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National Research UniversityHigher school of economicsZverev Oleg VladimirovichMinimax hedging of European option in incomplete marketSUMMARY of DISSERTATIONfor academic degree of Candidate of Sciencesin Applied Mathematics HSEAcademic superviser:Doctor of Physical-MathematicalSciences, ProfessorKhametov Vladimir MinirovichMoscow — 2018IntroductionThe PhD thesis is on theory of risky asset’s portfolio management in incompletemultidimensional markets without transaction costs with discrete time and final horizon. In the PhDthesis this theory was applied to solve some calculation problems for European options inincomplete multidimensional markets of risky assets.To outline the approach used in the PhD thesis let us give necessary definitions of option’stheory. Risky assets are objects with price evolving as adapted random sequences (stocks, forexample). Multidimensional market is a set of risky assets.
It is possible to describe fullymultidimensional market by probability distribution of these random sequences. Multidimensionalpredictable random sequences (with the same dimensions as markets) are called portfolio. Onlymarkets without transaction costs are considered, i.e. one doesn’t have to pay when transfer asset ofone type to another.European option is a contract, under which the Seller of assets (the Issuer) sells and theBuyer has right (but not obliged) to buy at fixed in advance price at the future moment stated in thecontract (called execution moment). Herewith to obtain this right the Buyer has to pay to the Issuera fee (money, for example) at the contract conclusion moment.
The fee is called option premium oroption value. When option is executed the Issuer has to deliver the assets to the Buyer, i.e. at themoment of execution the Issuer’s obligation appears. The Issuer has to fulfill the obligation. Thisobligation is called payoff. The payoff is a measurable function, possible, depending on all riskyasset’s prices up to the execution moment. So, to fulfill the obligation the Issuer has to construct aportfolio with capital not less than the value of obligation with given probability. Here a capital ofportfolio at any moment is a sum of products of quantities of risky assets and their prices, i.e.
thevalue of the portfolio.Note, there are arbitrage and arbitrage-free markets. As it is defined, in an arbitrage marketsthere is positive probability to make a profit with zero investments. Otherwise market is calledarbitrage-free. Simple criterion for arbitrage-free markets is known1: a market is arbitrage-free thenand only then risky asset’s prices do not change in mean while evolve.
This means, that randomsequences describing risky asset’s price evolution are martingales 2 . Appropriate probabilitymeasures are called martingale or risk-neutral measures. Arbitrage-free markets encompasscomplete and incomplete markets. Complete markets are defined by the fact that in such market anypayoff might be fulfilled almost surely. This means, that there is a portfolio of risky assets withcapital equal to value of the Issuer’s obligation.
Criterion for market completeness is known1: amarket is complete if and only if there is unique martingale measure. A complete market is1Shiryaev A.N. Essentials of Stochastic Finance. Vol. 2. Theory. Moscow: Fazis. 1998. - 1056 p. (inRussian)2Shiryaev A.N. Probability.
Moscow: Nauka. 1980. - 576 p. (in Russian).1idealization, which, as a rule, doesn’t have place, i.e. real multidimensional markets are incomplete.This means, that probability measure defining incomplete market is not unique. That is why theIssuer to fulfill the obligation has to: 1) chose probability measure with respect to which he or shewill calculate European option; 2) construct portfolio of risky assets assuring fulfillment of theobligation with given probability; 3) generate option’s value.Relevance of the topicTo calculate European option in incomplete market fair value principle is generally used 1,3.In contrast to above stated works, in the PhD thesis minimax principle has been used. This principlewas chose for the following reasons.
