. The financial market consists of a cash bond, an indexed bond and a stock. The manager aims to maximize the expected S-shaped utility of the terminal wealth exceeding a minimum guarantee. We apply the dual control method to solve the problem and derive the representations of the optimal wealth process and trading strategies in terms of the dual controlled process and the dual value function. We also perform some numerical tests and show how the S-shaped utility, the short-selling constraints and the portfolio insurance impact the optimal terminal wealth.

Journal article

Bian B, Zheng H, 2019, Turnpike property and convergence rate for an investment and consumption model, Mathematics and Financial Economics, Vol: 13, Pages: 227-251, ISSN: 1862-9660

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.

Journal article

Jia L, Pistorius M, Zheng H, 2019, Dynamic portfolio optimization with looping contagion risk, SIAM Journal on Financial Mathematics, Vol: 10, Pages: 1-36, ISSN: 1945-497X

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. Weprove that the value function is the unique viscosity solution of the HJB equation. We also perform somenumerical tests to compare and analyse the statistical distributions of the terminal wealth of log utility andpower utility based on two strategies, one using the full information of intensity process and the other a proxyconstant intensity process.

Journal article

Jang HJ, Na YH, Zheng H, 2018, Contingent convertible bonds with the default risk premium, International Review of Financial Analysis, Vol: 59, Pages: 77-93, ISSN: 1057-5219

Contingent convertible bonds (CoCos) are hybrid instruments characterized by both debt and equity. CoCos are automatically converted into equity or written down when a predefined trigger event occurs. The present study quantifies the issuing bank's default risk that only manifests in the post-conversion period for pricing CoCos depending on a loss-absorbing method. This work aims to reflect the distinct features of equity-conversion CoCos - in contrast to a write-down CoCos - in a valuation framework. Accordingly, we propose a model to compute the ratio of common equity Tier 1 (CET1), which is composed of core capital and risky assets, by employing a geometric Brownian motion and a random variable. Then, we formulate the post-conversion risk premium by measuring the probability with which the bank's CET1 ratio breaches a regulatory default threshold after conversion. Finally, we empirically examine a positive value of the post-conversion risk premium embedded in the market prices of equity-conversion CoCos.

Journal article

li Y, Zheng H, 2018, Dynamic convex duality in constrained utility maximization, Stochastics: An International Journal of Probability and Stochastic Processes, Vol: 90, Pages: 1145-1169, ISSN: 1744-2508

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 forward and backward stochastic differential equations (FBSDEs) plus some 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. We also find that the optimal 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 problem directly.

Journal article

Gu J-W, Steffensen M, Zheng H, 2018, Optimal dividend strategies of two collaborating businesses in the diffusion approximation model, Mathematics of Operations Research, Vol: 43, Pages: 377-398, ISSN: 0364-765X

In this paper, we consider the optimal dividend payment strategy for an insurance company that has two collaborating business lines. The surpluses of the business lines are modeled by diffusion processes. The collaboration between the two business lines permits that money can be transferred from one line to another with or without proportional transaction costs, while money must be transferred from one line to another to help both business lines keep running before simultaneous ruin of the two lines eventually occurs.

Journal article

Li Y, Zheng H, 2018, Constrained quadratic risk minimization via forward and backward stochastic differential equations, SIAM Journal on Control and Optimization, Vol: 56, Pages: 1130-1153, ISSN: 0363-0129

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 forward and backward stochastic differential equations (FBSDEs) together with other conditions. We characterize 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.

