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1
Content available Application of the Trefftz method for option pricing
Brzozowska-Rup Katarzyna, Hożejowska Sylwia, Hożejowski Leszek
Zeszyty Naukowe. Organizacja i Zarządzanie / Politechnika Śląska
|
2020
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z. 146
37--49
EN
Purpose: Option pricing is hardly a new topic, however, in many cases they lack an analytical solution. The article proposes a new approach to option pricing based on the semi-analytical Trefftz method. Design/methodology/approach: An appropriate transformation makes it possible to reduce the Black-Scholes equation to the heat equation. This admits the Trefftz method (which has shown its effectiveness in heat conduction problems) to be employed. The advantage of such an approach lies in its computational simplicity and in the fact that it delivers a solution satisfying the governing equation. Findings: The theoretical option pricing problem being considered in the paper has been solved by means of the Trefftz method, and the results achieved appear to comply with those taken from the Black-Scholes formula. Numerical simulations have been carried out and compared, which has confirmed the accuracy of the proposed approach. Originality/value: Although a number of solutions to the Black-Scholes model have appeared, the paper presents a thoroughly novel idea of implementation of the Trefftz method for solving this model. So far, the method has been applied to problems having nothing in common with finance. Therefore the present approach might be a starting point for software development, competitive to the existing methods of pricing options.
2
Content available Lower Precision calculation for option pricing
Ścibisz-Mordelska K., Nielek R.
Computer Science
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2017
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Vol. 18 (4)
429--446
EN
The problem of options pricing is one of the most critical issues and fundamental building blocks in mathematical finance. The research includes deployment of lower precision type in two options pricing algorithms: Black-Scholes and Monte Carlo simulation. We make an assumption that the shorter the number used for calculations is (in bits), the more operations we are able to perform in the same time. The results are examined by a comparison to the outputs of single and double precision types. The major goal of the study is to indicate whether the lower precision types can be used in financial mathematics. The findings indicate that Black-Scholes provided more precise outputs than the basic implementation of Monte Carlo simulation. Modification of the Monte Carlo algorithm is also proposed. The research shows the limitations and opportunities of the lower precision type usage. In order to benefit from the application in terms of the time of calculation improved algorithms can be implemented on GPU or FPGA. We conclude that under particular restrictions the lower precision calculation can be used in mathematical finance.
3
Content available A fuzzy approach to option pricing in a Levy process setting
Nowak P., Romaniuk M.
International Journal of Applied Mathematics and Computer Science
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2013
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Vol. 23, no. 3
613--622
EN
In this paper the problem of European option valuation in a Levy process setting is analysed. In our model the underlying asset follows a geometric Levy process. The jump part of the log-price process, which is a linear combination of Poisson processes, describes upward and downward jumps in price. The proposed pricing method is based on stochastic analysis and the theory of fuzzy sets.We assume that some parameters of the financial instrument cannot be precisely described and therefore they are introduced to the model as fuzzy numbers. Application of fuzzy arithmetic enables us to consider various sources of uncertainty, not only the stochastic one. To obtain the European call option pricing formula we use the minimal entropy martingale measure and Levy characteristics.
4
Content available remote Quantum physics methods in share option valuation
Wróblewski M.
Czasopismo Techniczne. Automatyka
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2013
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R. 110, z. 2-AC
23--40
EN
This paper deals with European share option pricing using quantum physics methods. These contingent claims are usually priced using the Black-Scholes equation. This nonlinear parabolic equation is based on geometric Brownian motion model of the stock price stochastic process. Similar processes also appear among quantum particles and are described by the time-dependent Schrödinger equation. In this paper, the option pricing based on the Schrödinger equation approach is proposed. Using Wick transformation, the Black-Scholes equation is transformed into the equivalent Schrödinger equation. The Fourier separation method is used to find analytical solutions to this equation. The last square method is used to calibrate the Schrödinger model based on real market data. Numerical results are provided and discussed.
PL
Artykuł dotyczy wyceny europejskich opcji na akcje z użyciem metod fizyki kwantowej. Tego typu obliczenia zazwyczaj przeprowadza się z wykorzystaniem równania Blacka-Scholesa. To nieliniowe, paraboliczne równanie, oparte jest na geometrycznym modelu ruchu Browna procesu stochastycznego cen akcji. Podobne procesy dotyczą także cząstek kwantowych i są opisane zależnym od czasu równaniem Schrödingera. Zaproponowano wycenę opcji na akcje z wykorzystaniem równania Schrödingera. Używając transformacji Wicka, równanie Blacka-Scholesa przekształcone jest do równoważnej postaci równania Schrödingera. W celu znalezienia analitycznego rozwiązania tego równania, zastosowano metodę separacji zmiennych Fouriera. Metoda najmniejszych kwadratów została użyta w celu kalibracji modelu Schrödingera dla danych giełdowych. Dostarczono i przedyskutowano wyniki numeryczne.
