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Algorithmic optimization and its application in finance
Algorithmic optimization and its application in finance
The goal of this thesis is to examine different issues in the area of finance and application of financial and mathematical models under consideration of optimization methods. Prior to the application of a model to its scope, the model results have to be adjusted according to the observed data. For this reason a target function is defined which is being minimized by using optimization algorithms. This allows finding the optimal model parameters. This procedure is called model calibration or model fitting and requires a suitable model for this application. In this thesis we apply financial and mathematical models such as Heston, CIR, geometric Brownian motion, as well as inverse transform sampling, and Chi-square test. Moreover, we test the following optimization methods: Genetic algorithms, Particle-Swarm, Levenberg-Marquardt, and Simplex algorithm. The first part of this thesis deals with the problem of finding a more accurate forecasting approach for market liquidity by using a calibrated Heston model for the simulation of the bid/ask paths instead of the standard Brownian motion and the inverse transformation method instead of compound Poisson process for the generation of the bid/ask volume distributions. We show that the simulated trading volumes converge to one single value which can be used as a liquidity estimator and we find that the calibrated Heston model as well as the inverse transform sampling are superior concerning the use of the standard Brownian motion, resp. compound Poisson process. In the second part, we examine the price markup for hedging or liquidity costs, that customers have to pay when they buy structured products by replicating the payoff of ten different structured products and comparing their fair values with the prices actually traded. For this purpose we use parallel computing, a new technology that was not possible in the past. This allows us to use a calibrated Heston model to calculate the fair values of structured products over a longer period of time. Our results show that the markup that clients pay for these ten products ranges from 0.9%-2.9%. We can also observe that products with higher payoff levels, or better capital protection, require higher costs. We also identify market volatility as a statistically significant driver of the markup. In the third part, we show that the tracking error of an passively managed ETF can be significantly reduced through the use of optimization methods if the correlation factor between Index and ETF is used as target function. By finding optimal weights of a self-constructed bond- and the DAX- index, the number of constituents can be reduced significantly, while keeping the tracking error small. In the fourth part, we develop a hedging strategy based on fuel prices that can be applied primarily to the end users of petrol and diesel fuels. This enables the fuel consumer to buy fuel at a certain price for a certain period of time by purchasing a call option. To price the American call option we use a geometric Brownian motion combined with a binomial model.
Market Liquidity, Heston model, Geometric Brownian motion, Calibration, Optimization techniques, Compound Poisson process, Market Liquidity, Inverse transformation sampling, Pricing, Structured products, ETF tracking, Fuel price, Fuel consumption, Inflation, American call option, Binomial option pricing
Avdiu, Kujtim
2021
English
Universitätsbibliothek der Ludwig-Maximilians-Universität München
Avdiu, Kujtim (2021): Algorithmic optimization and its application in finance. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics
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Abstract

The goal of this thesis is to examine different issues in the area of finance and application of financial and mathematical models under consideration of optimization methods. Prior to the application of a model to its scope, the model results have to be adjusted according to the observed data. For this reason a target function is defined which is being minimized by using optimization algorithms. This allows finding the optimal model parameters. This procedure is called model calibration or model fitting and requires a suitable model for this application. In this thesis we apply financial and mathematical models such as Heston, CIR, geometric Brownian motion, as well as inverse transform sampling, and Chi-square test. Moreover, we test the following optimization methods: Genetic algorithms, Particle-Swarm, Levenberg-Marquardt, and Simplex algorithm. The first part of this thesis deals with the problem of finding a more accurate forecasting approach for market liquidity by using a calibrated Heston model for the simulation of the bid/ask paths instead of the standard Brownian motion and the inverse transformation method instead of compound Poisson process for the generation of the bid/ask volume distributions. We show that the simulated trading volumes converge to one single value which can be used as a liquidity estimator and we find that the calibrated Heston model as well as the inverse transform sampling are superior concerning the use of the standard Brownian motion, resp. compound Poisson process. In the second part, we examine the price markup for hedging or liquidity costs, that customers have to pay when they buy structured products by replicating the payoff of ten different structured products and comparing their fair values with the prices actually traded. For this purpose we use parallel computing, a new technology that was not possible in the past. This allows us to use a calibrated Heston model to calculate the fair values of structured products over a longer period of time. Our results show that the markup that clients pay for these ten products ranges from 0.9%-2.9%. We can also observe that products with higher payoff levels, or better capital protection, require higher costs. We also identify market volatility as a statistically significant driver of the markup. In the third part, we show that the tracking error of an passively managed ETF can be significantly reduced through the use of optimization methods if the correlation factor between Index and ETF is used as target function. By finding optimal weights of a self-constructed bond- and the DAX- index, the number of constituents can be reduced significantly, while keeping the tracking error small. In the fourth part, we develop a hedging strategy based on fuel prices that can be applied primarily to the end users of petrol and diesel fuels. This enables the fuel consumer to buy fuel at a certain price for a certain period of time by purchasing a call option. To price the American call option we use a geometric Brownian motion combined with a binomial model.