Journal article

Wong KC, Yam SCP, Zheng H, 2017, Utility-deviation-risk portfolio selection, SIAM Journal on Control and Optimization, Vol: 55, Pages: 1819-1861, ISSN: 0363-0129

We here provide a comprehensive study of the utility-deviation-risk portfolio selection problem. By considering the first-order condition for the corresponding objective function, we first derive the necessary condition that the optimal terminal wealth satisfying two mild regularity conditions solves for a primitive static problem, called the nonlinear moment problem. We then illustrate the application of this general necessity result by revisiting the nonexistence of the optimal solution for the mean-semivariance problem. Second, we establish an alternative version of the verification theorem serving as the sufficient condition that the solution, which satisfies another mild condition different from that for necessity, of the nonlinear moment problem is the optimal terminal wealth of the original utility-deviation-risk portfolio selection problem. We then apply this general sufficiency result to revisit the various well-posed mean-risk problems already known in the literature and to also establish the existence of the optimal solutions for both utility-downside-risk and utility-strictly-convex-risk problems under the assumption that the underlying utility satisfies the Inada condition. To the best of our knowledge, positive answers to the latter two problems have long been absent in the literature. In particular, the existence result in the utility-downside-risk problem is in contrast to the well-known nonexistence of an optimal solution for the mean-downside-risk problem. As a corollary, the existence result in utility-semivariance problem allows us to utilize the semivariance as a proper risk measure in the classical portfolio management paradigm.

Journal article

Ma J, Li W, Zheng H, 2017, Dual control Monte-Carlo method for tight bounds of value function in regime switching utility maximization, European Journal of Operational Research, Vol: 262, Pages: 851-862, ISSN: 0377-2217

In this paper, we study the dual control approach for the optimal asset allocation problem in a continuous-time regime-switching market. We find the lower and upper bounds of the value function that is a solution to a system of fully coupled nonlinear partial differential equations. These bounds can be tightened with additional controls to the dual process. We suggest a Monte-Carlo algorithm for computing the tight lower and upper bounds and show the method is effective with a variety of utility functions, including power, non-HARA and Yaari utilities. The latter two utilities are beyond the scope of any current methods available in finding the value function.

Journal article

Huang YT, Song Q, Zheng H, 2017, Weak Convergence of Path-Dependent SDEs in Basket Credit Default Swap Pricing with Contagion Risk, SIAM Journal on Financial Mathematics, Vol: 8, Pages: 1-27, ISSN: 1945-497X

We investigate computational aspects of basket credit default swap pricing with counterparty credit risk under a multiname contagion model. This model enables us to capture systematic volatility increases in the market triggered by particular bankruptcies. A drawback of this model is its analytical intractability due to a combination of path-dependent coefficients and a path-dependent functional, which furthermore causes potential failure of convergence of numerical approximations under standing assumptions. In this paper, we find sufficient conditions for the desired convergence of functionals associated with approximated solution of certain path-dependent stochastic differential equations.

Journal article

Bian B, Wu N, Zheng H, 2016, Optimal liquidation in a finite time regime switching model with permanent and temporary pricing impact, Discrete and Continuous Dynamical Systems - Series B, Vol: 21, Pages: 1401-1420, ISSN: 1553-524X

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.

Journal article

Liu C, Zheng H, 2015, Asymptotic analysis for target asset portfolio allocation with small transaction costs, Insurance Mathematics & Economics, Vol: 66, Pages: 59-68, ISSN: 0167-6687

In this paper we discuss the asset allocation in the presence of small proportional transaction costs. The objective is to keep the asset portfolio close to a target portfolio and at the same time to reduce the trading cost in doing so. We derive the variational inequality and prove a verification theorem. Furthermore, we apply the second order asymptotic expansion method to characterize explicitly the optimal no transaction region when the transaction cost is small and show that the boundary points are asymmetric in relation to the target portfolio position, in contrast to the symmetric relation when only the first order asymptotic expansion method is used, and the leading order is a constant proportion of the cubic root of the small transaction cost. In addition, we use the asymptotic results for the boundary points and obtain an expansion for the value function. The results are illustrated in the numerical example.

