Least-squares Monte-Carlo methods for optimal stopping investment under CEV models

The optimal stopping investment is a kind of mixed expected utility maximization problems with optimal stopping time. The aim of this paper is to develop the least-squares Monte-Carlo methods to solve the optimal stopping investment under the constant elasticity of variance (CEV) model. Such a problem has no closed-form solutions for the value functions, optimal strategies and optimal exercise boundaries due to the early exercised feature. The dual optimal stopping problem is first derived and then the strong duality between the dual and prime problems is established. The least-squares Monte-Carlo methods based on the dual control theory are developed and numerical simulations are provided. Both the power and non-HARA utilities are studied.     

Optimal constrained investment in the Cramer-Lundberg model

We consider an insurance company whose surplus is represented by the classical Cramer-Lundberg process. The company can invest its surplus in a risk-free asset and in a risky asset, governed by the Black-Scholes equation. There is a constraint that the insurance company can only invest in the risky asset at a limited leveraging level; more precisely, when purchasing, the ratio of the investment amount in the risky asset to the surplus level is no more than a; and when short-selling, the proportion of the proceeds from the short-selling to the surplus level is no more than b. The objective is to find an optimal investment policy that minimizes the probability of ruin. The minimal ruin probability as a function of the initial surplus is characterized by a classical solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation. We study the optimal control policy and its properties. The interrelation between the parameters of the model plays a crucial role in the qualitative behavior of the optimal policy. For example, for some ratios between a and b, quite unusual and at first ostensibly counterintuitive policies may appear, like short-selling a stock with a higher rate of return to earn lower interest, or borrowing at a higher rate to invest in a stock with lower rate of return. This is in sharp contrast with the unrestricted case, first studied in Hipp and Plum, or with the case of no short-selling and no borrowing studied in Azcue and Muler.     

Optimal investment and consumption under a continuous-time cointegration model with exponential utility

In this paper, we study the effects of cointegration on optimal investment and consumption strategies for an investor with exponential utility. A Hamilton-Jacobi-Bellman (HJB) equation is derived first and then solved analytically. Both the optimal investment and consumption strategies are expressed in closed form. A verification theorem is also established to demonstrate that the solution of the HJB equation is indeed the solution of the original optimization problem under an integrability condition. In addition, a simple and sufficient condition is proposed to ensure that the integrability condition is satisfied. Financially, the optimal investment and consumption strategies are decomposed into two parts: the myopic part and the hedging demand caused by cointegration. Discussions on the hedging demand are carried out first, based on analytical formulae. Then numerical results show that ignoring the information about cointegration results in a utility loss.     

Optimal portfolios and Heston's stochastic volatility model: an explicit solution for power utility

Given an investor maximizing utility from terminal wealth with respect to a power utility function, we present a verification result for portfolio problems with stochastic volatility. Applying this result, we solve the portfolio problem for Heston's stochastic volatility model. We find that only under a specific condition on the model parameters does the problem possess a unique solution leading to a partial equilibrium. Finally, it is demonstrated that the results critically hinge upon the specification of the market price of risk. We conclude that, in applications, one has to be very careful when exogenously specifying the form of the market price of risk.     

Optimal execution strategies in limit order books with general shape functions

We consider optimal execution strategies for block market orders placed in a limit order book (LOB). We build on the resilience model proposed by Obizhaeva and Wang (2005 Obizhaeva, A and Wang, J. 2005. Optimal trading strategy and supply/demand dynamics, Preprint Available online at: http://www.rhsmith.umd.edu/faculty/obizhaeva/OW060408.pdf (accessed 16 February 2009)[Crossref] , [Google Scholar]) but allow for a general shape of the LOB defined via a given density function. Thus, we can allow for empirically observed LOB shapes and obtain a nonlinear price impact of market orders. We distinguish two possibilities for modelling the resilience of the LOB after a large market order: the exponential recovery of the number of limit orders, i.e. of the volume of the LOB, or the exponential recovery of the bid–ask spread. We consider both of these resilience modes and, in each case, derive explicit optimal execution strategies in discrete time. Applying our results to a block-shaped LOB, we obtain a new closed-form representation for the optimal strategy of a risk-neutral investor, which explicitly solves the recursive scheme given in Obizhaeva and Wang (2005 Obizhaeva, A and Wang, J. 2005. Optimal trading strategy and supply/demand dynamics, Preprint Available online at: http://www.rhsmith.umd.edu/faculty/obizhaeva/OW060408.pdf (accessed 16 February 2009)[Crossref] , [Google Scholar]). We also provide some evidence for the robustness of optimal strategies with respect to the choice of the shape function and the resilience-type.     

