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Trinomska drevesa in metoda končnih diferenc za vrednotenje opcij : delo diplomskega seminarja
Založnik, Jan (Author), Bernik, Janez (Mentor) More about this mentor... This link opens in a new window, Kanduč, Tadej (Co-mentor)

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Abstract
Eden pomembnejših dogodkov v finančnem svetu je zagotovo, ko so leta 1973 Black, Scholes in Merton izdali revolucionarno formulo za vrednotenje opcij. Trgovanje z opcijami se je s tem zelo razširilo, saj je končno obstajala analitična formula za izračun premij opcij. Vendar so se zaradi potreb na trgu oblikovale zahtevnejše opcije, ki niso imele analitične rešitve. Zato se je par let kasneje razvila numerična metoda z binomskimi drevesi za računanje opcij. Metodo so oblikovali tako, da se lahko cena finačnega instrumenta v vsakem časovnem koraku dvigne ali pade in da se z manjšanjem koraka rešitev približuje k analitični reštivi. Zaradi želje po hitrejši konvergenci binomskega modela je nastal trinomski model, ki ima v vsakem koraku tri odločitve, poleg dviga in padca še dodamo možnost, da se cena ne spremeni. Prav tako so se pojavile številne možnosti parametrizacije, ki bi naj odpravljale določene težave prvotnih modelov in seveda zagotovile hitrejšo konvergenco. V podoben razred sodi tudi metoda končnih diferenc, kjer zmanjšamo domeno parcilane diferencialne enačbe Black-Scholes-Mertonovega modela na končno množico točk in s pomočjo končnih diferenc izračunamo vrednost premije. Seveda se tudi tu z večanjem števila točk numerična rešitev približuje analitični rešitvi, vendar s povečano časovno zahtevnostjo.

Language:Slovenian
Keywords:Opcije, Black-Scholesov model, Black-Scholes-Mertonov model, binomska drevesa, trinomska drevesa, metoda končnih diferenc, implicitna metoda končnih diferenc, eksplicitna metoda končnih diferenc.
Work type:Bachelor thesis/paper (mb11)
Organization:FMF - Faculty of Mathematics and Physics
Year:2021
UDC:519.8
COBISS.SI-ID:76694787 This link opens in a new window
Views:95
Downloads:20
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Secondary language

Language:English
Title:Trinomial and finite difference option pricing
Abstract:
One of the most important event in the financial world is definitely the invention of the revolutionary option pricing formula by Black, Scholes and Merton in 1973. Trading with options has thus become very widespread, as there was finally an analytical formula for calculating option premiums. However, due to market needs, more sophisticated options were formed that did not have an analytical solution. Therefore, a couple of years later, a numerical method with binomial trees for calculating options was developed. The method was designed in such a way that the price of a financial instrument could rise or fall at each time step and that by reducing the step size, the solution approached the analytical solution. Due to the desire for faster convergence of the binomial model, a trinomial model was created, which has three decisions in each step, in addition to the rise and fall, we also add the possibility that the price does not change. Numerous parameterizations for these methods have also emerged, which are supposed to eliminate certain problems of the original models and, of course, ensure faster convergence.A similar class of method is also the finite difference method, where we reduce the domain of the partial differential equation of the Black-Scholes-Merton model to a finite set of points and calculate the value of the premium with the help of finite differences. Of course with the increase of the number of points, it approaches the analytical solution, but with the increased time complexity.

Keywords:Options, Black-Scholes model, Black-Scholes-Merton model, binomial trees, trinomial trees, finite difference method, implicit finite difference method, explicit finite difference method.

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