Journal article

Ma J, Deng D, Zheng H, 2015, Convergence analysis and optimal strike choice for static hedges of general path-independent pay-offs, Quantitative Finance, Vol: 16, Pages: 593-603, ISSN: 1469-7696

In this paper, we propose a new algorithm to find the optimal static replicating portfolios for general path-independent nonlinear pay-off functions and give an estimate for the rate of convergence that is absent in the literature. We choose the static replication by designing an adaptation function arising in the error bound between the nonlinear pay-off function and the linear spline approximation and derive the equidistribution equation for selecting the optimal strikes. The numerical tests for variance swaps, swaptions, static quadratic hedges and also for a jump-diffusion process, allowing for the default of the underlying asset, show that the proposed iterative equidistribution equation algorithm is simple, fast and accurate. The paper generalizes and improves the results on static replication and approximation in the literature.

Journal article

Westray N, Zheng H, 2015, CONSTRAINED NONSMOOTH UTILITY MAXIMIZATION ON THE POSITIVE REAL LINE, Publisher: AMER INST MATHEMATICAL SCIENCES-AIMS

Scholarly edition

Bian B, Hu S, Yuan Q, Zhang Het al., 2015, Constrained viscosity solution to the HJB equation arising in perpetual American employee stock options pricing, Discrete and Continuous Dynamical Systems, Vol: 35, Pages: 5413-5433, ISSN: 1553-5231

Journal article

Dong X, Zheng H, 2015, Intensity process for a pure jump Levy structural model with incomplete information, Stochastic Processes and Their Applications, Vol: 125, Pages: 1307-1322, ISSN: 0304-4149

Journal article

Li Y, Zheng H, 2015, Weak Necessary and Sufficient Stochastic Maximum Principle for Markovian Regime-Switching Diffusion Models, APPLIED MATHEMATICS AND OPTIMIZATION, Vol: 71, Pages: 39-77, ISSN: 0095-4616

Journal article

Gu J-W, Jiang B, Ching W-K, Zheng Het al., 2014, On modeling economic default time: a reduced-form model approach, Computational Economics, Vol: 47, Pages: 157-177, ISSN: 1572-9974

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. (Rev Deriv Res 11(3): 171–204, 2008) shows that the time difference between the economic and recorded default dates has a significant impact on recovery rate estimates. Guo et al. (http://arxiv.org/abs/1012.0843, 2011) develop a theoretical structural firm asset value model for a firm default process that embeds the distinction of these two default times. In this paper, we assume the market participants cannot observe the firm asset value directly and we develop reduced-form models for characterizing the economic and recorded default times. We derive the probability distributions of these two default times. Numerical experiments with empirical data are given to demonstrate the proposed models. Our approach helps researchers to gain a new perspective for economic and recorded defaults and is more feasible in general practice compared with current method. Our results can also contribute to the understanding of the impacts of various parameters on the economic and recorded default times.

Journal article

Bian B, Zheng H, 2014, Turnpike property and convergence rate for an investment model with general utility functions, Journal of Economic Dynamics & Control, Vol: 51, Pages: 28-49, ISSN: 0165-1889

Journal article

Gu J-W, Ching W-K, Siu T-K, Zheng Het al., 2014, On reduced-form intensity-based model with 'trigger' events, JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY, Vol: 65, Pages: 331-339, ISSN: 0160-5682

Journal article

Gu J-W, Ching W-KC, Zheng H, 2014, A Hidden Markov Reduced-form Risk Model, IEEE Conference on Computational Intelligence for Financial Engineering and Economics (CIFEr), Publisher: IEEE, Pages: 190-196, ISSN: 2380-8454

Conference paper

Zhu D-M, Xie Y, Ching W-K, Zheng Het al., 2014, On Pricing and Hedging Basket Credit Derivatives with Dependent Structure, IEEE Conference on Computational Intelligence for Financial Engineering and Economics (CIFEr), Publisher: IEEE, Pages: 435-440, ISSN: 2380-8454

Conference paper

Gu J-W, Ching W-K, Siu T-K, Zheng Het al., 2013, On pricing basket credit default swaps, QUANTITATIVE FINANCE, Vol: 13, Pages: 1845-1854, ISSN: 1469-7688

Journal article

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