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[{"key": "dc.contributor.advisor", "value": "Geiss, Hannah", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Laitinen, Aleksanteri", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2025-06-16T08:35:35Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2025-06-16T08:35:35Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2025", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/103544", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa tutkitaan optiohinnoittelua hypyt mukaan ottaen. Tutkielmassa k\u00e4ytet\u00e4\u00e4n kehyst\u00e4 joka on esitetty artikkelissa 'Hedging with Options in Models with Jumps' jonka on kirjoittanut Rama Cont, Peter Tankov ja Ekaterina Voltchkova, t\u00e4m\u00e4 kehys antaa matemaattisen esityksen option hinnalle hyppyjen l\u00e4sn\u00e4 ollessa. Esitys joka saadaan sis\u00e4lt\u00e4\u00e4 alkup\u00e4\u00e4oman, integraalin Brownin liikkeen suhteen ja integraalin kompensoidun Poisson satunnaismitan suhteen. Integraali kompensoidun Poisson satunnaismitan suhteen on osa joka sis\u00e4lt\u00e4\u00e4 ep\u00e4jatkuvuudet (hypyt). N\u00e4in pystyt\u00e4\u00e4n saamaan option hinnalle kyseinen esitys k\u00e4ytt\u00e4m\u00e4ll\u00e4 It\u00f4n kaavaa funktioon joka koostuu hy\u00f6dykkeen hinnasta.\n\nFokuksena on n\u00e4ytt\u00e4\u00e4 kuinka pystyt\u00e4\u00e4n muodostamaan esitys optiohinnoittelulle kun hyppyj\u00e4 on mukana, ja kuinka ratkaisu suojauksen varianssin minimoinnille saadaan. N\u00e4m\u00e4 k\u00e4sitteet on esitelty edell\u00e4 mainitulla paperilla, jonka kehyst\u00e4 seurataan tiiviisti.\n\nTutkielma alkaa palauttamalla lyhyesti mieleen k\u00e4sitteit\u00e4 todenn\u00e4k\u00f6isyysteoriasta ja antamalla m\u00e4\u00e4ritelmi\u00e4 liittyen Hilbert avaruuksiin. Seuraavaksi k\u00e4sitell\u00e4\u00e4n teoriaa koskien stokastisia prosesseja, satunnaismittoja ja stokastisia integraaleja. Tutkielmassa esitet\u00e4\u00e4n k\u00e4sitteit\u00e4 jotka liittyv\u00e4t rahoitukseen, p\u00e4\u00e4asiassa tapauksiin joissa jatkuvat prosessit ovat oletuksena, sill\u00e4 kirjallisuus ei ole viel\u00e4 t\u00e4ysin vakiintunut rahoitusk\u00e4sitteille joihin liittyy hyppyj\u00e4. P\u00e4\u00e4ty\u00f6kalut jotka saadaan edellisist\u00e4 matemaattisista k\u00e4sitteist\u00e4 ovat eritoten ehdollisiin odotusarvoihin liittyv\u00e4t ominaisuudet, ominaisuudet liittyen martingaaleihin, It\u00f4n isometria ja It\u00f4n kaava, joita k\u00e4ytet\u00e4\u00e4n perinpohjaisesti l\u00e4pi koko t\u00e4m\u00e4n tutkielman.\n\nT\u00e4ss\u00e4 tutkielmassa on kaksi jo aiemmin mainittua p\u00e4\u00e4tavoitetta. Ensin n\u00e4ytet\u00e4\u00e4n kuinka d-ulotteista It\u00f4n kaavaa k\u00e4ytt\u00e4en pystyt\u00e4\u00e4n muodostamaan esitys option hinnalle tietyill\u00e4 ehdoilla. T\u00e4m\u00e4 It\u00f4n kaava sis\u00e4lt\u00e4\u00e4 kaksi osaa, jatkuvan osan ja ep\u00e4jatkuvan osan jotka vastaavat esityst\u00e4 option hinnasta kun hypyt ovat mukana. Sen j\u00e4lkeen kun It\u00f4n kaavaa on k\u00e4ytetty funktioon joka koostuu hy\u00f6dykkeiden hinnoista, huomataan ett\u00e4 tulos joka saadaan muistuttaa osittain ratkaisua jota etsittiin, mutta tuloksessa on ylim\u00e4\u00e4r\u00e4isi\u00e4 osia. K\u00e4ytt\u00e4m\u00e4ll\u00e4 ominaisuuksia liittyen martingaaleihin pystyt\u00e4\u00e4n n\u00e4ytt\u00e4m\u00e4\u00e4n ett\u00e4 ylim\u00e4\u00e4r\u00e4iset osat ovat nolla ja saadaan esitys option hinnalle. Toisessa tavoitteessa n\u00e4ytet\u00e4\u00e4n miten suojauksen varianssin minimoinnin ratkaisu pystyt\u00e4\u00e4n saamaan. Ratkaisu suojauksen varianssin minimoinnille on pari, eli alkup\u00e4\u00e4oma ja suojausstrategia, niin ett\u00e4 suojauksen virhe mitattuna keskineli\u00f6ss\u00e4 on minimaalinen. Asetelmassa oletetaan martingaalimitta annettuna. T\u00e4ll\u00f6in l\u00f6ytyy yksik\u00e4sitteinen ratkaisu. Jotta pystyt\u00e4\u00e4n ratkaisemaan suojauksen minimoinnin ratkaisu, k\u00e4ytet\u00e4\u00e4n ominaisuuksia liittyen martingaaleihin, Brownin liikkeen ominaisuuksia, It\u00f4n isometriaa ja It\u00f4n kaavaa. Lopulta saadaan strategia ja alkup\u00e4\u00e4oma mitk\u00e4 minimoivat suojauksen virheen riitt\u00e4v\u00e4sti tavalliselle eurooppalaiselle optiolle. Tutkielmassa my\u00f6s osoitetaan kuinka on mahdollista k\u00e4ytt\u00e4\u00e4 kyseisi\u00e4 metodeja muihin asetelmiin kuten aasialaisiin optioihin ja kuinka stokastisen differentiaaliyht\u00e4l\u00f6n kaava poikkeaa barrier optiosta.