5
Content available remote Actuarial Approach to Option Pricing in a Fractional Black–Scholes Model with Time-Dependent Volatility
Falkowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2013
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Vol. 61, no 2
181--193
EN
We study actuarial methods of option pricing in a fractional Black–Scholes model with time-dependent volatility. We interpret the option as a potential loss and we show that the fair premium needed to insure this loss coincides with the expectation of the discounted claim payoff under the average risk neutral measure.
6
Content available Option pricing by Esscher transforms in the cases of normal inverse Gaussian and variance gamma processes
Troush N. N., Kuzmina A. V.
Studia Informatica : systems and information technology
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2012
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Vol. 1-2(16)
35--43
EN
The class of Esscher transforms is an important tool for option pricing Gerber and Shiu (1994) showed that the Esscher transform is an efficient technique for valuing derivative securities if the log returns of the underlying securities are governed by certain stochastic processes with stationary and independent increments. Levy processes are the processes of such type. Special cases of the Levy processes such as the normal inverse Gaussian process and the variance gamma process are considered at this paper. Values of these processes parameters for the existence of Esscher transform are deduced. A new algorithm of a normal inverse Gaussian process and variance gamma process simulation is also presented in this paper. These algorithm is universal and simpler one compared with analogous algorithms.
7
Content available Zastosowanie sieci neuronowych do wyceny kontraktów opcyjnych na indeks WIG20
Kraszewska M.
Pomiary Automatyka Robotyka
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2011
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R. 15, nr 12
220-222
PL
W artykule zaprezentowano możliwość zaadoptowania sztucznych sieci neuronowych do wyceny kontraktów opcyjnych na indeks WIG20 Giełdy Papierów Wartościowych w Warszawie. Analizując dane rzeczywiste z lat 2005-2009 zbudowano szereg modeli sieci neuronowych z wykorzystaniem programu Statistica. Uzyskane rezultaty porównano z wynikami otrzymanymi z modelu Blacka-Scholesa. Do pomiaru dokładności prognoz modeli użyto powszechnie znane miary błędów.
EN
The possibility of adopting artificial neural networks to the valuation of option contracts on WIG20 Warsaw Stock Exchange is presented. Using real data from 2005-2009 several models of neural networks were examined in Statistica. The results were compared with results received using the Black-Scholes formula. To measure the accuracy of forecasting models commonly known measurement errors were used.
8
Content available remote Modelowanie polskiego rynku energii elektrycznej
Opalski K., Kacprzak K., Maciejczyk K., Pawłowski M.
Matematyka Stosowana : matematyka dla społeczeństwa
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2011
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T. 13/54
105-114
PL
W artykule opisano modelowanie rynku energii elektrycznej handlowanej na Towarowej Giełdzie Energii S.A. w Warszawie. W pracy opisano wybrany model dynamiki cen energii elektrycznej,a następnie opisano proces kalibracji powstałego modelu do danych rynkowych tak,a by był spełniony warunek braku arbitrażu. Tak skalibrowany model został wykorzystany do symulacji chwilowej ceny energii. Pozwoliło to na przeprowadzenie wyceny kontraktu terminowego typu forward na dostawę energii,a także waniliowej opcji zakupu energii elektrycznej. Odpowiednie wyceny zostały zaimplementowane w statystycznym pakiecie R.
EN
The paper deals with mathematical modelling of energy market. First,t he model of price dynamics has been chosen. This model has been calibrated to data from the Polish Power Exchange in Warsaw. The calibrated model has been used to simulate energy prices which was used to price forward contracts and vanilla put options for energy supply. All simulations has been made in R package.
9
Content available remote The modified tempered stable distribution, GARCH models and option pricing
Kim Y. S., Rachev S. T., Chung D. M., Bianchi M. L.
Probability and Mathematical Statistics
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2009
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Vol. 29, Fasc. 1
91--117
EN
We introduce a new variant of the tempered stable distribution, named the modified tempered stable (MTS) distribution and we develop a GARCH option pricing model with MTS innovations. This model allows the description of some stylized empirical facts observed in financial markets, such as volatility clustering, skewness, and heavy tails of stock returns. To demonstrate the advantages of the MTS-GARCH model, we present the results of the parameter estimation.
10
Content available remote Princing European options on instruments with a constant dividend yield : The randomized discrete-time approach
Weron R.
Probability and Mathematical Statistics
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2002
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Vol. 22, Fasc. 2
407--418
EN
Due to the well-known fact that market returns are not normally distributed, we use generalized hyperbolic distributions for pricing options in a randomized discrete-time setup. The obtained formulas can be used for pricing options on stock indexes, currencies and futures contracts. We test them on options written on the Nikkei 225 index futures and conclude that a proper calibration scheme could give us an advantage in the financial market.
11
Content available remote Existence of variance-optimal hedging strategy in discrete time model with transaction costs
Motoczyński M.
Bulletin of the Polish Academy of Sciences. Mathematics
|
1999
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Vol. 47, no 2
191-207
EN
In the paper, a variance-optimal hedging of a contingent claim for a discrete time model with transaction costs is considered. Existence of an optimal hedging strategy is shown.
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