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There are a lot of works dealingwith European option’s calculation theory in one dimensional complete market without transactioncosts in discreet time. These are works by Cox J.C., Ross R.A., Rubinstein M. 4, Harrison J.M.,Kreps D.5, Shiryaev A. N., Kabanov Yu. M., Kramkov O.
D. and Mel’nikov A. V.6, Föllmer H.,Schied A.7. There in the works they established uniqueness of martingale measure and obtained it’sexplicit form. It is proved that payoff allows S-representation. These results gave an opportunity toobtain option’s value and to construct perfect hedging portfolio.
There are also works of someauthors on the theory of quantile hedging for European option in one-dimensional complete market.For example, there is in the article by Novikov A.A. 8for the case of one-dimensional completemarket method is justified for calculation of option’s value and of hedging strategy. In H.
Föllmer’sand P. Leukert’s article 9 they consider static problem to minimize option’s value under givenprobability of payoff being fulfilled. They claim, that the solution of this problem coincides with thesolution for the problem of European option’s calculation with some modified payoff.
To construct4Cox J. C., Ross R.A., Rubinstein M. Option pricing: a simplified approach. / Journal of FinancialEconomics. - 1979. - v.7. - №3. - p.229-2635Harrison, J.M., Kreps, D. Martingales and arbitrage in multiperiod security markets. / Journal of EconomicTheory. - 1979. - v.20. - p.381-4086Shiryaev A.N., KabanovYu.M., Kramkov O.D., Mel’nikov A.V.
Toward the Theory of Pricing of Optionsof Both European and American Types. I. Discrete time / Theory of Probability and its Applications. –1994. – 39. – 1. – p.23-79 (in Russian)7Föllmer H., Schied A. Stochastic Finance. An Introduction in Discrete Time.
Moscow: MTsNMO. 2008. 496 p. (in Russian) (in Russian)8Novikov A.A. Hedging of options with given probability / Theory of Probability and its Applications.1998. – 43. –1. – p.152-161 (in Russian)9Fёllmer H., Leukert, P. Quantile hedging. / Finance and Stochastics. - 1999. - v.3. - 3. - p.251-2734the latter they use the Neyman-Pearson lemma. In the article by P.V. Grigor’ev, Yu.S. Kan10twostep optimal control problem is considered with two types of assets and with quantile criterionunder assumption that risk asset’s yield is uniformly distributed. Based on Yu.S.
Kan’sarticle11analytical solution was constructed for the problem to manage portfolio of securities thenstrategy belongs to the set of Markov strategies. In the article by A.I. Kibzun, A.V. Naumov andV.I. Norkin12for one-dimensional market with time horizon equal to one they establish conditionswhen it is possible to reduce quantile hedging problem to a partially integer programming problem.Theory of European option’s calculation in incomplete market without transaction costs indiscrete time was considered in some works.
For example, in articles by V. Naik 13, Delbaen F. andSchachermayer W.14for a semimartingale model of a market with limited number of assets andbounded from below payoff f it is proved that upper value of the optionС0 allowsrepresentationС0 = sup ∈()whereM(S)is the set of equivalent locally martingale measures specified on trajectories of riskassets’s prices. In the article by F. Delbaen and W. Schachermayer 15 the validity of the aboveformula is established, where supremum is taken over the set of σ-martingale probability measures.In the works by Shiryaev A.N.1, Föllmer H.
and Schied A.7 formula for upper value of option isderived in the case when payoff is a nonnegative bounded function. More over they establishexistence conditions for superhedging portfolio in the case of equivalent martingale measures. Inthe article by A. Bizid, E.
Jouni16 in semimartingale market model where short selling is forbiddenand payoff is bounded formula for upper value of an option is obtained. In L. Ruschendorf’sarticle 17 formulas are derived allowing bottom and upper estimations for option’s value. In the10Grigor’ev P.V., Kan Yu. S. Optimal Control of the Investment Portfolio with Respect to the QuantileCriterion / Automation and Remote Control.
– 2004. – 2. – p. 179-197 (in Russian)11Kan Yu. S. Control Optimization by the Quantile Criterion / Automation and Remote Control. – 2001. –5. – p.77-88 (in Russian)12Kibzun A.I., Naumov A.V., Norkin V.I. On reducing a quantile optimization problem with discretedistribution to a mixed integer programming problem / Automation and Remote Control. – 2013.
– 6. –p.66-86 (in Russian)13Naik V., Uppal R. Leverage constraints and the optimal hedging of stock and bond options / Journal ofFinancial and Quantitative Analysis. - 1994. - v.29. -№2. - p.199-22214Delbaen F., Schachermayer W. The no-arbitrage property under a change of numeraire / Stochastics andStochastic Reports.
- 1995. - v.53. - p.213-26615Delbaen F., Schachermayer W. The Fundamental Theorem of Asset Pricing for Undounded StochasticProcesses / Mathematische Annalen. - 1998. - v.312. - №2. - р.215-25016Bizid A., Jouini E. Incomplete markets and short-sales constraints: an equilibrium approach. / Int. J. ofTheoretical and Applied Finance. - 2001. - v.4. - №2.