Optimal dividend strategy with transaction costs for an upward jump model

In this paper, we consider the optimal dividend problem with transaction costs when the incomes of a company can be described by an upward jump model. Both fixed and proportional costs are considered in the problem. The value function is defined as the expected total discounted dividends up to the time of ruin. Although the same problem has already been studied in the pure diffusion model and the spectrally negative Lévy process, the optimal dividend problem in an upward jump model has two different aspects in determining the optimal dividends barrier and in the property of the value function. First, the value function is twice continuous differentiable in the diffusion case, but it is not in the jump model. Second, under the spectrally negative Lévy process, downward jumps will not cause any payment actions; however, it might trigger dividend payments when there are upward jumps. In deriving the optimal barriers, we show that the value function is bounded by a linear function. Using this property, we establish the verification theorem for the value function. By solving the quasi-variational inequalities associated with this problem, we obtain the closed-form solution to the value function and hence the optimal dividend strategy when the income sizes follow a common exponential distribution. In the presence of a fixed transaction cost, it is shown that the optimal strategy is a two-barrier policy, and the optimal barriers are only dependent on the fixed cost and not the proportional cost. A numerical example is used to illustrate how the fixed cost plays a significant role in the optimal dividend strategy and also the value function. Moreover, an increased fixed cost results in larger but less frequent dividend payments.     

Time-consistent reinsurance and investment strategies for an AAI under smooth ambiguity utility

This paper investigates time-consistent reinsurance(excess-of-loss, proportional) and investment strategies for an ambiguity averse insurer(abbr. AAI). The AAI is ambiguous towards the insurance and financial markets. In the AAI's attitude, the intensity of the insurance claims' number and the market price of risk of a stock can not be estimated accurately. This formulation of ambiguity is similar to the uncertainty of different equivalent probability measures. The AAI can purchase excess-of-loss or proportional reinsurance to hedge the insurance risk and invest in a financial market with cash and an ambiguous stock. We investigate the optimization goal under smooth ambiguity given in Klibanoff, P., Marinacci, M., & Mukerji, S. [(2005). A smooth model of decision making under ambiguity. Econometrica 73, 1849–1892], which aims to search the optimal strategies under average case. The utility function does not satisfy the Bellman's principle and we employ the extended HJB equation proposed in Björk, T. & Murgoci, A. [(2014). A theory of Markovian time-inconsistent stochastic control in discrete time. Finance and Stochastics 18(3), 545–592] to solve this problem. In the end of this paper, we derive the equilibrium reinsurance and investment strategies under smooth ambiguity and present the sensitivity analysis to show the AAI's economic behaviors.     

Optimal investment of an insurer with regime-switching and risk constraint

We investigate an optimal investment problem of an insurance company in the presence of risk constraint and regime-switching using a game theoretic approach. A dynamic risk constraint is considered where we constrain the uncertainty aversion to the ‘true’ model for financial risk at a given level. We describe the surplus of an insurance company using a general jump process, namely, a Markov-modulated random measure. The insurance company invests the surplus in a risky financial asset whose dynamics are modeled by a regime-switching geometric Brownian motion. To incorporate model uncertainty, we consider a robust approach, where a family of probability measures is cosidered and the insurance company maximizes the expected utility of terminal wealth in the ‘worst-case’ probability scenario. The optimal investment problem is then formulated as a constrained two-player, zero-sum, stochastic differential game between the insurance company and the market. Different from the other works in the literature, our technique is to transform the problem into a deterministic differential game first, in order to obtain the optimal strategy of the game problem explicitly.     

