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[{"key": "dc.contributor.advisor", "value": "Geiss, Christel", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Helin, Santeri", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2021-03-10T06:30:28Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2021-03-10T06:30:28Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2021", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/74559", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Optiohinnoittelun teoria on keskeisess\u00e4 osassa tutkielmaamme ja tavoitteenamme on saada optiohinnoittelun teoriaa k\u00e4ytt\u00e4en teoreettinen estimaatti option reilusta hinnoittelusta. T\u00e4t\u00e4 option reilua hintaa sijoittajat voivat k\u00e4ytt\u00e4\u00e4 my\u00f6hemmin salkkujensa arvon maksimointiin. Yksi kuuluisimmista malleista optioiden hinnoittelussa on Black-Scholes-malli.\n\nBlack-Scholes-malli on keskeisess\u00e4 roolissa modernissa finanssiteoriassa ja on k\u00e4yt\u00f6ss\u00e4 my\u00f6s t\u00e4ll\u00e4 hetkell\u00e4. Mallin k\u00e4ytt\u00e4misess\u00e4 yksi suurimmista eduista on, ett\u00e4 malli riippuu ainoastaan yhdest\u00e4 ei havaittavissa olevasta parametrista \"sigma\" nimelt\u00e4\u00e4n volatiliteetti. T\u00e4m\u00e4 huomataan tutkielmassa johdettaessa Black-Scholes-yht\u00e4l\u00f6\u00e4. T\u00e4m\u00e4n volatiliteetin johtamiseen on olemassa my\u00f6s keinoja, mutta emme keskity niihin tutkielman aikana.\n\nOletamme tutkielman aikana, ett\u00e4 volatiliteetti pysyy vakiona, jotta laskut voitaisiin tehd\u00e4. T\u00e4m\u00e4 ei kuitenkaan vastaa oikeaa tilannetta sijoittamisessa, sill\u00e4 volatiliteetti voi vaihdella ajan kuluessa. Black-Scholes-yht\u00e4l\u00f6\u00e4 johdettaessa oletamme my\u00f6s, ett\u00e4 sijoittaessa ei ilmaannu veroja tai rahansiirron aikana tulevia kustannuksia. Lis\u00e4ksi tutkimme Black-Scholes-mallissa ainoastaan Euroopan optioita, koska kyseisess\u00e4 mallissa optiot voidaan suorittaa ainoastaan niiden ennalta s\u00e4\u00e4dellyn viimeisen k\u00e4ytt\u00f6p\u00e4iv\u00e4n ajanhetkell\u00e4.\n\nTutkimuksemme koostuu kahdesta p\u00e4\u00e4tavoitteesta. N\u00e4ist\u00e4 ensimm\u00e4inen on Euroopan put ja call optioiden reilun hinnan m\u00e4\u00e4ritt\u00e4minen, jolla tarkoitetaan, ett\u00e4 kenenk\u00e4\u00e4n ei tulisi saada riskit\u00f6nt\u00e4 voittoa. T\u00e4m\u00e4n tavoitteen suorittamista varten k\u00e4yt\u00e4mme Black-Scholes-mallia. Aloitamme mallin esittelyll\u00e4 kappaleessa 5 ja jatkamme t\u00e4st\u00e4 esittelem\u00e4ll\u00e4 todenn\u00e4k\u00f6isyysmitan vaihtamisen kappaleessa 6. Kolmannessa kappaleessa on esitelty t\u00e4rkeimm\u00e4t stokastiikan perusty\u00f6kalut laskemista varten. Koska stokastinen integrointi on t\u00e4rke\u00e4ss\u00e4 roolissa tutkielmassamme, esittelemme my\u00f6s yhden kuuluisimmista stokastisista integraaleista nimelt\u00e4 Ito integraali. Stokastinen integrointi ja Iton lause esitell\u00e4\u00e4n nelj\u00e4nness\u00e4 kappaleessa. Kappaleessa 7 k\u00e4yt\u00e4mme aiemmin esittelemi\u00e4mme teorioita, kuten todenn\u00e4k\u00f6isyysmitan vaihtoa ja stokastista laskentaa, Black-Scholes-yht\u00e4l\u00f6n ratkaisemiseen.\n\nKuten ensimm\u00e4isess\u00e4 p\u00e4\u00e4tavoitteessa, oletamme my\u00f6s toisessa p\u00e4\u00e4tavoitteessamme, ett\u00e4 mahdollisia veroja tai rahansiirron kustannuksia ei ole. Toisen p\u00e4\u00e4tavoitteen tarkoituksena on mallintaa optimaalista investoimista. T\u00e4ss\u00e4 meill\u00e4 on k\u00e4yt\u00f6ss\u00e4 yleisempi malli, joka koostuu monesta erilaisesta riskialttiista komponentista ja riskitt\u00f6m\u00e4st\u00e4 sijoittajan omaisuudesta pankkitilill\u00e4. Valitsemme sopivan rahastohallinnon ja yrit\u00e4mme l\u00f6yt\u00e4\u00e4 sille optimaalisen strategian h^* maksimoimalla valitun apuv\u00e4line funktion. Apuv\u00e4line funktioita (utility function) on valittavana monenlaisia ja siten yht\u00e4 oikeaa valintaa ei voi m\u00e4\u00e4ritell\u00e4. Tutkimusta tehdess\u00e4 valitsemme usein funktion, jota on matemaattisesti helppo k\u00e4sitell\u00e4 ja jolla on mielekk\u00e4it\u00e4 matemaattisia ominaisuuksia.