scholarly journals Guaranteed Deterministic Approach to Superhedging: Case of Binary European Option

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Sergey N. Smirnov ◽  
Andrey Yu. Zanochkin
Keyword(s):  
A Priori ◽  
Payoff Function ◽  
Market Model ◽  
Solution Properties ◽  
Deterministic Approach ◽  
European Option ◽  
Bellman Equations ◽  
Contingent Liability ◽  
Game Theoretic ◽  
Trading Constraints

For the superreplication problem with discrete time, a guaranteed deterministic formulation is considered: the problem is to guarantee coverage of the contingent liability on sold option under all admissible scenarios. These scenarios are defined by means of a priori defined compacts dependent on price prehistory: the price increments at each point in time must lie in the corresponding compacts. In a general case, we consider a market with trading constraints and assume the absence of transaction costs. The formulation of the problem is game theoretic and leads to the Bellman–Isaacs equations. This paper analyses the solution to these equations for a specific pricing problem, i.e., for a binary option of the European type, within a multiplicative market model, with no trading constraints. A number of solution properties and an algorithm for the numerical solution of the Bellman equations are derived. The interest in this problem, from a mathematical prospective, is related to the discontinuity of the option payoff function.

scholarly journals A guaranteed deterministic approach to superhedging: mixed strategies and game equilibrium

2020 ◽  
Vol 12 (1) ◽  
pp. 60-90
Author(s):  
Сергей Николевич Смирнов ◽  
Sergey Smirnov
Keyword(s):  
Transaction Costs ◽  
A Priori ◽  
Contingent Claim ◽  
Mixed Strategies ◽  
Deterministic Approach ◽  
Game Equilibrium ◽  
Game Theoretic ◽  
Price History ◽  
Pure Strategies ◽  
Trading Constraints

For a discrete-time superreplication problem, a guaranteed deterministic formulation is considered: the problem is to ensure a cheapest coverage of the contingent claim on an option under all scenarios which are set using a priori defined compacts, depending on the price history: price increments at each moment of time must lie in the corresponding compacts. The market is considered with trading constraints and without transaction costs. The statement of the problem is game-theoretic in nature and leads directly to the Bellman - Isaacs equations. In this article, we introduce a mixed extension of the ``market'' pure strategies. Several results concerning game equilibrium are obtained.

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

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1246 ◽  
Author(s):  
Sergey Smirnov
Keyword(s):  
A Priori ◽  
Contingent Claim ◽  
Mixed Strategies ◽  
Deterministic Approach ◽  
Deterministic Problem ◽  
History Of ◽  
Game Theoretic ◽  
Payoff Functions ◽  
Special Case ◽  
Problem Setting

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.

AN EXTENSION OF THE BRODY–HUGHSTON–MACRINA APPROACH TO MODELING OF DEFAULTABLE BONDS

2007 ◽  
Vol 10 (03) ◽  
pp. 557-589 ◽  
Author(s):  
MAREK RUTKOWSKI ◽  
NANNAN YU
Keyword(s):  
Credit Risk ◽  
Interest Rates ◽  
Explicit Expression ◽  
Market Model ◽  
Derivative Securities ◽  
Risk Modeling ◽  
European Option ◽  
Bond Price ◽  
Information Process ◽  
Defaultable Bond

The innovative information-based framework for credit risk modeling, proposed recently by Brody, Hughston, and Macrina, is extended to a more general and practically important setup of random interest rates. We first introduce the market model, and we derive an explicit expression for defaultable bond price. Next, the dynamics of the information process and dynamics of defaultable bond are found for both deterministic and random interest rates. Finally, the valuation and hedging of derivative securities are briefly examined. In particular, the valuation formula for a European option on a defaultable bond is established.

scholarly journals Seeking Negative Alphas Through Shorting

2017 ◽  
Vol 18 (6) ◽  
pp. 1488-1506
Author(s):  
Azhar Mohamad
Keyword(s):  
Factor Model ◽  
Stock Exchange ◽  
A Priori ◽  
Abnormal Returns ◽  
Market Model ◽  
Capital Asset Pricing ◽  
Short Sellers ◽  
Complete Market ◽  
Short Interest ◽  
Buy Back

