Optimal investment and reinsurance for an insurer under Markov-modulated financial market

We consider a spread financial market defined by the multidimensional Ornstein–Uhlenbeck (OU) process. We study the optimal consumption/investment problem for logarithmic utility functions using a stochastic dynamical programming method. We show a special verification theorem for this case. We find the solution to the Hamilton–Jacobi–Bellman (HJB) equation in explicit form and as a consequence we construct optimal financial strategies. Moreover, we study the constructed strategies with numerical simulations.

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HEDGING OPTIONS IN A DOUBLY MARKOV-MODULATED FINANCIAL MARKET VIA STOCHASTIC FLOWS

International Journal of Theoretical and Applied Finance ◽

10.1142/s021902491950047x ◽

2019 ◽

Vol 22 (08) ◽

pp. 1950047 ◽

Cited By ~ 1

Author(s):

TAK KUEN SIU ◽

ROBERT J. ELLIOTT

Keyword(s):

Markov Chain ◽

Financial Market ◽

Continuous Time ◽

Regime Switching ◽

Stochastic Flows ◽

Finite State ◽

Markov Modulated ◽

Appreciation Rate ◽

The Interest Rate ◽

Over Time

The hedging of a European-style contingent claim is studied in a continuous-time doubly Markov-modulated financial market, where the interest rate of a bond is modulated by an observable, continuous-time, finite-state, Markov chain and the appreciation rate of a risky share is modulated by a continuous-time, finite-state, hidden Markov chain. The first chain describes the evolution of credit ratings of the bond over time while the second chain models the evolution of the hidden state of an underlying economy over time. Stochastic flows of diffeomorphisms are used to derive some hedge quantities, or Greeks, for the claim. A mixed filter-based and regime-switching Black–Scholes partial differential equation is obtained governing the price of the claim. It will be shown that the delta hedge ratio process obtained from stochastic flows is a risk-minimizing, admissible mean-self-financing portfolio process. Both the first-order and second-order Greeks will be considered.

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MINIMIZING THE PROBABILITY OF LIFETIME RUIN: TWO RISKLESS ASSETS WITH TRANSACTION COSTS

Astin Bulletin ◽

10.1017/asb.2019.23 ◽

2019 ◽

Vol 49 (03) ◽

pp. 847-883

Author(s):

Xiaoqing Liang ◽

Virginia R. Young

Keyword(s):

Transaction Costs ◽

Financial Market ◽

Optimal Investment ◽

Investment Strategy ◽

Money Market ◽

The Other ◽

Proportional Transaction Costs ◽

Optimal Investment Strategy ◽

The Individual ◽

Riskless Asset

AbstractWe compute the optimal investment strategy for an individual who wishes to minimize her probability of lifetime ruin. The financial market in which she invests consists of two riskless assets. One riskless asset is a money market, and she consumes from that account. The other riskless asset is a bond that earns a higher interest rate than the money market, but buying and selling bonds are subject to proportional transaction costs. We consider the following three cases. (1) The individual is allowed to borrow from both riskless assets; ruin occurs if total imputed wealth reaches zero. Under the optimal strategy, the individual does not sell short the bond. However, she might wish to borrow from the money market to fund her consumption. Thus, in the next two cases, we seek to limit borrowing from that account. (2) We assume that the individual pays a higher rate to borrow than she earns on the money market. (3) The individual is not allowed to borrow from either asset; ruin occurs if both the money market and bond accounts reach zero wealth. We prove that the borrowing rate in case (2) acts as a parameter connecting the two seemingly unrelated cases (1) and (3).

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Malliavin method for optimal investment in financial markets with memory

Open Mathematics ◽

10.1515/math-2016-0027 ◽

2016 ◽

Vol 14 (1) ◽

pp. 286-299 ◽

Cited By ~ 2

Author(s):

Qiguang An ◽

Guoqing Zhao ◽

Gaofeng Zong

Keyword(s):

Financial Market ◽

Mean Field ◽

Optimal Investment ◽

Volterra Equations ◽

Necessary Condition ◽

Dynamic Programming Method ◽

Classical Dynamic ◽

Stochastic Volterra Equation ◽

Optimal Investment Problem ◽

With Memory

AbstractWe consider a financial market with memory effects in which wealth processes are driven by mean-field stochastic Volterra equations. In this financial market, the classical dynamic programming method can not be used to study the optimal investment problem, because the solution of mean-field stochastic Volterra equation is not a Markov process. In this paper, a new method through Malliavin calculus introduced in [1], can be used to obtain the optimal investment in a Volterra type financial market. We show a sufficient and necessary condition for the optimal investment in this financial market with memory by mean-field stochastic maximum principle.

