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## A Guaranteed Deterministic Approach to Superhedging—The Case of Convex Payoff Functions on Options

This paper considers super-replication in a guaranteed deterministic problem setting with discrete time. The aim of hedging a contingent claim is to ensure the coverage of possible payoffs under the option contract for all admissible scenarios. These scenarios are given by means of a priori given compacts that depend on the history of prices. The increments of the price at each moment in time must lie in the corresponding compacts. The absence of transaction costs is assumed. The game–theoretic interpretation of pricing American options implies that the corresponding Bellman–Isaacs equations hold for both pure and mixed strategies. In the present paper, we study some properties of the least favorable (for the “hedger”) mixed strategies of the “market” and of their supports in the special case of convex payoff functions.

Internal rate of return *IRR *is one of the key criteria for justifying and choosing capital investments with conventional cash flows. However, this criterion is not practically used when the rate of return of investment instruments (short sales, options, futures, swaps) is calculated because these instruments create non-conventional cash flows. The author previously showed that *IRR* problems were observed when the present value of cash flows changed sign from period to period. This paper offers a criterion to evaluate the rate of return of investment instruments with non-conventional cash flows, i.e. General Rate of Return (*GRR*).

For matrix games we study how small nonzero probability must be used in optimal strategies. We show that for n×n win–lose–draw games (i.e. (−1,0,1) matrix games) nonzero probabilities smaller than n−O(n)are never needed. We also construct an explicit n×n win–lose game such that the unique optimal strategy uses a nonzero probability as small as n−Ω(n). This is done by constructing an explicit (−1,1)nonsingular n×n matrix, for which the inverse has only nonnegative entries and where some of the entries are of value nΩ(n).

For the discrete-time superreplication problem, a guaranteed deterministic formulation is proposed: the problem is to ensure the cheapest coverage of the contingent claim on an American option under all admissible scenarios. These scenarios are set by a priori defined compacts depending on the price history; the price increment at each moment of time must lie in the corresponding compact. The market is considered without trading constraints and transaction costs. The problem statement is game-theoretic in nature and leads directly to the Bellman–Isaacs equations of a special form under the assumption of no trading constraints. In the present study, we estimate the modulus of continuity of uniformly continuous solutions, including the Lipschitz case.

We study game equilibria in a model of production and externalities in network with three types of agents who possess different productivities. Each agent may invest a part of her endowment (for instance, time or money) on the first stage; consumption on the second period depends on her own investment and productivity as well as on the investments of her neighbors in the network. Three ways of agent’s behavior are possible: passive (no investment), active (a part of endowment is invested) and hyperactive (the whole endowment is invested). We introduce adjustment dynamics and study what equilibria are possible and under which conditions. We study also which of these equilidria are stable and under which correlations of parameters of the network. We introduce adjustment dynamics described by differential equations and find conditions of existence and dynamical stability of equilibria in regular networks with three types of agents.

In the paper the investment and speculative strategies based on selling naked options are discussed. The risk of such decisions was assessed , advantages and disadvantages of these investment decisions compared to other option strategies was analyzed.

The paper examines the structure, governance, and balance sheets of state-controlled banks in Russia, which accounted for over 55 percent of the total assets in the country's banking system in early 2012. The author offers a credible estimate of the size of the country's state banking sector by including banks that are indirectly owned by public organizations. Contrary to some predictions based on the theoretical literature on economic transition, he explains the relatively high profitability and efficiency of Russian state-controlled banks by pointing to their competitive position in such functions as acquisition and disposal of assets on behalf of the government. Also suggested in the paper is a different way of looking at market concentration in Russia (by consolidating the market shares of core state-controlled banks), which produces a picture of a more concentrated market than officially reported. Lastly, one of the author's interesting conclusions is that China provides a better benchmark than the formerly centrally planned economies of Central and Eastern Europe by which to assess the viability of state ownership of banks in Russia and to evaluate the country's banking sector.

The paper examines the principles for the supervision of financial conglomerates proposed by BCBS in the consultative document published in December 2011. Moreover, the article proposes a number of suggestions worked out by the authors within the HSE research team.