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We consider the problem of optimal consumption of multiple goods in incomplete semimartingale markets. We formulate the dual problem and identify conditions that allow for the existence and uniqueness of the solution, and provide a characterization of the optimal consumption strategy in terms of the dual optimizer. We illustrate our results with examples in both complete and incomplete models. In particular, we construct closed-form solutions in some incomplete models.
We considered a pension fund that needs to hedge uncertain long-term liabilities. We modeled the pension fund as a robust investor facing an incomplete market and fearing model uncertainty for the evolution of its liabilities. The robust agent is assumed to minimize the shortfall between the assets and liabilities under an endogenous worst-case scenario by means of solving a min–max robust optimization problem. When the funding ratio is low, robustness reduces the demand for risky assets. However, cherishing the hope of covering the liabilities, a substantial risk exposure is still optimal. A longer investment horizon or a higher funding ratio weakens the investor's fear of model misspecification. If the expected equity return is overestimated, the initial capital requirement for hedging can be decreased by following the robust strategy.
We consider the problem of optimal investment with intermediate consumption in a general semimartingale model of an incomplete market, with preferences being represented by a utility stochastic field. We show that the key conclusions of the utility maximization theory hold under the assumptions of no unbounded profit with bounded risk and of the finiteness of both primal and dual value functions.
On an incomplete financial market, the stocks are modeled as pure jump processes subject to defaults. The exponential utility maximization problem is investigated characterizing the value function in term of Backward Stochastic Differential Equations (BSDEs), driven by pure jump processes. In general, in this setting, there is no unique solution. This is the reason why, the value function is proven to be the limit of a sequence of processes. Each of them is the solution of a Lipschitz BSDE and it corresponds to the value function associated with a subset of bounded admissible strategies. Given a representation of the jump processes driving the model, the aim of this note is to give a recursive backward scheme for the value function of the initial problem.
Consider a financial market in which an agent trades with utility-induced restrictions on wealth. For a utility function which satisfies the condition of reasonable asymptotic elasticity at -∞, we prove that the utility-based superreplication price of an unbounded (but sufficiently integrable) contingent claim is equal to the supremum of its discounted expectations under pricing measures with finite loss-entropy. For an agent whose utility function is unbounded from above, the set of pricing measures with finite loss-entropy can be slightly larger than the set of pricing measures with finite entropy. Indeed, the former set is the closure of the latter under a suitable weak topology. Central to our proof is a proof of the duality between the cone of utility-based superreplicable contingent claims and the cone generated by pricing measures with finite loss-entropy.
In this paper we propose a discrete-time model with fixed maximum time to maturity of traded bonds. At each trading time, a bond matures and a new bond is introduced in the market, such that the number of traded bonds is constant. The entry price of the newly issued bond depends on the prices of the bonds already traded and a stochastic term independent of the existing bond prices. Hence, we obtain a bond market model for the reinvestment risk, which is present in practice, when hedging long term contracts. In order to determine optimal hedging strategies we consider the criteria of super-replication and risk-minimization.
For the martingale case Föllmer and Sondermann (1986) introduced a unique admissible risk-minimizing hedging strategy for any square-integrable contingent claim H. Schweizer (1991) developed their theory further to the semimartingale case introducing the notion of local risk-minimization. Møller (2001) extended the theory of Föllmer and Sondermann (1986) to hedge general payment processes occurring mainly in insurance. We expand local risk-minimization to the theory of hedging general payment processes and derive such a hedging strategy for general unit-linked life insurance contracts in a general Lévy process financial market.
The aim of the paper is twofold. Firstly, to analyze the historical data of the earthquakes in the boarder area of Greece and then to produce a reliable model for the risk dynamics of the magnitude of the earthquakes, using advanced techniques from the Extreme Value Theory. Secondly, to discuss briefly the relevant theory of incomplete markets and price earthquake catastrophe bonds, combining the model found for the earthquake risk and an appropriate model for the interest rate dynamics in an incomplete market framework. The paper ends by providing some numerical results using Monte Carlo simulation techniques and stochastic iterative equations.
In an incomplete financial market in which the dynamics of the asset prices is driven by a d-dimensional continuous semimartingale X, we consider the problem of pricing European contingent claims embedded in a power utility framework. This problem reduces to identifying the p-optimal martingale measure, which can be given in terms of the solution to a semimartingale backward equation. We use this characterization to examine two extreme cases. In particular, we find a necessary and sufficient condition, written in terms of the mean-variance trade-off, for the p-optimal martingale measure to coincide with the minimal martingale measure. Moreover, if and only if an exponential function of the mean-variance trade-off is a martingale strongly orthogonal to the asset price process, the p-optimal martingale measure can be simply expressed in terms of a Doléans-Dade exponential involving X.
We consider an incomplete market model whose stock price fluctuation is given by a jump diffusion process. For this model, we calculate the density process of the minimal martingale measure. Also, we state the relation to a locally risk-minimizing strategy.
This paper reviews methods for hedging and valuation of insurance claims with an inherent financial risk, with special emphasis on quadratic hedging approaches and indifference pricing principles and their applications in insurance. It thus addresses aspects of the interplay between finance and insurance, an area which has gained considerable attention during the past years, in practice as well as in theory. Products combining insurance risk and financial risk have gained considerable market shares. Special attention is paid to unit-linked life insurance contracts, and it is demonstrated how these contracts can be valued and hedged by using traditional methods as well as more recent methods from incomplete financial markets such as risk-minimisation, mean-variance hedging, super-replication and indifference pricing with mean-variance utility functions.
A unit-linked life insurance contract is a contract where the insurance benefits depend on the price of some specific traded stocks. We consider a model describing the uncertainty of the financial market and a portfolio of insured individuals simultaneously. Due to incompleteness the insurance claims cannot be hedged completely by trading stocks and bonds only, leaving some risk to the insurer. The theory of risk-minimization is briefly reviewed and applied after a change of measure. Risk-minimizing trading strategies and the associated intrinsic risk processes are determined for different types of unit-linked contracts. By extending the model to the situation where certain reinsurance contracts on the insured lives are traded, the direct insurer can eliminate the risk completely. The corresponding self-financing strategies are determined.
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