Christos E. Kountzakis

Restricted Coherent Risk Measures and Actuarial Solvency

We prove a general dual representation form for restricted coherent risk measures, and we apply it to a minimization problem of the required solvency capital for an insurance company.

Nonreplication of Options

In this paper, we study the replication of options in security markets X with a finite number of states. Specifically, we prove that in security markets without binary vectors, for any portfolio, at most options can be replicated where m is the number of states. This is an essential improvement of the result of Baptista where it is proved that the set of replicated options is of measure zero. Additionally, we extend the results of Aliprantis and Tourky on the nonreplication of options by generalizing their condition that markets are strongly resolving. Our results are based on the theory of lattice‐subspaces and positive bases.

Minimum Regret Pricing of Contingent Claims in Incomplete Markets

In this paper we propose a contingent claim pricing scheme between two counterparties in an incomplete one period market. According to our approach the two counterparties of a non-marketed contingent claim select a pair of pricing kernels, in order to agree on a common price, by minimizing their joint regret function, which quantifies the departure from their initial beliefs. The joint regret function is a convex combination of entropy-like or norm-dependent functionals. The relevant optimization problem is posed in terms of a partially finite convex programming problem in the space of pricing kernels.

A use of Black-Scholes model in market risk

The aim of this paper is to use the Black-Scholes model for market risk by using the estimated drift and volatility of a stock-return for a relatively small time-horizon, in case where the historical returns are better-fitted to a normal distribution. In this case, we may use the infinitesimal operator of the complete market formulated by the stock and the numeraire by taking the interest rate also constant and equal to the mean reference rate for the turning of theoretical and historical VaR to an economic capital functional.

RAROC in portfolio optimization

In this paper, we provide the implications of using the performance ratio being defined by the expected shortfall-well known as RAROC-in static portfolio optimization, by giving the proof of the relevant results. We also use RAROC as a primary function in the AUGMECON algorithm, providing the main scheme of such an application on data from Athens Stock Exchange. The use of RAROC in AUGMECON as a primary function is also faced as an optimization problem under a general nonconvex optimization framework.

Coherent Risk Measures Under Dominated Variation

We study the relation between the properties of the coherent risk measures and of the heavy-tailed distributions from radial subsets of random variables. As a result, a new risk measure is introduced for this type of random variable. Under the assumptions of the Lundberg and renewal risk models, the solvency capital in the class of distributions with dominatedly varying tails is calculated. Further, the existence and uniqueness of the solution in the optimisation problem, associated to the minimisation of the risk over a set of financial positions, is investigated. The optimisation results hold on the \(L^{1+\varepsilon }\)-spaces, for any \(\varepsilon \ge 0\), but the uniqueness collapses on \(L^{1}\), the canonical space for the law-invariant coherent risk measures.

Monetary Risk Functionals on Orlicz Spaces Produced by Set-Valued Risk Maps and Random Measures

In this article we study the construction of coherent or convex risk functionals defined either on an Orlicz heart, either on an Orlicz space, with respect to a Young loss function. The Orlicz heart is taken as a subset of \(L^{0}(\varOmega, \mathcal{F}, \mu)\) endowed with the pointwise partial ordering. We define set-valued risk maps related to this partial ordering. We also derive monetary risk functionals both by the class of coherent set-valued risk maps defined on them. We also use random measures related to heavy-tailed distributions in order to define monetary risk functionals on Orlicz spaces, whose properties are also compared to the previous ones.

The Completion of Real-Asset Markets by Options

We combine the theory of finite-dimensional lattice subspaces and the theory of regular values for maps between smooth manifolds in order to study the completion of real asset markets by options. The strike asset of the options is supposed to be a nominal asset. The main result of the paper is like in the case of the completion of a nominal asset market by options that if the strike asset of the options is the riskless asset, then the completion of a real asset market is generically equal to ℝ𝑆.

No arbitrage pricing of non-marketed claims in multi-period markets

In this paper, we modify the arbitrage-free interval of prices for a non-marketed contingent claim in the finite event-tree model of financial markets, according to the perfect hedging approach being well-known for the two-period model. We prove the existence of solution to the corresponding sellers and buyers price problem for such a claim under no-arbitrage prices for the marketed contracts and we show that each of these problems can be solved by decomposing it into a finite number of one-period linear programming problems solved backwards. Finally, we indicate that the set of the no-arbitrage prices for a non-marketed contingent claim is the interval of the real numbers whose supremum and infimum is the sellers and the buyers price of the claim, respectively. The determination of the set of no-arbitrage prices for a non-marketed contingent claim is related to the utility pricing of such a claim.