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Stochastic Models in Financial Mathematics
Waczulík, Oliver ; Hurt, Jan (advisor) ; Večeř, Jan (referee)
Title: Stochastic Models in Financial Mathematics Author: Bc. Oliver Waczulík Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Jan Hurt, CSc., Department of Probability and Mathe- matical Statistics Abstract: This thesis looks into the problems of ordinary stochastic models used in financial mathematics, which are often influenced by unrealistic assumptions of Brownian motion. The thesis deals with and suggests more sophisticated alternatives to Brownian motion models. By applying the fractional Brownian motion we derive a modification of the Black-Scholes pricing formula for a mixed fractional Bro- wnian motion. We use Lévy processes to introduce subordinated stable process of Ornstein-Uhlenbeck type serving for modeling interest rates. We present the calibration procedures for these models along with a simulation study for estima- tion of Hurst parameter. To illustrate the practical use of the models introduced in the paper we have used real financial data and custom procedures program- med in the system Wolfram Mathematica. We have achieved almost 90% decline in the value of Kolmogorov-Smirnov statistics by the application of subordinated stable process of Ornstein-Uhlenbeck type for the historical values of the monthly PRIBOR (Prague Interbank Offered Rate) rates in...
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Continuous market models with stochastic volatility
Petrovič, Martin ; Maslowski, Bohdan (advisor) ; Hlubinka, Daniel (referee)
Vilela Mendes et al. (2015), based on the discovery of long-range dependence in the volatility of stock returns, proposed a stochastic volatility continuous mar- ket model where the volatility is given as a transform of the fractional Brownian motion (fBm) and studied its No-Arbitrage and completeness properties under va- rious assumptions. We investigate the possibility of generalization of their results from fBm to a wider class of Hermite processes. We have reworked and completed the proofs of the propositions in the cited article. Under the assumption of indepen- dence of the stock price and volatility driving processes the model is arbitrage-free. However, apart from a case of a special relation between the drift and the volatility, the model is proved to be incomplete. Under a different assumption that there is only one source of randomness in the model and the volatility driving process is bounded, the model is arbitrage-free and complete. All the above results apply to any Hermite process driving the volatility. 1
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Stochastic Models in Financial Mathematics
Waczulík, Oliver ; Hurt, Jan (advisor) ; Večeř, Jan (referee)
Title: Stochastic Models in Financial Mathematics Author: Bc. Oliver Waczulík Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Jan Hurt, CSc., Department of Probability and Mathe- matical Statistics Abstract: This thesis looks into the problems of ordinary stochastic models used in financial mathematics, which are often influenced by unrealistic assumptions of Brownian motion. The thesis deals with and suggests more sophisticated alternatives to Brownian motion models. By applying the fractional Brownian motion we derive a modification of the Black-Scholes pricing formula for a mixed fractional Bro- wnian motion. We use Lévy processes to introduce subordinated stable process of Ornstein-Uhlenbeck type serving for modeling interest rates. We present the calibration procedures for these models along with a simulation study for estima- tion of Hurst parameter. To illustrate the practical use of the models introduced in the paper we have used real financial data and custom procedures program- med in the system Wolfram Mathematica. We have achieved almost 90% decline in the value of Kolmogorov-Smirnov statistics by the application of subordinated stable process of Ornstein-Uhlenbeck type for the historical values of the monthly PRIBOR (Prague Interbank Offered Rate) rates in...
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