Random Liabilities Research Articles

Forward Backward SDEs Systems for Utility Maximization in Jump Diffusion Models

We consider the classical problem of maximizing the expected utility of terminal net wealth with a final random liability in a simple jump-diffusion model. In the spirit of Horst et al. (Stoch Process Appl 124(5):1813–1848, 2014) and Santacroce and Trivellato (SIAM J Control Optim 52(6):3517–3537, 2014), under suitable conditions the optimal strategy is expressed in implicit form in terms of a forward backward system of equations. Some explicit results are presented for the pure jump model and for exponential utilities.

Robust asset-liability management games for n players under multivariate stochastic covariance models

This paper investigates a non-zero-sum stochastic differential game among n competitive CARA asset-liability managers, who are concerned about the potential model ambiguity and aim to seek the robust investment strategies. The ambiguity-averse managers are subject to uncontrollable and idiosyncratic random liabilities driven by generalized drifted Brownian motions and have access to an incomplete financial market consisting of a risk-free asset, a market index and a stock under a multivariate stochastic covariance model. The market dynamics permit not only stochastic correlation between the risky assets but also path-dependent and time-varying risk premium and volatility, depending on two affine-diffusion factor processes. The objective of each manager is to maximize the expected exponential utility of his terminal surplus relative to the average among his competitors under the worst-case scenario of the alternative measures. We manage to solve this robust non-Markovian stochastic differential game by using a backward stochastic differential equation approach. Explicit expressions for the robust Nash equilibrium investment policies, the density generator processes under the well-defined worst-case probability measures and the corresponding value functions are derived. Conditions for the admissibility of the robust equilibrium strategies are provided. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium investment strategies and draw some economic interpretations from these results.

Robust optimal asset-liability management with mispricing and stochastic factor market dynamics

This paper investigates a robust optimal asset-liability management problem under an expected utility maximization criterion. More specifically, the manager is concerned about the potential model uncertainty and aims to seek the robust optimal investment strategies. We incorporate an uncontrollable random liability described by a generalized drifted Brownian motion. Also, the manager has access to an incomplete financial market consisting of a risk-free asset, a market index with potentially path-dependent, time-varying risk premium and volatility, and a pair of mispriced stocks. The market dynamics are assumed to rely on an affine-form, square-root factor process and the price error is modeled by a co-integrated system. We adopt a backward stochastic differential equation approach hinging on the martingale optimality principle to solve this non-Markovian robust control problem. Closed-form expressions for the robust optimal investment strategies, the probability perturbation process under the well-defined worst-case scenario and the corresponding value function are derived. The admissibility of the robust optimal controls is verified under some technical conditions. Finally, we perform some numerical examples to illustrate the effects of model parameters on the robust investment strategies and draw some economic interpretations from these results.

Systemic-Risk and Evolutionary Stable Strategies in a Financial Network

We consider a financial network represented at any time instance by a random liability graph which evolves over time. The agents connect through credit instruments borrowed from each other or through direct lending, and these create the liability edges. These random edges are modified (locally) by the agents over time, as they learn from their experiences and (possibly imperfect) observations. The settlement of the liabilities of various agents at the end of the contract period (at any time instance) can be expressed as solutions of random fixed point equations. Our first step is to derive the solutions of these equations (asymptotically and one for each time instance), using a recent result on random fixed point equations. The agents, at any time instance, adapt one of the two available strategies, risky or less risky investments, with an aim to maximize their returns. We aim to study the emerging strategies of such replicator dynamics that drives the financial network. We theoretically reduce the analysis of the complex system to that of an appropriate ordinary differential equation (ODE). Using the attractors of the resulting ODE we show that the replicator dynamics converges to one of the two pure evolutionary stable strategies (all risky or all less risky agents); one can have mixed limit only when the observations are imperfect. We verify our theoretical findings using exhaustive Monte Carlo simulations. The dynamics avoid the emergence of the systemic-risk regime (where majority default). However, if all the agents blindly adapt risky strategy it can lead to systemic risk regime.

