Utility Indifference Option Pricing Model with a Non-Constant Risk-Aversion under Transaction Costs and Its Numerical Approximation

Our goal is to analyze the system of Hamilton-Jacobi-Bellman equations arising in derivative securities pricing models. The European style of an option price is constructed as a difference of the certainty equivalents to the value functions solving the system of HJB equations. We introduce the transformation method for solving the penalized nonlinear partial differential equation. The transformed equation involves possibly non-constant the risk aversion function containing the negative ratio between the second and first derivatives of the utility function. Using comparison principles we derive useful bounds on the option price. We also propose a finite difference numerical discretization scheme with some computational examples.

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Policy iteration for Hamilton–Jacobi–Bellman equations with control constraints

Computational Optimization and Applications ◽

10.1007/s10589-021-00278-3 ◽

2021 ◽

Author(s):

Sudeep Kundu ◽

Karl Kunisch

Keyword(s):

Linear Equations ◽

Hjb Equation ◽

Policy Iteration ◽

Optimal Feedback Control ◽

Iteration Step ◽

Control Constraints ◽

Hjb Equations ◽

Bellman Equations ◽

Hamilton Jacobi Bellman ◽

Feedback Control Theory

AbstractPolicy iteration is a widely used technique to solve the Hamilton Jacobi Bellman (HJB) equation, which arises from nonlinear optimal feedback control theory. Its convergence analysis has attracted much attention in the unconstrained case. Here we analyze the case with control constraints both for the HJB equations which arise in deterministic and in stochastic control cases. The linear equations in each iteration step are solved by an implicit upwind scheme. Numerical examples are conducted to solve the HJB equation with control constraints and comparisons are shown with the unconstrained cases.

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Optimal Execution considering Trading Signal and Execution Risk Simultaneously

Mathematical Problems in Engineering ◽

10.1155/2021/5514413 ◽

2021 ◽

Vol 2021 ◽

pp. 1-12

Author(s):

Yuan Cheng ◽

Lan Wu

Keyword(s):

Kernel Function ◽

Price Impact ◽

Value Functions ◽

Feedback Form ◽

Bellman Equations ◽

Impact Function ◽

Optimal Execution ◽

Dirac Function ◽

Hamilton Jacobi Bellman ◽

Optimal Execution Problem

In this paper, we study the optimal execution problem by considering the trading signal and the transaction risk simultaneously. We propose an optimal execution problem by taking into account the trading signal and the execution risk with the associated decay kernel function and the transient price impact function being of generalized forms. In particular, we solve the stochastic optimal control problems under the assumptions that the decay kernel function is the Dirac function and the transient price function is a linear function. We give the optimal executing strategies in state-feedback form and the Hamilton‐Jacobi‐Bellman equations that the corresponding value functions satisfy in the cases of a constant execution risk and a linear execution risk. We also demonstrate that our results can recover previous results when the process of the trading signal degenerates.

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Equilibrium Investment Strategy for DC Pension Plan with Inflation and Stochastic Income under Heston’s SV Model

Mathematical Problems in Engineering ◽

10.1155/2016/2391849 ◽

2016 ◽

Vol 2016 ◽

pp. 1-18 ◽

Cited By ~ 2

Author(s):

Jingyun Sun ◽

Zhongfei Li ◽

Yongwu Li

Keyword(s):

Pension Plan ◽

Investment Strategy ◽

Investment Strategies ◽

Value Functions ◽

Defined Contribution ◽

Hjb Equations ◽

Plan Member ◽

The Mean ◽

Hamilton Jacobi Bellman ◽

Mean Variance

We consider a portfolio selection problem for a defined contribution (DC) pension plan under the mean-variance criteria. We take into account the inflation risk and assume that the salary income process of the pension plan member is stochastic. Furthermore, the financial market consists of a risk-free asset, an inflation-linked bond, and a risky asset with Heston’s stochastic volatility (SV). Under the framework of game theory, we derive two extended Hamilton-Jacobi-Bellman (HJB) equations systems and give the corresponding verification theorems in both the periods of accumulation and distribution of the DC pension plan. The explicit expressions of the equilibrium investment strategies, corresponding equilibrium value functions, and the efficient frontiers are also obtained. Finally, some numerical simulations and sensitivity analysis are presented to verify our theoretical results.

