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Alexander Gairat

Publications and source records attributed to Alexander Gairat.

5 recordsLinked to original sources
arXiv2026

Explicit local volatility formula for Cheyette-type interest rate models

This paper addresses the approximation of the local volatility function in the Cheyette interest rate model. Its main contribution is an explicit analytical formula for approximating local volatility, derived by extending the classical Dupire framework to interest rate models. In particular, an implicit Dupire-like expression for local volatility is first derived for options written on the short rate. This expression is then approximated using a combination of perturbation methods and probabilistic techniques, resulting in a formula expressed in terms of time and strike derivatives of the Bachelier implied variance. The final formula naturally extends to multi-factor Cheyette models and provides a practical tool for model calibration.

q-fin.PR↗
arXiv2024

Extreme ATM skew in a local volatility model with discontinuity: joint density approach

This paper concerns a local volatility model in which volatility takes two possible values, and the specific value depends on whether the underlying price is above or below a given threshold value. The model is known, and a number of results have been obtained for it. In particular, option pricing formulas and a power law behaviour of the implied volatility skew have been established in the case when the threshold is taken at the money. In this paper we derive an alternative representation of option pricing formulas. In addition, we obtain an approximation of option prices by the corresponding Black-Scholes prices. Using this approximation streamlines obtaining the aforementioned behaviour of the skew. Our approach is based on the natural relationship of the model with Skew Brownian motion and consists of the systematic use of the joint distribution of this stochastic process and some of its functionals.

q-fin.MF↗
arXiv2022

Discrete SIR model on a homogeneous tree and its continuous limit

We study a discrete Susceptible-Infected-Recovered (SIR) model for the spread of infectious disease on a homogeneous tree and the limit behavior of the model in the case when the tree vertex degree tends to infinity. We obtain the distribution of the time it takes for a susceptible vertex to get infected in terms of a solution of a non-linear integral equation under broad assumptions on the model parameters. Namely, infection rates are assumed to be time-dependent, and recovery times are given by random variables with a fairly arbitrary distribution. We then study the behavior of the model in the limit when the tree vertex degree tends to infinity, and infection rates are appropriately scaled. We show that in this limit the integral equation of the discrete model implies an equation for the susceptible population compartment. This is a master equation in the sense that both the infectious and the recovered compartments can be explicitly expressed in terms of its solution. In other words, the master equation implies a continuous SIR model for the joint time evolution of all three population compartments.

math.PR↗
arXiv2022

Skew Brownian motion with dry friction: joint density approach

This note concerns distributions of Skew Brownian motion with dry friction and its occupation time. These distributions were obtained in [2] by using the Laplace transform and joint characteristic functions. We provide an alternative approach to deriving these distributions. Our approach is based on using the results for Skew Brownian motion obtained in [3].

math.PR↗