arXiv subjects

Grzegorz Krzyżanowski

Publications and source records attributed to Grzegorz Krzyżanowski.

6 recordsLinked to original sources

arXiv2022

About subordinated generalizations of 3 classical models of option pricing

In this paper, we investigate the relation between Bachelier and Black-Scholes models driven by the infinitely divisible inverse subordinators. Such models, in contrast to their classical equivalents, can be used in markets where periods of stagnation are observed. We introduce the subordinated Cox-Ross-Rubinstein model and prove that the price of the underlying in that model converges in distribution and in Skorokhod space to the price of underlying in the subordinated Black-Scholes model defined in [31]. Motivated by this fact we price the selected option contracts using the binomial trees. The results are compared to other numerical methods.

math.NA↗

arXiv2022

A tempered subdiffusive Black-Scholes model

In this paper, we focus on the tempered subdiffusive Black-Scholes model. The main part of our work consists of the finite difference method as a numerical approach to the option pricing in the considered model. We derive the governing fractional differential equation and the related weighted numerical scheme. The proposed method has the $2-α$ order of accuracy with respect to time, where $α\in(0,1)$ is the subdiffusion parameter, and $2$ with respect to space. Furthermore, we provide the stability and convergence analysis. Finally, we present some numerical results.

math.NA↗

arXiv2020

A computational weighted finite difference method for American and barrier options in subdiffusive Black-Scholes model

Subdiffusion is a well established phenomenon in physics. In this paper we apply the subdiffusive dynamics to analyze financial markets. We focus on the financial aspect of time fractional diffusion model with moving boundary i.e. American and barrier option pricing in the subdiffusive Black-Scholes (B-S) model. Two computational methods for valuing American options in the considered model are proposed - the weighted finite difference (FD) and the Longstaff-Schwartz method. In the article it is also shown how to valuate numerically wide range of barrier options using the FD approach.

q-fin.CP↗

arXiv2020

Performance analysis of Zero Black-Derman-Toy interest rate model in catastrophic events: COVID-19 case study

In this paper we continue the research of our recent interest rate tree model called Zero Black-Derman-Toy (ZBDT) model, which includes the possibility of a jump at each step to a practically zero interest rate. This approach allows to better match to risk of financial slowdown caused by catastrophic events. We present how to valuate a wide range of financial derivatives for such a model. The classical Black-Derman-Toy (BDT) model and novel ZBDT model are described and analogies in their calibration methodology are established. Finally two cases of applications of the novel ZBDT model are introduced. The first of them is the hypothetical case of an S-shape term structure and decreasing volatility of yields. The second case is an application of the ZBDT model in the structure of United States sovereign bonds in the current $2020$ economic slowdown caused by the Coronavirus pandemic. The objective of this study is to understand the differences presented by the valuation in both models for different derivatives.

q-fin.PR↗

arXiv2020

Zero Black-Derman-Toy interest rate model

We propose a modification of the classical Black-Derman-Toy (BDT) interest rate tree model, which includes the possibility of a jump with small probability at each step to a practically zero interest rate. The corresponding BDT algorithms are consequently modified to calibrate the tree containing the zero interest rate scenarios. This modification is motivated by the recent 2008-2009 crisis in the United States and it quantifies the risk of a future crises in bond prices and derivatives. The proposed model is useful to price derivatives. This exercise also provides a tool to calibrate the probability of this event. A comparison of option prices and implied volatilities on US Treasury bonds computed with both the proposed and the classical tree model is provided, in six different scenarios along the different periods comprising the years 2002-2017.

econ.EM↗

arXiv2020

A weighted finite difference method for subdiffusive Black Scholes Model

In this paper we focus on the subdiffusive Black Scholes model. The main part of our work consists of the finite difference method as a numerical approach to the option pricing in the considered model. We derive the governing fractional differential equation and the related weighted numerical scheme being a generalization of the classical Crank-Nicolson scheme. The proposed method has $2-α$ order of accuracy with respect to time where $α\in(0,1)$ is the subdiffusion parameter, and $2$ with respect to space. Further, we provide the stability and convergence analysis. Finally, we present some numerical results.

cs.CE↗