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The Dynamic Programming Method of Stochastic Differential Game for Functional Forward-Backward Stochastic System

Mathematical Problems in Engineering ◽

10.1155/2013/958920 ◽

2013 ◽

Vol 2013 ◽

pp. 1-14 ◽

Cited By ~ 1

Author(s):

Shaolin Ji ◽

Chuanfeng Sun ◽

Qingmeng Wei

Keyword(s):

Dynamic Programming ◽

Differential Game ◽

Backward Stochastic Differential Equation ◽

Transformation Method ◽

Dynamic Programming Method ◽

Stochastic Differential Game ◽

Value Functions ◽

Girsanov Transformation ◽

Path Dependent ◽

Hamilton Jacobi Bellman

This paper is devoted to a stochastic differential game (SDG) of decoupled functional forward-backward stochastic differential equation (FBSDE). For our SDG, the associated upper and lower value functions of the SDG are defined through the solution of controlled functional backward stochastic differential equations (BSDEs). Applying the Girsanov transformation method introduced by Buckdahn and Li (2008), the upper and the lower value functions are shown to be deterministic. We also generalize the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations to the path-dependent ones. By establishing the dynamic programming principal (DPP), we derive that the upper and the lower value functions are the viscosity solutions of the corresponding upper and the lower path-dependent HJBI equations, respectively.

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Pricing Basket Weather Derivatives on Rainfall and Temperature Processes

International Journal of Financial Studies ◽

10.3390/ijfs7030035 ◽

2019 ◽

Vol 7 (3) ◽

pp. 35

Author(s):

Nelson Christopher Dzupire ◽

Philip Ngare ◽

Leo Odongo

Keyword(s):

Weather Derivatives ◽

Fair Price ◽

Market Pricing ◽

No Arbitrage ◽

Bellman Equations ◽

Arbitrage Opportunities ◽

Indifference Price ◽

Utility Indifference ◽

Hamilton Jacobi Bellman ◽

Indifference Prices

This paper follows an incomplete market pricing approach to analyze the evaluation of weather derivatives and the viability of a weather derivatives market in terms of hedging. A utility indifference method is developed for the specification of indifference prices for the seller and buyer of a basket of weather derivatives written on rainfall and temperature. The agent’s risk preference is described by an exponential utility function and the prices are derived by dynamic programming principles and corresponding Hamilton Jacobi-Bellman equations from the stochastic optimal control problems. It is found the indifference measure is equal to the physical measure as there is no correlation between the capital market and weather. The fair price of the derivative should be greater than the seller’s indifference price and less than the buyer’s indifference price for market viability and no arbitrage opportunities.

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Some Results on Bellman Equations of Optimal Production Control in a Stochastic Manufacturing System

Journal of Probability and Statistics ◽

10.1155/2009/370217 ◽

2009 ◽

Vol 2009 ◽

pp. 1-23

Author(s):

Azizul Baten ◽

Anton Abdulbasah Kamil

Keyword(s):

Production Cost ◽

Manufacturing System ◽

Production Control ◽

Inventory Problem ◽

Present Value ◽

Hjb Equations ◽

Bellman Equations ◽

Production Inventory ◽

Production Cost Control ◽

Hamilton Jacobi Bellman

The paper studies the production inventory problem of minimizing the expected discounted present value of production cost control in a manufacturing system with degenerate stochastic demand. We establish the existence of a unique solution of the Hamilton-Jacobi-Bellman (HJB) equations associated with this problem. The optimal control is given by a solution to the corresponding HJB equation.

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Stability and convergence of second order backward differentiation schemes for parabolic Hamilton–Jacobi–Bellman equations

Numerische Mathematik ◽

10.1007/s00211-021-01202-x ◽

2021 ◽

Author(s):

Olivier Bokanowski ◽

Athena Picarelli ◽

Christoph Reisinger

Keyword(s):

Parabolic Equations ◽

Linear Equations ◽

Second Order ◽

Stability And Convergence ◽

Classical Solutions ◽

Hjb Equations ◽

Bellman Equations ◽

Linear Parabolic Equations ◽

Viscosity Sense ◽

Hamilton Jacobi Bellman

AbstractWe study a second order Backward Differentiation Formula (BDF) scheme for the numerical approximation of linear parabolic equations and nonlinear Hamilton–Jacobi–Bellman (HJB) equations. The lack of monotonicity of the BDF scheme prevents the use of well-known convergence results for solutions in the viscosity sense. We first consider one-dimensional uniformly parabolic equations and prove stability with respect to perturbations, in the $$L^2$$ L 2 norm for linear and semi-linear equations, and in the $$H^1$$ H 1 norm for fully nonlinear equations of HJB and Isaacs type. These results are then extended to two-dimensional semi-linear equations and linear equations with possible degeneracy. From these stability results we deduce error estimates in $$L^2$$ L 2 norm for classical solutions to uniformly parabolic semi-linear HJB equations, with an order that depends on their Hölder regularity, while full second order is recovered in the smooth case. Numerical tests for the Eikonal equation and a controlled diffusion equation illustrate the practical accuracy of the scheme in different norms.