The Issuer don’t know probability distribution of risky asset’sprices a priori. Suppose, that Issuer’s risk function is exponential and depends on his profit. TheIssuer minimizes expected value of exponential risk. This might be gained by portfolio, whichenforces the Issuer to parry any unfavorable for him probability distribution of risky asset’s prices.To implement this principle one need to justify applicability for stochastic version of dynamicprogramming in the case than adapted sequence is observed and objective functional ismultiplicative. So, there is minimax problem of optimal stochastic portfolio management. Thisproblem has not been considered in scientific literature yet.Within the approach it has been managed to establish new existence conditions for: 1)optimal portfolios being predictable random sequences and are invariant with respect to anyequivalent probability measure; 2) uniform Doob decomposition with respect to any equivalentprobability measure for measurable bounded functionals set on trajectories of adapted randomsequences; 3) extreme measures delivering maximum value to expected risk and to find propertiesof these measures, to prove (for the first time), that initial incomplete market is complete withrespect to extreme measure.Above stated results allow to calculate constructively European option in incompletemarket.This justifies the topic and the results of the PhD thesis.Purposes of research are to find:(1) minimax value of issuer’s expected exponential risk,(2) constructive existence condition of hedging (superhedging, quantile hedging, quantilesuperhedging) portfolio for European option in incomplete market without transaction costs.As a rule in theory of European option’s calculation static problem is considered.
There in thePhD thesis the problem is considered in dynamics. So the scientific novelty of the PhD thesis isrelated to the following:3Bertsekas D., Shreve S. Stochastic Optimal Control. Moscow: Nauka. 1985. - 280 p. (in Russian).2(1)it is the first time when the applicability of the dynamic programming method for nonMarkov systems with multiplicative risk function has been justified for the case of discreet time.This allows to find out that evolution of upper guaranteed value for expected exponential issuer’srisk is submitted to Bellman’s type recurrent relation even if risk asset’s prices are presented bysemimartingales;(2)new conditions for existence of uniform Doob decomposition have been obtained;(3)conditions have been obtained for existence of superhedging, quantile superhedgingportfolios of European options in incomplete multidimensional markets without transaction costswith respect to any equivalent probability measure;(4)criterion have been constructed for existence of extreme probability measure delivering themaximum of expected exponential issuer’s value, characteristics of the extreme measure have beenexamined.The PhD thesis is a theoretical one.
It’s results belongs to the field of optimal stochasticcontrol. It is possible to use them in stochastic theory of optimal control as well as in stochasticfinancial mathematics. Theoretical significance of the results is justified by the following:(1)conditions have been obtained under which evolution of upper guaranteed value forexpected exponential issuer’s risk is submitted to Bellman’s type recurrent relation when riskasset’s prices are presented by semimartingales,(2)it is proved that any bounded payoff allows uniform Doob decomposition with respect toany probability measure from the set of equivalent probability measures,(3)criterion have been constructed for existence of extreme probability measure and portfoliodelivering minimax value for expected exponential issuer’s risk, moreover it has been proved thatwith respect to the measure initial incomplete market is a complete one,(4)it has been proved that for the case of incomplete multidimensional markets withouttransaction costs in discrete time it is possible to reduce problem of quantile hedging (quantilesuperhedging) to two problems of perfect hedging (superhedging).In the PhD thesis methods of functional analysis, probability theory, theory of stochasticprocesses and stochastic analysis were used.The practical significance of the results obtained is as follows:(1) since the sequences of risk asset’s prices, as a rule, are semimartingales, obtained statementsmight be used to select the minimax portfolio management;(2) the criterion has been established for the existence of extreme probability measure with respectto which: (a) the initial market is complete; (b) the upper bound of the option’s value has beenfound, (c) a hedging portfolio has been constructed,3(3) for incomplete markets without transaction costs the results obtained allow to construct aquantile hedging portfolio.Results to defend:(1)Bellman’s type recurrent relation for the sequence of upper guaranteed values for expectedexponential issuer’s risk in incomplete multidimensional market without transaction costs;(2)existence conditions for superhedging, quantile superhedging portfolio of European optionin incomplete multidimensional market without transaction costs with respect to any measure formthe set of equivalent probability measures;(3)existence criterion for probability measure (the worst-case measure) delivering essentialsupremum for expected exponential issuer’s risk, characteristics of the measure;(4)existence conditions for minimax and quantile minimax portfolios.The degree of development for the research problem.
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