Mean-variance asset-liability management under CIR interest rate and the family of 4/2 stochastic volatility models with derivative trading

<p style='text-indent:20px;'>This paper investigates the effects of derivative trading on the performance of asset-liability management in the presence of stochastic interest rate and stochastic volatility under the mean-variance criterion. Specifically, the asset-liability manager can invest not only in a money market account, a zero-coupon (rollover) bond, and a stock index but also in stock derivatives. It is assumed that the interest rate follows a Cox-Ingersoll-Ross (CIR) process, and the instantaneous variance of the stock index is governed by the family of 4/2 stochastic volatility models, which embraces the Heston model and 3/2 model, as particular cases. By solving a system of three backward stochastic differential equations, closed-form expressions for the optimal strategies and optimal value functions are derived in two cases: with and without the stock derivatives. Moreover, we consider the special cases without random liabilities. Numerical examples are provided to illustrate theoretical results and explore the effects of derivative trading on efficient frontiers.</p>

The Study of Mean-Variance Risky Asset Management with State-Dependent Risk Aversion under Regime Switching Market

How do investors require a distribution of the wealth among multiple risky assets while facing the risk of the uncontrollable payment for random liabilities? To cope with this problem, firstly, this paper explores the approach of asset-liability management under the state-dependent risk aversion with only risky assets, which has been considered under a continuous-time Markov regime-switching setting. Next, based on this realistic modelling, an extended Hamilton-Jacob-Bellman (HJB) system has been necessarily established for solving the optimization problem of asset-liability management. It has been derived closed-form analytical expressions applied in the time-inconsistent investment with optimal control theory to see that happens to the optimal value of the function. Ultimately, numerical examples presented with comparisons of the analytical results under different market conditions are exposed to analyse numerically the developed mean variance asset liability management strategy. We find that our proposed model can explain the financial phenomena more effectively and accurately.

Mean-Variance Asset-Liability Management in a Non-Markovian Regime-Switching Jump-Diffusion Market with Random Horizon

This paper investigates the problem of mean-variance asset-liability management (ALM) in a market where the dynamics of assets are non-Markovian regime-switching models driven by a Brownian motion, a Poisson random measure and a continuous time finite-state Markov chain. It is assumed that the time horizon is uncertain relying not only on asset prices and liability values, but also on other factors. The insurer aims to minimize the variance of the terminal surplus given an expected terminal surplus subject to the risk of paying out random liabilities of an extensive Cramer-Lundberg model. By further developing the solvability of a linear backward stochastic differential equation (BSDE) with two types of jumps, namely jumps modelled by a Poisson random measure and by basic martingales related to a Markov chain, closed-form expressions for both the efficient strategy and the efficient frontier are obtained and represented by unique solutions to several relevant BSDEs.

Dynamic asset-liability management problem in a continuous-time model with delay

ABSTRACT This paper investigates a dynamic continuous-time asset-liability management (ALM) problem with delay under the mean-variance criterion. The investor allocates her wealth in a financial market consisting of one risk-free asset and one risky asset, and she is subject to a random liability. The historical information of the wealth and liability affects the investor's wealth process, which is then governed by a stochastic differential delay equation. Firstly, a general ALM problem with delay is formulated and the extended Hamilton-Jacobi-Bellman system of equations is obtained. Secondly, we focus on a linear model and derive the closed-form expressions of the equilibrium investment strategy and the corresponding equilibrium value function. Meanwhile, we also derive the pre-commitment strategy for the mean-variance ALM problem with delay using the maximum principle. Finally, some numerical examples and sensitivity analysis are presented to illustrate the equilibrium investment strategies and the efficient frontiers under the equilibrium and pre-commitment frameworks.