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Option pricing theory is a concept where we aim to value an option theoretically by using variables such as stock price, exercise price, volatility, interest rate and expiration date. \nBy using option pricing theory we can obtain the theoretical estimation of an options fair value which can be used later by, for example, traders to maximize profits. One commonly used model in option pricing that we are going to introduce is called the Black-Scholes model. \n\nThe Black-Scholes model has a big role in the modern financial theory and is still widely used today. This model was first developed in 1973 by Fischer Black, Robert Merton and Myron Scholes. Due to its success the creators of the model Robert Merton and Myron Scholes were even given the Nobel price award. Fisher Black were also in close collaboration with Robert Merton and Myron Scholes but since he died before the Noble price was granted he did not have enough time to get the reward. One of the main features of the Black-Scholes model is that the pricing formula depends only on one non-observable parameter \"sigma\", the so called volatility. The volatility can be evaluated for example by using the historical method or the implied method. This is one of the main reasons behind the success of the Black-Scholes formula. \n\nThe focus of the thesis is the modelling of the two basic activities on a financial market. The first one we discuss is the option pricing and the second one is the optimal investment. The prices of both of these activities on certain underlyings are modelled by the same processes, exponential diffusion processes, and both actions can be performed on the same underlyings at the same time.\n\nTo be more precise we have two main objectives to accomplish. The first one is to determine a fair price for the European call and put options, which is done by using the Black-Scholes model. We start by introducing our model in chapter 5 and then continue by introducing the change of measure technique in the chapter 6. The basic tools needed for the computations are in the third chapter. Since stochastic integration is used in our theorems we also introduce one of the most popular stochastic integrals, the Ito integral, ensuring the foundation for our theorems and main results. Stochastic integration and Ito's formula will be introduced in the fourth chapter. In chapter 7 we finally show that how one can derive the Black-Scholes formula by using the change of measure technique and stochastic calculus. \n\nIn the final part our second objective is to find the most suitable strategy for a given utility function. Like in the Black-Scholes model we also need to assume that there are no transaction costs or taxes but in this case we can have many possible solutions depending on the utility function. We consider the Risk-sensitive asset management criterion in the special case, where asset and factor risks are not correlated. Here our main objective is to maximise the expected log return of the portfolio by using the risk-sensitive asset management criterion. This criterion is known for giving penalty for high variance, negative skewness and high kurtosis while rewarding positive skewness (see \\cite{Mark} Chapter 2.2). \n\nChoosing logarithm of the portfolio value as a reward function provides us with a setting where the calculations can be carried out. This leads to a risk-sensitive asset management criterion, which is a great choice when managing portfolio value. For example this criterion works well with Markowitz' mean-variance analysis. and is consistent with utility theory (see \\cite{Mark} Chapter 2.2). We can also show that the risk-sensitive asset management criterion is a log coherent optimization criterion meaning that is satisfies the four axioms that we are going to introduce in the chapter 8. The Appendix part discusses existence and uniqueness of solutions for the SDEs we use in the Risk-sensitive asset management part.", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2021-03-10T06:30:28Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2021-03-10T06:30:28Z (GMT). No. of bitstreams: 0\n Previous issue date: 2021", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "45", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "eng", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "option pricing", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "stochastics", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "The Black-Scholes model and risk-sensitive asset management", 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