- p.211-24317Rüschendorf L. On Upper and Lower Prices in Discrete-Time Models / Tr. Mat. Inst. Steklova. - 2002. V.237. - p.143-1485article by A.A. Gushchin and E. Mordecki 18in one-dimensional semimartingale model of (B,S)market conditions are established under which upper and lower option’s values are attainable. In thework by E. Eberlein, A. Papapantoleon, A.
N. Shiryaev 19 for one-dimensional semimartingalemarket model when risk assets’ prices are specified by process with independent increments theyestablished existence conditions for call-put parity for options of European, American and Asiantypes. R.V. Khasanov’s in his PhD thesis20 consider static European option’s calculation problem inmultidimensional market. Assuming that risk assets’ prices are specified by semimartingales, authorderives formula for option’s upper valueС0 = sup∈ = sup∈ ,where are sets of locally martingale and σ-martingale densities respectively. It was shown thatseparating measure is finitely additive.
The problem to calculate European option with quantilecriterion in incomplete market without transaction costs was considered in articles of some authors:Föllmer H., Schied A., Leukert P.21, Karatzas I.22, Cvitanic J.23, Leung T., Song Q. and Yang J.24They considered static problem to minimaze option’s value under given probability of payoff’sfulfillment. It is stated that solution of the problem coincides with solution of European option’scalculation problem with some modified payoff equal to product of initial payoff f and indicator ofsome set.
In the work by Azanov V.M. and Kan Yu. S. 25 they consider maximization problem forprobability to achieve a given capital value under fixed initial capital. Relations are established foroptimal strategy.Note that in most works European option’s calculation problem in incomplete marketwithout transaction costs is considered as a static problem. This made it possible to derive formulasfor upper (lower) option’s value or it’s estimation. But this approach can’t give formula for hedgingformula and correspondent capital.18Gushchin A.A., Mordecki E. Bounds for options’ values for semimartingale market models / Proceedingsof the Steklov Institute of Mathematics. – 2002.
- 237, 80-122. (in Russian).19Eberlein E., Papapantoleon A., Shiryaev A. N. On the duality principle in option pricing: semimartingalesetting. / Finance and Stochastics. - 2008. - v.12. - 2. - p.265-29220Khasanov R.V. Maximization of utility with random contribution and hedging of payoffs, thesis to obtaindegree of candidate of physical and mathematical sciences. Moscow. 2013. 91p.
(in Russian).21Föllmer H., Leukert P. Efficient hedging: Cost versus shortfall risk. / Finance and Stochastics. - 2000. - v.4. - 2. - p.117-14622Cvitanić J., Karatzas I. On dynamic measures of risk. / Finance and Stochastics. - 1999. - v.3. - 4. - p.45148223Cvitanić J. Minimizing expected loss of hedging in incomplete and constraint markets. / SIAM Journal onControl and Optimization. - 2000.
- v.38 - 4. - p.1050-1066.24Leung T., Song Q., Yang J. Outperformance portfolio optimization via the equivalence of pure andrandomized hypothesis testing. / Finance and Stochastics. - 2013. - v.17. - 4. - p.839-87025Azanov V.M., Kan Yu. S. Bilateral Estimation of the Bellman Function in the Problems of OptimalStochastic Control of Discrete Systems by the Probabilistic Performance Criterion / Automation andRemote Control.
- 2018. - 2. - p.3-18 (in Russian)6Personal contribution of the author in problem’s development: the results of the articleshave been obtained by the dissertator personally, Khametov V.M. contributed by problem statementand by general guidance.The list of publications on the theme of the PhD thesisThe results of the PhD thesis are published in peer-reviewed scientific editions from the list byHigher Attestation Commission and the list of high level by National Research University «HighSchool of Economics».The articles from the list of leading peer-reviewed scientific editions from the list by HigherAttestation Commission of Ministry of science and education of Russian Federation:1.Zverev O.V. On conditions on fairness of optional decomposition.
/ Zverev O.V., KhametovV.M. // 2009. Surveys on Applied and Industrial Mathematics. vol. 16, issue 6. p. 1067-1068. (inRussian) (personal contribution of the author 0,04 рр).2.Zverev O.V. Quantile hedging of European typo options in incomplete markets withouttransaction costa.
Part 1. Superhrdging. / Zverev O.V., Khametov V.M. // 2014. Controlproblems.vol. 6. p. 31-44. (in Russian) (personal contribution of the author 0,7рр).3.Zverev O.V. Quantile hedging of European typo options in incomplete markets withouttransaction costa. Part 2. Minimax hedging. / Zverev O.V., Khametov V.M. // 2015.
Controlproblems. vol. 1, p. 47-52. (in Russian) (personal contribution of the author 0,3рр).Other publications by the author:4.Zverev O.V. Minimax hedging for European type options in incomplete markets (Discreettime). /Zverev O.V., Khametov V.M. // 2011. Surveys on Applied and Industrial Mathematics. vol.18.issue 1. p. 26-54. (in Russian) (personal contribution of the author 0,8рр).5.Zverev O.V.
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