Portfolio Selection with Random Liability and Affine Interest Rate in the Mean-Variance Framework

Abstract This paper studies a dynamic mean-variance portfolio selection problem with random liability in the affine interest rate environment, where the financial market consists of three assets: one risk-free asset, one risky asset and one zero-coupon bond. Assume that short rate is driven by affine interest rate model and liability process is described by the drifted Brownian motion, in addition, stock price dynamics is affected by interest rate dynamics. The investors expect to look for an optimal strategy to minimize the variance of the terminal surplus for a given expected terminal surplus. The efficient strategy and the efficient frontier are explicitly obtained by applying dynamic programming principle and Lagrange duality theorem. A numerical example is given to illustrate our results and some economic implications are analyzed.

Mean-Variance Asset Liability Management with State-Dependent Risk Aversion

ABSTRACTThis article investigates the asset liability management problem with state-dependent risk aversion under the mean-variance criterion. The investor allocates the wealth among multiple assets including a risk-free asset and multiple risky assets governed by a system of geometric Brownian motion stochastic differential equations, and the investor faces the risk of paying uncontrollable random liabilities. The state-dependent risk aversion is taken into account in our model, linking the risk aversion to the current wealth held by the investor. An extended Hamilton-Jacobi-Bellman system is established for the optimization of asset liability management, and by solving the extended Hamilton-Jacobi-Bellman system, the analytical closed-form expressions for the time-inconsistent optimal investment strategies and the optimal value function are derived. Finally, numerical examples are presented to illustrate our results.

Evaluation and default time for companies with uncertain cash flows

In this study, we propose a modelling framework for evaluating companies financed by random liabilities, such as insurance companies or commercial banks. In this approach, earnings and costs are driven by double exponential jump–diffusion processes and bankruptcy is declared when the income falls below a default threshold, which is proportional to the charges. A change of numeraire, under the Esscher risk neutral measure, is used to reduce the dimension. A closed form expression for the value of equity is obtained in terms of the expected present value operators, with and without disinvestment delay. In both cases, we determine the default threshold that maximizes the shareholder’s equity. Subsequently, the probabilities of default are obtained by inverting the Laplace transform of the bankruptcy time. In numerical applications of the proposed model, we apply a procedure for calibration based on market and accounting data to explain the behaviour of shares for two real-world examples of insurance companies.

Mean–variance asset–liability management with asset correlation risk and insurance liabilities

Consider an insurer who invests in the financial market where correlations among risky asset returns are randomly changing over time. The insurer who faces the risk of paying stochastic insurance claims needs to manage her asset and liability by taking into account of the correlation risk. This paper investigates the impact of correlation risk to the optimal asset–liability management (ALM) of an insurer. We employ the Wishart process to model the stochastic covariance matrix of risky asset returns. The insurer aims to minimize the variance of the terminal wealth given an expected terminal wealth subject to the risk of paying out random liabilities of compound Poisson process. This ALM problem then becomes a linear–quadratic stochastic optimal control problem with stochastic volatilities, stochastic correlations and jumps. The recognition of an affine form in the solution process enables us to derive the explicit closed-form solution to the optimal ALM portfolio policy, obtain the efficient frontier, and identify the condition that the solution is well behaved.

Forward Backward Semimartingale Systems for Utility Maximization

We consider the problem of maximizing the expected utility of terminal wealth with a terminal random liability when the underlying asset price process is a continuous semimartingale. The optimal strategy is characterized in terms of a forward backward semimartingale system of equations. The results cover the cases of exponential, logarithmic, and power utilities, which we analyze as illustrative examples.

Mean-variance principle of managing cointegrated risky assets and random liabilities

Using the diffusion limit of the discrete-time error correction model of cointegration for risky assets and geometric Brownian motion for the value of liabilities, we solve the asset-liability management (ALM) problem using the theory of backward stochastic differential equations. The solutions of the ALM policy and the efficient frontier in terms of surplus are obtained as closed-form formulas. We numerically examine the impact of cointegration to the trade-off between risk and return in managing cointegrated risky assets and random liabilities.