A stochastic differential game for optimal investment of an insurer with regime switching

We introduce a model to discuss an optimal investment problem of an insurance company using a game theoretic approach. The model is general enough to include economic risk, financial risk, insurance risk, and model risk. The insurance company invests its surplus in a bond and a stock index. The interest rate of the bond is stochastic and depends on the state of an economy described by a continuous-time, finite-state, Markov chain. The stock index dynamics are governed by a Markov, regime-switching, geometric Brownian motion modulated by the chain. The company receives premiums and pays aggregate claims. Here the aggregate insurance claims process is modeled by either a Markov, regime-switching, random measure or a Markov, regime-switching, diffusion process modulated by the chain. We adopt a robust approach to model risk, or uncertainty, and generate a family of probability measures using a general approach for a measure change to incorporate model risk. In particular, we adopt a Girsanov transform for the regime-switching Markov chain to incorporate model risk in modeling economic risk by the Markov chain. The goal of the insurance company is to select an optimal investment strategy so as to maximize either the expected exponential utility of terminal wealth or the survival probability of the company in the ‘worst-case’ scenario. We formulate the optimal investment problems as two-player, zero-sum, stochastic differential games between the insurance company and the market. Verification theorems for the HJB solutions to the optimal investment problems are provided and explicit solutions for optimal strategies are obtained in some particular cases.     

Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria

In this paper, an ambiguity-averse insurer (AAI) whose surplus process is approximated by a Brownian motion with drift, hopes to manage risk by both investing in a Black–Scholes financial market and transferring some risk to a reinsurer, but worries about uncertainty in model parameters. She chooses to find investment and reinsurance strategies that are robust with respect to this uncertainty, and to optimize her decisions in a mean-variance framework. By the stochastic dynamic programming approach, we derive closed-form expressions for a robust optimal benchmark strategy and its corresponding value function, in the sense of viscosity solutions, which allows us to find a mean-variance efficient strategy and the efficient frontier. Furthermore, economic implications are analyzed via numerical examples. In particular, our conclusion in the mean-variance framework differs qualitatively, for certain parameter ranges, with model-uncertainty robustness conclusions in the framework of utility functions: model uncertainty does not always result in an agent deciding to reduce risk exposure under mean-variance criteria, opposite to the conclusions for utility functions in Maenhout and Liu. Our conclusion can be interpreted as saying that the mean-variance problem for the AAI explains certain counter-intuitive investor behaviors, by which the attitude to risk exposure, for an AAI facing model uncertainty, depends on positive past experience.     

Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps

This paper considers a robust optimal excess-of-loss reinsurance-investment problem in a model with jumps for an ambiguity-averse insurer (AAI), who worries about ambiguity and aims to develop a robust optimal reinsurance-investment strategy. The AAI’s surplus process is assumed to follow a diffusion model, which is an approximation of the classical risk model. The AAI is allowed to purchase excess-of-loss reinsurance and invest her surplus in a risk-free asset and a risky asset whose price is described by a jump-diffusion model. Under the criterion for maximizing the expected exponential utility of terminal wealth, optimal strategy and optimal value function are derived by applying the stochastic dynamic programming approach. Our model and results extend some of the existing results in the literature, and the economic implications of our findings are illustrated. Numerical examples show that considering ambiguity and reinsurance brings utility enhancements.     

A general framework for the derivation of asset price bounds: an application to stochastic volatility option models

We present a generalization of Cochrane and Saá-Requejo’s good-deal bounds which allows to include in a flexible way the implications of a given stochastic discount factor model. Furthermore, a useful application to stochastic volatility models of option pricing is provided where closed-form solutions for the bounds are obtained. A calibration exercise demonstrates that our benchmark good-deal pricing results in much tighter bounds. Finally, a discussion of methodological and economic issues is also provided.      

Numerical methods for dynamic Bertrand oligopoly and American options under regime switching

Bertrand oligopolies are competitive markets in which a small number of firms producing similar goods use price as their strategic variable. In particular, each firm wants to determine the optimal price that maximizes its expected discounted lifetime profit. The oligopoly problem can be modeled as nonzero-sum games which can be formulated as systems of Hamilton–Jacobi–Bellman (HJB) partial differential equations (PDEs). In this paper, we propose fully implicit, positive coefficient finite difference schemes that converge to the viscosity solution for the HJB PDE from dynamic Bertrand monopoly and the two-dimensional HJB system from dynamic Bertrand duopoly. Furthermore, we develop fast multigrid methods for solving these systems of discrete nonlinear HJB PDEs. The new multigrid methods are general and can be applied to other systems of HJB and HJB-Isaacs PDEs arising from American options under regime switching and American options with unequal lending/borrowing rates and stock borrowing fees under regime switching, respectively. We provide a theoretical analysis for the smoother, restriction and interpolation operators of the multigrid methods. Finally, we demonstrate the effectiveness of our method by numerical examples from the dynamic Bertrand problem and pricing American options under regime switching.     