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "In this thesis we study option pricing with jumps. We use the framework proposed in 'Hedging with Options in Models with Jumps' by Rama Cont, Peter Tankov and Ekaterina Voltchkova which gives a representation for the option price in the presence of jumps. The representation we get includes the initial capital, an integral with respect to the Brownian motion and an integral with respect to the compensated Poisson random measure. The integral with respect to the compensated Poisson random measure is the one that includes discontinuities (jumps). We can obtain the representation for the option price by applying It\u00f4's formula on the function of the asset price.\n\nOur focus is in showing how the representation for option pricing with jumps is formed, and how the solution for minimal variance hedging is obtained. These concepts are introduced in the paper introduced above, and we will be following the introduced framework closely.\n\nThe thesis starts with briefly recalling the basic concepts related to probablility theory, and gives definitions related to Hilbert spaces. We continue with the theory of stochastic processes, random measures and stochastic integrals. We introduce concepts related to finance, mainly in the continuous process case setting since the literature is not well established for financial concepts related to jumps. The main tools we obtain from the previous mathematical concepts are especially properties of conditional expectation, properties related to martingales, It\u00f4's isometry and It\u00f4's formula which we will use throughout this thesis.\n\nThere are two main objectives in this thesis as noted earlier. First we will show by using a d-dimensional It\u00f4 formula how a representation for the option price can be obtained under certain conditions. This It\u00f4 formula consists of two parts, a continuous and discontinuous part which corresponds to the representation for the option price when jumps are included. After applying It\u00f4's formula on the formula of asset prices we notice that the result which we get is partly corresponding to the solution which we are looking for, but there seem to be additional terms. By using properties related to martingales we are able to show that the additional terms are zero and get a representation for the option price.\nIn our second objective we will show how the minimal variance hedging solution can be obtained. A solution to the minimal variance hedging would be a pair consisting of the initial capital and a hedging strategy such that the hedging error measured in mean square is minimal. In our setting we assume that the martingale measure is given. Then the solution is unique. In order to solve the minimal variance hedging problem we use properties related to martingales, properties of the Brownian motion, It\u00f4's isometry, and It\u00f4's formula. Finally we get a strategy and initial capital which minimizes the hedging error for sufficient regular European options. We will also indicate how one can apply such methods to other settings such as hedging Asian options and how the stochastic differential equation formula differs for barrier options.", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2025-06-16T08:35:35Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2025-06-16T08:35:35Z (GMT). No. of bitstreams: 0\n Previous issue date: 2025", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "54", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "eng", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": null, "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Option pricing and hedging in models with jumps", "language": null, "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202506165357", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": null, "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": null, "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Stokastiikka ja todenn\u00e4k\u00f6isyysteoria", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Stochastics and Probability", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.copyright", "value": "\u00a9 The Author(s)", "language": "fi", "element": "rights", "qualifier": "copyright", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.yso", "value": "optiot", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "stokastiset prosessit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "rahoitusmarkkinat", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matemaattiset mallit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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