Optimal investment for an insurer with cointegrated assets: CRRA utility

This paper considers the optimal investment problem for an insurer that invests in cointegrated assets subject to the random payments of insurance claims. The insurer’s objective is to maximize the expected utility of the terminal wealth subject to the cointegration dynamics of risky assets and the risk of paying out random liabilities with a compound Poisson process. We solve the continuous-time investment problems for the class of the constant relative risk averse utility function using the framework of the HJB equation. An explicit solution is derived by recognizing an exponential affine form in the derivation process. We then investigate the risk-preference of insurers toward statistical arbitrage from pairs-trading using the analytical results. Although a financial market with cointegrated risky assets implies the existence of statistical arbitrage opportunities, insurers may not be interested in those opportunities due to the social responsibility of a high level of risk aversion. However, if insurers are forced to trade cointegrated assets, the derived optimal solution enhances the investment performance.

Mean–variance asset–liability management: Cointegrated assets and insurance liability

The cointegration of major financial markets around the globe is well evidenced with strong empirical support. This paper considers the continuous-time mean–variance (MV) asset–liability management (ALM) problem for an insurer investing in an incomplete financial market with cointegrated assets. The number of trading assets is allowed to be less than the number of Brownian motions spanning the market. The insurer also faces the risk of paying uncertain insurance claims during the investment period. We assume that the cointegration market follows the diffusion limit of the error-correction model for cointegrated time series. Using the Markowitz (1952) MV portfolio criterion, we consider the insurer’s problem of minimizing variance in the terminal wealth, given an expected terminal wealth subject to interim random liability payments following a compound Poisson process. We generalize the technique developed by Lim (2005) to tackle this problem. The particular structure of cointegration enables us to solve the ALM problem completely in the sense that the solutions of the continuous-time portfolio policy and efficient frontier are obtained as explicit and closed-form formulas.

Differentiability of quadratic BSDEs generated by continuous martingales

In this paper we consider a class of BSDEs with drivers of quadratic growth, on a stochastic basis generated by continuous local martingales. We first derive the Markov property of a forward–backward system (FBSDE) if the generating martingale is a strong Markov process. Then we establish the differentiability of a FBSDE with respect to the initial value of its forward component. This enables us to obtain the main result of this article, namely a representation formula for the control component of its solution. The latter is relevant in the context of securitization of random liabilities arising from exogenous risk, which are optimally hedged by investment in a given financial market with respect to exponential preferences. In a purely stochastic formulation, the control process of the backward component of the FBSDE steers the system into the random liability and describes its optimal derivative hedge by investment in the capital market, the dynamics of which is given by the forward component. The representation formula of the main result describes this delta hedge in terms of the derivative of the BSDEs solution process on the one hand and the correlation structure of the internal uncertainty captured by the forward process and the external uncertainty responsible for the market incompleteness on the other hand. The formula extends the scope of validity of the results obtained by several authors in the Brownian setting. It is designed to extend a genuinely stochastic representation of the optimal replication in cross hedging insurance derivatives from the classical Black–Scholes model to incomplete markets on general stochastic bases. In this setting, Malliavin’s calculus which is required in the Brownian framework, is replaced by new tools based on techniques related to a calculus of quadratic covariations of basis martingales.

On the Pricing of Contingent Capital Notes

A bank's stock price is modeled as a call option on the spread of random assets over random liabilities. The logarithm of assets and liabilities are jointly modeled as driven by four variance gamma processes and this model is estimated by calibrating to quoted equity options seen as compound spread options. On defining risk-weighted assets as asset value less the bid price plus the ask price of liabilities less the liability value we endogenize capital adequacy ratios following the methods of conic finance for the bid and ask prices. All computations are illustrated on CSGN.VX, ADRed into USD on March 29, 2011.