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This paper studies the method of constructing high order recombined multinomial tree based on fast Fourier transform (FFT), and applies multinomial tree option pricing under the Lévy process. First, the Lévy option pricing model and Fourier transform are introduced. Then, the network model based on FFT (Markov chain) is presented. After that, a method of constructing a recombined multinomial tree based on FFT is given. It is proved that the discrete random variables corresponding to the multinomial tree converge to the Lévy distributed continuous random variable. Next, we obtain the European option pricing formula of FFT multinomial tree pricing, and apply the reverse iteration method to the American option pricing. Finally, under the Jump-diffuse process, the difference between the computational accuracy and computational efficiency of the Semi-analytical solution of European Option and Merton European Call Option which are priced under FFT is compared. The results show that the method of constructing a high-order recombined multinomial tree based on FFT has very high calculation precision and calculation speed, which can solve the problem of traditional risk-neutral multinomial tree construction and it is a promising pricing method for derivative products.

Proces decentralizacije i dekarbonizacije elektroenergetskog sustava uzrokuje ekstremne volatilnosti na spot tržištu električne energije. Kako bi se sudionici na tržištu zaštitili tog od rizika i trgovali na učinkovito, važno je upotrijebiti prikladne matematičke modele za predikciju cijena kratkoročnih i dugoročnih tržišta. Glavne karakteristike cijena koje modeli moraju zadovoljiti su sezonalnost, mean reversion proces i iznenadne skokove u cijenama. U radu su obrađena tri modela, koja uspijevaju pokriti neke ili sve od njih. Prvi dva modela su jedno-faktorski modeli s Gaussovom razdiobom i sa inverznom Gaussovom razdiobom, bazirani na Ornstein-Uhlenbeck i Lévyevom procesu. Iako uspješno pokrivaju sezonalni trend i povratak u prosjek, ne uspijevaju replicirati skokove i osigurati odgovarajuću dinamiku cijena. Uvođenje dodatnog faktora u trećem modelu, baziranog na Poissonovom procesu, uspješno rješava problem iznenadnih skokova i povećava volatilnost u cijenama forward ugovora, no traži nešto kompleksniji način estimacije parametara. The processes of decentralization and decarbonisation of the electricity system causes extreme volatility in the spot electricity market. To protect market participants from price risks and make trading efficient, it is important to use appropriate mathematical models to predict the price on short-term and long-term markets. The main price characteristics that models must meet are seasonality, mean reversion and jumps process. The thesis deals with three models, which manage to cover some or all of them. The first two models are one-factor models with Gaussian distribution and with inverse Gaussian distribution, based on the Ornstein-Uhlenbeck and Lévy process. Although they successfully cover the seasonal trend and mean reversion process, they fail to replicate the jumps and ensure an appropriate dynamic price. By introducing an additional factor in the third model, based on the Poisson process, solving the problem of sudden jumps and increasing volatility in forward contract prices is done more successfully, with somewhat more complex way of estimating the parameters required.