Guaranteed Deterministic Approach to Superhedging: Lipschitz Properties of Solutions of the Bellman–Isaacs Equations

Smirnov, S. N.

doi:10.1007/978-3-030-23699-1_14

Cited by 18 publications

(24 citation statements)

References 8 publications

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“…The present article continues the series of publications by the author [1][2][3][4][5][6][7][8]. These works develop a model of the financial market with an uncertain deterministic evolution of prices and discrete time.…”

Section: Introductionmentioning

confidence: 91%

“…. , x t−1 ) presents the price history up to the current moment t. The conditions for the validity of (1) are formulated in Theorem 3.1 in [1]. Assume for convenience that g 0 = −∞, that is, there is no claim on the option at the initial moment; g t ≥ 0 for t = 1, .…”

Section: Introductionmentioning

confidence: 99%

“…Note that one of the important conditions for the validity of (1), formulated in Theorem 3.1 in Reference [1], is the boundedness of payoff functions g t , which ensures that functions v * t are bounded from above:…”

Section: Introductionmentioning

confidence: 99%

“…Let us now sketch the content of the above-mentioned series of our papers. In Reference [1], we describe in detail the guaranteed deterministic approach, refer to the relevant literature, and formulate a corresponding financial market model with trading constraints. We also frame the problem of superhedging contingent claims on options, obtain the Bellman-Isaacs equations (1), and discuss what kinds of economic assumptions are needed to ensure the equation's validity.…”

Section: Introductionmentioning

confidence: 99%

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

Smirnov

2019

Mathematics

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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.

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Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Smirnov

2019

Mathematics

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“…We adapt the framework of Smirnov [13,14] and introduce a portfolio optimization problem as a game against nature where an investor tries to maximize the expected worst-case reward from a portfolio by making general and non-restrictive assumptions about the market. The robust optimization techniques have been introduced for a variety of economic problems recently.…”

Section: Introductionmentioning

confidence: 99%

Robust Portfolio Optimization in an Illiquid Market in Discrete-Time

Andreev

2019

Mathematics

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We present a robust dynamic programming approach to the general portfolio selection problem in the presence of transaction costs and trading limits. We formulate the problem as a dynamic infinite game against nature and obtain the corresponding Bellman-Isaacs equation. Under several additional assumptions, we get an alternative form of the equation, which is more feasible for a numerical solution. The framework covers a wide range of control problems, such as the estimation of the portfolio liquidation value, or portfolio selection in an adverse market. The results can be used in the presence of model errors, non-linear transaction costs and a price impact.

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