Shorting involves selling stocks that one does not own. Advocates of shorting argue that it is needed to make the financial market a two-way (complete) market in which investors with bearish opinions can participate. To gain from shorting, short sellers hope to buy back the shorted stocks at a lower price. Obtaining ‘negative’ alphas or abnormal returns is thus desirable for short sellers as they imply the underperformance of the stocks and that a profit has been realized. Abnormal returns, according to Fama (1998), are anomalies that tend to disappear when reasonable changes are made to the methodology used to measure them. Diamond and Verrecchia (1987), however, theorize and argue a priori that an unusually large increase in short interest will be followed by a period of negative abnormal returns. Short interest is equal to the number of shorted shares divided by the number of shares available to be shorted. Using daily short interest data for stocks traded on the London Stock Exchange for the period of September 2003 to April 2010, we employ an event study to investigate the effects that follow shorting. Alphas and abnormal returns are measured according to the Market Model (MM), the Capital Asset Pricing Model (CAPM) and the Fama–French Three-factor Model (FF3F), and are estimated using different estimation windows of 60 and 120 days. In all the methodologies under study, we find significant negative alphas following shorting.

scholarly journals A non-deterministic approach to forecasting the trophic evolution of lakes

2016 ◽  
Vol 75 (s1) ◽  
Author(s):  
Roberto Bertoni ◽  
Martino Bertoni ◽  
Giuseppe Morabito ◽  
Michela Rogora ◽  
Cristiana Callieri
Keyword(s):  
Genetic Algorithms ◽  
A Priori ◽  
Predictive Ability ◽  
Data Series ◽  
Deterministic Approach ◽  
Ecological Data ◽  
Natural Systems ◽  
Long Time ◽  
Negative Side Effects ◽  
Limnological Parameters

<p>Limnologists have long recognized that one of the goals of their discipline is to increase its predictive capability. In recent years, the role of prediction in applied ecology escalated, mainly due to man’s increased ability to change the biosphere. Such alterations often came with unplanned and noticeably negative side effects mushrooming from lack of proper attention to long-term consequences. Regression analysis of common limnological parameters has been successfully applied to develop predictive models relating the variability of limnological parameters to specific key causes. These approaches, though, are biased by the requirement of a priori cause-relation assumption, oftentimes difficult to find in the complex, nonlinear relationships entangling ecological data. A set of quantitative tools that can help addressing current environmental challenges avoiding such restrictions is currently being researched and developed within the framework of ecological informatics. One of these approaches attempting to model the relationship between a set of inputs and known outputs, is based on genetic algorithms and programming (GP). This stochastic optimization tool is based on the process of evolution in natural systems and was inspired by a direct analogy to sexual reproduction and Charles Darwin’s principle of natural selection. GP works through genetic algorithms that use selection and recombination operators to generate a population of equations. Thanks to a 25-years long time-series of regular limnological data, the deep, large, oligotrophic Lake Maggiore (Northern Italy) is the ideal case study to test the predictive ability of GP. Testing of GP on the multi-year data series of this lake has allowed us to verify the forecasting efficacy of the models emerging from GP application. In addition, this non-deterministic approach leads to the discovery of non-obvious relationships between variables and enabled the formulation of new stochastic models.</p>

Numerically Stable Design Optimization With Price Competition

2014 ◽  
Vol 136 (8) ◽  
Author(s):  
W. Ross Morrow ◽  
Joshua Mineroff ◽  
Kate S. Whitefoot
Keyword(s):  
Price Competition ◽  
Performance Metrics ◽  
Consumer Preferences ◽  
Market Model ◽  
Optimal Designs ◽  
Future Price ◽  
Design Decisions ◽  
Computational Optimization ◽  
Game Theoretic ◽  
The Impact

Researchers in decision-based design (DBD) have suggested that business objectives, e.g., profits, should replace engineering requirements or performance metrics as the objective for engineering design. This requires modeling market performance, including consumer preferences and competition between firms. Game-theoretic “design-then-pricing” models—i.e., product design anticipating future price competition–provide an important framework for integrating consumer preferences and competition when design decisions must be made before prices are decided by a firm or by its competitors. This article concerns computational optimization in a design-then-pricing model. We argue that some approaches may be fundamentally difficult for existing solvers and propose a method that exhibits both improved efficiency and reliability relative to existing methods. Numerical results for a vehicle design example validate our theoretical arguments and examine the impact of anticipating pricing competition on design decisions. We find that anticipating pricing competition, while potentially important for accurately forecasting profits, does not necessarily have a significant effect on optimal design decisions. Most existing examples suggest otherwise, anticipating competition in prices is important to choosing optimal designs. Our example differs in the importance of design constraints, that reduce the influence the market model has on optimal designs.

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