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Finite Difference Methods for the Hamilton–Jacobi–Bellman Equations Arising in Regime Switching Utility Maximization

Journal of Scientific Computing ◽

10.1007/s10915-020-01352-4 ◽

2020 ◽

Vol 85 (3) ◽

Author(s):

Jingtang Ma ◽

Jianjun Ma

Keyword(s):

Finite Difference ◽

Regime Switching ◽

Utility Maximization ◽

Finite Difference Methods ◽

Hjb Equations ◽

Functional Operator ◽

Bellman Equations ◽

Numerical Comparisons ◽

Difference Methods ◽

Hamilton Jacobi Bellman

AbstractFor solving the regime switching utility maximization, Fu et al. (Eur J Oper Res 233:184–192, 2014) derive a framework that reduce the coupled Hamilton–Jacobi–Bellman (HJB) equations into a sequence of decoupled HJB equations through introducing a functional operator. The aim of this paper is to develop the iterative finite difference methods (FDMs) with iteration policy to the sequence of decoupled HJB equations derived by Fu et al. (2014). The convergence of the approach is proved and in the proof a number of difficulties are overcome, which are caused by the errors from the iterative FDMs and the policy iterations. Numerical comparisons are made to show that it takes less time to solve the sequence of decoupled HJB equations than the coupled ones.

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STOCHASTIC DIFFERENTIAL GAMES BETWEEN TWO INSURERS WITH GENERALIZED MEAN-VARIANCE PREMIUM PRINCIPLE

Astin Bulletin ◽

10.1017/asb.2017.35 ◽

2018 ◽

Vol 48 (1) ◽

pp. 413-434 ◽

Cited By ~ 10

Author(s):

Shumin Chen ◽

Hailiang Yang ◽

Yan Zeng

Keyword(s):

Differential Game ◽

Game Problem ◽

Stochastic Differential Game ◽

Value Functions ◽

Programming Technique ◽

Bellman Equations ◽

Equilibrium Strategies ◽

Generalized Mean ◽

Hamilton Jacobi Bellman ◽

Mean Variance

AbstractWe study a stochastic differential game problem between two insurers, who invest in a financial market and adopt reinsurance to manage their claim risks. Supposing that their reinsurance premium rates are calculated according to the generalized mean-variance principle, we consider the competition between the two insurers as a non-zero sum stochastic differential game. Using dynamic programming technique, we derive a system of coupled Hamilton–Jacobi–Bellman equations and show the existence of equilibrium strategies. For an exponential utility maximizing game and a probability maximizing game, we obtain semi-explicit solutions for the equilibrium strategies and the equilibrium value functions, respectively. Finally, we provide some detailed comparative-static analyses on the equilibrium strategies and illustrate some economic insights.

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Precommitted Investment Strategy versus Time-Consistent Investment Strategy for a General Risk Model with Diffusion

Abstract and Applied Analysis ◽

10.1155/2014/358623 ◽

2014 ◽

Vol 2014 ◽

pp. 1-15

Author(s):

Lidong Zhang ◽

Ximin Rong ◽

Ziping Du

Keyword(s):

Risk Model ◽

Theoretical Perspective ◽

Optimal Strategies ◽

Investment Strategy ◽

Value Functions ◽

Lagrange Method ◽

Bellman Equations ◽

Strategy Versus ◽

Hamilton Jacobi Bellman ◽

Mean Variance

We mainly study a general risk model and investigate the precommitted strategy and the time-consistent strategy under mean-variance criterion, respectively. A lagrange method is proposed to derive the precommitted investment strategy. Meanwhile from the game theoretical perspective, we find the time-consistent investment strategy by solving the extended Hamilton-Jacobi-Bellman equations. By comparing the precommitted strategy with the time-consistent strategy, we find that the company under the time-consistent strategy has to give up the better current utility in order to keep a consistent satisfaction over the whole time horizon. Furthermore, we theoretically and numerically provide the effect of the parameters on these two optimal strategies and the corresponding value functions.

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