The supermartingale property of the optimal wealth process for general semimartingales

We consider an incomplete stochastic financial market where the price processes are described by a vector valued semimartingale that is possibly non locally bounded. We face the classical problem of utility maximization from terminal wealth, under the assumption that the utility function is finite-valued and smooth on the entire real line and satisfies reasonable asymptotic elasticity. In this general setting, it was shown in Biagini and Frittelli (Financ. Stoch. 9, 493–517, 2005) that the optimal claim admits an integral representation as soon as the minimax σ-martingale measure is equivalent to the reference probability measure. We show that the optimal wealth process is in fact a supermartingale with respect to every σ-martingale measure with finite generalized entropy, thus extending the analogous result proved by Schachermayer (Financ. Stoch. 4, 433–457, 2003) for the locally bounded case.      

On the numerical stability of simulation methods for SDEs under multiplicative noise in finance

When simulating discrete-time approximations of solutions of stochastic differential equations (SDEs), in particular martingales, numerical stability is clearly more important than some higher order of convergence. Discrete-time approximations of solutions of SDEs with multiplicative noise, similar to the Black–Scholes model, are widely used in simulation in finance. The stability criterion presented in this paper is designed to handle both scenario simulation and Monte Carlo simulation, i.e. both strong and weak approximations. Methods are identified that have the potential to overcome some of the numerical instabilities experienced when using the explicit Euler scheme. This is of particular importance in finance, where martingale dynamics arise frequently and the diffusion coefficients are often multiplicative. Stability regions for a range of schemes are visualized and analysed to provide a methodology for a better understanding of the numerical stability issues that arise from time to time in practice. The result being that schemes that have implicitness in the approximations of both the drift and the diffusion terms exhibit the largest stability regions. Most importantly, it is shown that by refining the time step size one can leave a stability region and may face numerical instabilities, which is not what one is used to experiencing in deterministic numerical analysis.     

Asymptotic optimal investment under interest rate for a class of subexponential distributions

We consider a classical risk model with the possibility of investment and positive interest rate for the riskless bond. The stock price movement is modelled as a geometric Brownian motion, the claim sizes are assumed to have a distribution belonging to a certain subclass of subexponential distributions. In this setting, we study the asymptotic behaviour of the optimal investment strategy under the ruin probability as a risk measure. This problem has been already considered before, but no results were obtained, for instance, for Weibull and Benktander-type-II distributions with certain parameters. We introduce a method which closes this gap.     

Change analysis of a dynamic copula for measuring dependence in multivariate financial data

This paper proposes a new approach to measure dependencies in multivariate financial data. Data in finance and insurance often cover a long time period. Therefore, the economic factors may induce some changes within the dependence structure. Recently, two methods have been proposed using copulas to analyse such changes. The first approach investigates changes within the parameters of the copula. The second determines the sequence of copulas using moving windows. In this paper we take into account the non-stationarity of the data and analyse the impact of (1) time-varying parameters for a copula family, and (2) the sequence of copulas, on the computations of the VaR and ES measures. We propose tests based on conditional copulas and the goodness-of-fit to decide the type of change, and further give the corresponding change analysis. We illustrate our approach using the Standard & Poor 500 and Nasdaq indices in order to compute risk measures using the two previous methods.     

Asymptotics of bond yields and volatilities for extended CIR models under the real-world measure

ABSTRACT

The Cox–Ingersoll–Ross CIR short rate model is a mean-reverting model of the short rate which, for suitably chosen parameters, permits closed-form valuation formulae of zero-coupon bonds and options on zero-coupon bonds. This article supplies proofs of the formulae for the expected present value of payoffs under the real-world probability measure, known as actuarial valuation. Importantly, we give formulae for asymptotic levels of bond yields and volatilities for extended CIR models when suitable conditions are imposed on the model parameters.