Određivanje vrijednosti američkih opcija

Žanetić, Marija (2016) Određivanje vrijednosti američkih opcija. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract

American call (put) option is defined as a contract which gives its owner the right, but not obligation to buy (sell) stock (or some other financial assets) under the strike price K in any time t before the expiration T. Payoff is defined as a function of the difference between stock price and strike price so that the difference is not less than zero. Before describing numerical methods, we had to set up some a priori bounds that the values of the options must satisfy. We also listed features of options and set the model of financial market which is actually an idealized form of real market. Then, we described assumptions on which the methods of calculation are based on. Among those assumptions is a claim that the asset price follows (geometric) Brownian motion and that the value function V (S; t) solves the Black - Scholes partial differential equation. Numerical methods used are the finite difference methods. Three of these methods are described: explicit, implicit and Crank - Nicolson method. To use these methods, we had to first discretize the problem; i.e., to induct the grid on which the solution will be approximated. Also, we had to derive the equations of central, backward and forward differences using Taylor expansions. In order to make the calculations easier, these methods are applied only on transformed variables. After those are calculated, from them we calculate the original variables such as stock price and value of the option. Stability and error order of each method are also described. We listed all boundary conditions and defined the contact point (which does not always have to exist). A Set of those points is called the early exercise curve. By those contact points we can explain when value of an option satisfies the Black - Scholes equation, and when inequation. Then, we introduced one simple problem and we want to find its solution because we can apply it on our linear complementary problem. After we find that solution, we can start calculating values of the options. Using all obtained results, we can construct the algorithm for computation of American options and, by it, we can write a computer program. There are the four main errors that occur during calculation. Since this work is primarily based on numerical analysis, only discretization errors, which we can control, are described. After numerical methods, two analytical methods are shortly described. We call them analytical although they include some numerical algorithms. In the last section, four examples are shown in which the algorithm and the program that calculates values of the options are tested.

Item Type: Thesis (Diploma thesis)
Supervisor: Bosner, Nela
Date: 2016
Number of Pages: 49
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 23 Nov 2016 13:27
Last Modified: 23 Nov 2016 13:27
URI: http://digre.pmf.unizg.hr/id/eprint/5335

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