In this article, we consider the problem of modeling the dynamics of cryptocurrencies using a wide class of tempered stable Levy processes. At the first step, the generalized Blumenthal-Getour index is estimated based on the realized power-law variation of a number of logarithmic returns of cryptocurrency rates. We consider the case when the jump activity index is less than one, which corresponds to Levy processes of bounded variation. The modeled process is then presented as a sequence of positive and negative Levy jumps over short periods of time. We calibrate the model to one-touch artificial digital options, which are the statistical probabilities of crossing predetermined barriers, and we suggest using the FFT in conjunction with the Wiener-Hopf method. The main advantage of our approach is the use of explicit Wiener-Hopf factorization formulas to determine the price of one-touch synthetic digital options within such models. The proposed technique simplifies the fitting of parameters of non-Gaussian Levy processes with jump activity not exceeding 1. In essence, we replace the original continuous-time process with a discrete process that approximates a tempered stable Levy process.

Keywords: mathematical modeling, cryptocurrencies, Levy models, Tempered Stable Levy models, CGMY modes, Blumenthal-Getoor index, options

In the paper, we consider the problem of modeling the dynamics of leading cryptocurrencies such as Bitcoin (BTC) and Etherium (ETH). We calculate the log-returns based on the time series of cryptocurrency rates and analyze the realized power variation to estimate the corresponding generalized Blumenthal-Getoor index. The analysis shows that tempered stable Levy processes without a diffusion component are suitable for modeling the cryptocurrency rates considered. To obtain a more accurate estimate of the parameter that describes the activity of jumps in the log-returns, we exclude the influence of drift by considering auxilary series of increments of the initial log-returns.

Keywords: mathematical modeling, cryptocurrencies, Levy models, Tempered Stable Levy models, CGMY modes, Blumenthal-Getoor index, realized power variation

The article investigates approaches to managing currency risks on the Moscow exchange based on static hedging and taking into account the features of the Russian derivatives market. For more delicate managing currency risks, it is proposed to use barrier options that are not traded on the Moscow exchange. It is shown that the barrier option can be replicated with the portfolio of European options, the strike prices of which are the barrier and the strike price of the barrier option. The main idea behind the approach applied is replication of zero price of the up-an-out call (in the case of crossing the barrier) by the linear combination of European call options with different time to expiry. Examples of constructing replicating portfolios of options on the futures contract for the US dollar - Russian ruble are given . Further analysis of the portfolio's value dynamics demonstrates the inadequacy of the classical Black-Scholes model for the Russian derivatives market. The approaches of static hedging barrier options in jump models are disscussed.

Keywords: mathematical modeling, numerical method, mathematical finance, barrier options, Black-Scholes model, static hedging, Levy models, derivatives market.

The article considers a problem of the Russian financial market modeling and suggests wide classes of diffusion and jump models. The authors examine the adequacy of the diffusion models by means of quadratic variation analysis on the Russian financial market, which is used to construct the volatility index, the most important quantitative risk indicator in the financial market. It is shown that the existing volatility indexes RTSVX and RVI worse approximate the realized variation than the alternative index, based on the implied variation integration. Further analysis of power variation of log-returns for RTS index shows that Levy processes with unbounded variation without diffusion component better describe the dynamics of the Russian financial market. A new model-free formula for the volatility index is proposed in the class of Lévy processes. The new formula suggested is based on the variation representation via market option prices.

Keywords: mathematical modeling, numerical method, mathematical finance, volatility index, option, Levy process, diffusion model, quadratic variation, RTS index, derivatives market.