To Örebro University

We propose several new models in finance known as the Fractal Activity Time Geometric Brownian Motion (FATGBM) models with Student marginals. We summarize four models that construct stochastic processes of underlying prices with short-range and long-range dependencies. We derive solutions of option Greeks and compare with those in the Black-Scholes model. We analyse performance of delta hedging strategy using simulated time series data and verify that hedging errors are biased particularly for long-range dependence cases. We also apply underlying model calibration on S&P 500 index (SPX) and the U.S./Euro rate, and implement delta hedging on SPX options.

This article presents a comprehensive study on developing a predictive product pricing model using LightGBM, a machine learning method optimized for regression challenges in situations with limited historical data. It begins by detailing the core principles of LightGBM, including gradient descent, and then delves into the method's unique features like Gradient-based One-Side Sampling (GOSS) and Exclusive Feature Bundling (EFB). The model's efficacy is demonstrated through a comparative analysis with XGBoost, highlighting Light-GBM's enhanced efficiency and slight improvement in prediction accuracy. This research offers valuable insights into the application of LightGBM in developing fast and accurate product pricing models, crucial for businesses in the rapidly evolving data landscape.

The article focuses on Value-at-risk measuring for options in situations characterized by the lack of liquidity when the underlying stock price has motionless periods. A similar behavior can be observed in physical systems exhibiting sub-diffusion. In the considered sub-diffusive model, the bond movement and stock process are time-changed by the stochastic clock with gamma subordinator. In the model, the two techniques for option pricing were considered. The first very common approach for the time-changed model is to find option prices as the discounted expected payoff under the risk-neutral measure. The second technique for option pricing is based on a fractional version of what is called Dupire's equation. The Value-at-Risk evaluating procedure for the proposed model was discussed and we show that this procedure is based on the Fractional Fokker-Planck equation (FFPE).

The prediction of stochastic processes and the estimation of random fields of different natures is becoming an increasingly common field of research among scientists of various specialties. However, an analysis of papers across different estimating problems shows that a dynamic approach over an iterative and recursive interpolation of random fields on fractal is still an open area of investigation. There are many papers related to the interpolation problems of stationary sequences, estimation of random fields, even on the perforated planes, but all of this still provides a place for an investigation of a more complicated structure like a fractal, which might be more beneficial in appliances of certain industry fields. For example, there has been a development of mobile phone and WiFi fractal antennas based on a first few iterations of the Sierpinski carpet. In this paper, we introduce an estimation for random fields on the Sierpinski carpet, based on the usage of the known spectral density, and calculation of the spectral characteristic that allows an estimation of the optimal linear functional of the omitted points in the field. We give coverage of an idea of stationary sequence estimating that is necessary to provide a basic understanding of the approach of the interpolation of one or a set of omitted values. After that, the expansion to random fields allows us to deduce a dynamic approach on the iteration steps of the Sierpinski carpet. We describe the numerical results of the initial iteration steps and demonstrate a recurring pattern in both the matrix of Fourier series coefficients of the spectral density and the result of the optimal linear functional estimation. So that it provides a dependency between formulas of the different initial sizes of the field as well as a possible generalizing of the solution for N-steps in the Sierpinski carpet. We expect that further evaluation of the mean squared error of this estimation can be used to identify the possible iteration step when further estimation becomes irrelevant, hence allowing us to reduce the cost of calculations and make the process viable.

Fractional Brownian motion (fBM) belongs to the class of long-range dependent systems with self-similarity property and has been widely used in different applications. Mathematically speaking, the scaled fBm is fully characterised by its scaling parameter σ > 0 and Hurst parameter H ∈ (0, 1). But in spite of many obvious advantages, for many real-life data with long-range dependence, the classical fBM with Gaussian property cannot be considered an appropriate model. For example the classical fBM cannot model the real time series with apparent constant time periods (called also trapping events), which are often observed in data sets recorded within various fields. One of the possible solutions is the time-changed fBM BHLt with α-stable L ´evy subordinator (Lt)t≥0.

