Optimal portfolio selection in nonlinear arbitrage spreads

This paper analytically solves the portfolio optimization problem of an investor faced with a risky arbitrage opportunity (e.g. relative mispricing in equity pairs). Unlike the extant literature, which typically models mispricings through the Ornstein–Uhlenbeck (OU) process, we introduce a nonlinear generalization of OU which jointly captures several important risk factors inherent in arbitrage trading. While these factors are absent from the standard OU, we show that considering them yields several new insights into the behavior of rational arbitrageurs: Firstly, arbitrageurs recognizing these risk factors exhibit a diminishing propensity to exploit large mispricings. Secondly, optimal investment behavior in light of these risk factors precipitates the gradual unwinding of losing trades far sooner than is entailed in existing approaches including OU. Finally, an empirical application to daily FTSE100 pairs data shows that incorporating these risks renders our model's risk-management capabilities superior to both OU and a simple threshold strategy popular in the literature. These observations are useful in understanding the role of arbitrageurs in enforcing price efficiency.     

Traditional versus non-traditional reinsurance in a dynamic setting

We consider a stochastic risk reserve process whose risk exposure can be controlled dynamically by applying proportional reinsurance and by issuing CAT Bonds. The CAT Bond payments are only partly correlated with the insurers losses. The aim is to minimize the probability of ruin. Using a two-dimensional diffusion approximation we obtain a controlled diffusion problem which can be solved explicitly with the help of the HJB equation. We present some numerical results and discuss to which extend the proportional reinsurance can be replaced by issuing CAT Bonds.     

HARA utility maximization in a Markov-switching bond–stock market

We present a flexible multidimensional bond–stock model incorporating regime switching, a stochastic short rate and further stochastic factors, such as stochastic asset covariance. In this framework we consider an investor whose risk preferences are characterized by the hyperbolic absolute risk-aversion utility function and solve the problem of optimizing the expected utility from her terminal wealth. For the optimal portfolio we obtain a constant-proportion portfolio insurance-type strategy with a Markov-switching stochastic multiplier and prove that it assures a lower bound on the terminal wealth. Explicit and easy-to-use verification theorems are proven. Furthermore, we apply the results to a specific model. We estimate the model parameters and test the performance of the derived optimal strategy using real data. The influence of the investor’s risk preferences and the model parameters on the portfolio is studied in detail. A comparison to the results with the power utility function is also provided.     

Optimal investment-reinsurance with dynamic risk constraint and regime switching

We study an optimal investment–reinsurance problem for an insurer who faces dynamic risk constraint in a Markovian regime-switching environment. The goal of the insurer is to maximize the expected utility of terminal wealth. Here the dynamic risk constraint is described by the maximal conditional Value at Risk over different economic states. The rationale is to provide a prudent investment–reinsurance strategy by taking into account the worst case scenario over different economic states. Using the dynamic programming approach, we obtain an analytical solution of the problem when the insurance business is modeled by either the classical Cramer–Lundberg model or its diffusion approximation. We document some important qualitative behaviors of the optimal investment–reinsurance strategies and investigate the impacts of switching regimes and risk constraint on the optimal strategies.     

Optimal Proportional Reinsurance Policies in a Dynamic Setting

We consider dynamic proportional reinsurance strategies and derive the optimal strategies in a diffusion setup and a classical risk model. Optimal is meant in the sense of minimizing the ruin probability. Two basic examples are discussed.     

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.     

On a renewal risk process with dependence under a Farlie–Gumbel–Morgenstern copula

In this article, we consider an extension to the renewal or Sparre Andersen risk process by introducing a dependence structure between the claim sizes and the interclaim times through a Farlie–Gumbel–Morgenstern copula proposed by Cossette et al. (2010) for the classical compound Poisson risk model. We consider that the inter-arrival times follow the Erlang(n) distribution. By studying the roots of the generalised Lundberg equation, the Laplace transform (LT) of the expected discounted penalty function is derived and a detailed analysis of the Gerber–Shiu function is given when the initial surplus is zero. It is proved that this function satisfies a defective renewal equation and its solution is given through the compound geometric tail representation of the LT of the time to ruin. Explicit expressions for the discounted joint and marginal distribution functions of the surplus prior to the time of ruin and the deficit at the time of ruin are derived. Finally, for exponential claim sizes explicit expressions and numerical examples for the ruin probability and the LT of the time to ruin are given.