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Dynamic Index Optimal Investment Strategy Based on Stochastic Differential Equations in Financial Market Options

Wireless Communications and Mobile Computing ◽

10.1155/2021/5545956 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Jun Zhang

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Financial Market ◽

Capital Investment ◽

Optimal Investment ◽

Investment Strategy ◽

Insurance Companies ◽

Investment Efficiency ◽

Bank Deposits ◽

Optimal Investment Strategy

With the gradual development and improvement of the financial market, financial derivatives such as futures and options have also become the objects of competition in the financial market. Therefore, how to make the most favorable and optimized investment and consumption when options are included? It has become a problem facing investors. Aiming at the optimal investment problem of investors, this paper studies the calculation of an optimal investment strategy in stochastic differential equations in financial market options on the basis of fuzzy theory. Now, stochastic calculus has become an important branch of stochastic analysis, finance, control, and other fields. The study of introducing stochastic differential equations is mainly to solve the stochastic control problem, which is the principle of the stochastic maximum. In finance, some hedging or pricing problems of contingent rights can eventually be transformed into a series of stochastic differential equations. Based on the historical data of five aspects of bank deposits, bonds, funds, stocks, and real estate of four listed insurance companies, the paper analyzes the optimization strategy of the capital investment of listed insurance companies based on the investment yield of the domestic investment market. According to the final results, the historical proportion of bank deposits of the superior company is 27%, and the optimal proportion given by the model is 25%; the total proportion of funds and stocks is 15%, and the optimal proportion of funds analyzed in the model is 20% and the optimal proportion of stocks is 10%. Therefore, the final results show that the investment efficiency of listed insurance companies can actually increase investment in stocks and funds and reduce the proportion of bank deposits to obtain greater investment returns.

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Optimal Investment for the Insurers in Markov-Modulated Jump-Diffusion Models

Computational Economics ◽

10.1007/s10614-014-9454-7 ◽

2014 ◽

Vol 46 (1) ◽

pp. 143-156 ◽

Cited By ~ 1

Author(s):

Jinzhi Li ◽

Haiying Liu

Keyword(s):

Optimal Investment ◽

Jump Diffusion ◽

Diffusion Models ◽

Markov Modulated

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Portfolio optimization with unobservable Markov-modulated drift process

Journal of Applied Probability ◽

10.1239/jap/1118777176 ◽

2005 ◽

Vol 42 (2) ◽

pp. 362-378 ◽

Cited By ~ 60

Author(s):

Ulrich Rieder ◽

Nicole Bäuerle

Keyword(s):

Portfolio Optimization ◽

Drift Rate ◽

Optimization Problems ◽

Optimal Investment ◽

Power Utility ◽

Portfolio Strategy ◽

Markov Modulated ◽

Hamilton Jacobi Bellman ◽

The Value Function ◽

Complete Observation

We study portfolio optimization problems in which the drift rate of the stock is Markov modulated and the driving factors cannot be observed by the investor. Using results from filter theory, we reduce this problem to one with complete observation. In the cases of logarithmic and power utility, we solve the problem explicitly with the help of stochastic control methods. It turns out that the value function is a classical solution of the corresponding Hamilton-Jacobi-Bellman equation. As a special case, we investigate the so-called Bayesian case, i.e. where the drift rate is unknown but does not change over time. In this case, we prove a number of interesting properties of the optimal portfolio strategy. In particular, using the likelihood-ratio ordering, we can compare the optimal investment in the case of observable drift rate to that in the case of unobservable drift rate. Thus, we also obtain the sign of the drift risk.

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Portfolio optimization with unobservable Markov-modulated drift process

Journal of Applied Probability ◽

10.1017/s0021900200000401 ◽

2005 ◽

Vol 42 (02) ◽

pp. 362-378 ◽

Cited By ~ 12

Author(s):

Ulrich Rieder ◽

Nicole Bäuerle

Keyword(s):

Portfolio Optimization ◽

Drift Rate ◽

Optimization Problems ◽

Optimal Investment ◽

Power Utility ◽

Portfolio Strategy ◽

Markov Modulated ◽

Hamilton Jacobi Bellman ◽

The Value Function ◽

Complete Observation

We study portfolio optimization problems in which the drift rate of the stock is Markov modulated and the driving factors cannot be observed by the investor. Using results from filter theory, we reduce this problem to one with complete observation. In the cases of logarithmic and power utility, we solve the problem explicitly with the help of stochastic control methods. It turns out that the value function is a classical solution of the corresponding Hamilton-Jacobi-Bellman equation. As a special case, we investigate the so-called Bayesian case, i.e. where the drift rate is unknown but does not change over time. In this case, we prove a number of interesting properties of the optimal portfolio strategy. In particular, using the likelihood-ratio ordering, we can compare the optimal investment in the case of observable drift rate to that in the case of unobservable drift rate. Thus, we also obtain the sign of the drift risk.

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