We construct consistent estimators of the parameters (σ, α, H) for the time-changed fBM Xt =BHLt . Our approach is based on the limit theory for stationary increments of a linear fractional stable motion [1]. We use these techniques, combine negative power variation statistics and their empirical expectations and covariances to obtain consistent estimates of (σ, α, H). We show that Xt is a symmetric H/α-stable L ´evy process and for p < α we deduce the law of large numbers. The above law of large numbers immediately gives a consistent estimator of the self-similarity parameter H/α of Xt. In order to estimate the other parameters of the model we use the some identities, which has been shown in [2]. Finally we present the statistical inference and prove some weak limit theorems for the all parameters (σ, α, H) using classical delta-method.

References:

[1] Mazur, S., Otryakhin D. and Podolskij M., Estimation of the linear fractional stable motion, Bernoulli, 26(1), (2020) 226-252.

[2] Dang, T.T.N., Istas, J., Estimation of the Hurst and the stability indices of a H-self-similar stable process, Electronic Journal of Statistics, 11(2), (2018) 4103-4150.

We propose several new models in ecophysics known as the Fractal Activity Time Geometric Brownian Motion (FATGBM) models with Student marginals. We summarize four models that construct stochastic processes of underlying prices with short-range and long-range dependencies. We derive solutions of option Greeks and compare with those in the Black-Scholes model. We analyse perfor-mance of delta hedging strategy using simulated time series data and verify that hedging errors are biased particularly for long-range dependence cases. We also apply underlying model calibration on S&P 500 index (SPX) and the U.S./Euro rate, and implement delta hedging on SPX options.

In this  paper we propose an approach to option pricing which is based on the solution of the investor problem. We demonstrate that the link between optimal option pricing from investor’s point of view and risk measuring is especially close, and it is given by stochastic optimization. We consider the optimal option pricing X∗ as the optimal decision of the investor, who should maximize the expected profit. It is possible because the average value-at-risk AV@R is related to the simple stochastic optimization problem with a piecewise linear profit/cost function and as it was proved in [12], maximal value is attained. If we consider investing in a European option, then the profit/cost function is a payoff function Y(S) of a European call or put option and the optimal decision can be found as X∗=V@Rα(Y), where parameter α can be computed using interest rates for borrowing and lending and reflects the level of the real economic environment. We illustrate our results for GBM model and Student-like models with dependence (FAT models) and determine optimal option price as the optimal amount to invest for these cases. Meanwhile we measure and manage risk for these models.

The paper focuses on option pricing under unusual behaviour of the market, when the price may not be changed for some time what is quite a common situation on the modern financial markets. There are some patterns that can cause permanent price gaps to form and lead to illiquidity. For example, global changes that have a negative impact on financial activity, or a small number of market participants, or the market is quite young and is just in the process of developing, etc.

In the paper discrete and continuous time approaches for modelling market with illiquidity and evaluation option pricing were considered.Trinomial discrete time model improves upon the binomial model by allowing a stock price not only to move up, down but stay the same with certain probabilities, what is a desirable feature for the illiquid modelling. In the paper parameters for real financial data were identified and the backward induction algorithm for building call option price trinomial tree was applied.Subdiffusive continuous time model allows successfully apply the physical models for describing the trapping events to model financial data stagnation's periods. In this paper the Inverse Gaussian process IG was proposed as a subordinator for the subdiffusive modelling of illiquidity and option pricing. The simulation of the trajectories for subordinator, inverse subordinator and subdiffusive GBM were performed. The Monte Carlo method for option evaluation was applied.

Our aim was not only to compare these two models each with other, but also to show that both models adequately describe the illiquid market and can be used for option pricing on this market. For this purpose absolute relative percentage (ARPE) and root mean squared error (RMSE) for both models were computed and analysed.

Thanks to the proposed approaches, the investor gets a tools, which allows him to take into account the illiquidity.