arXiv research

Motivated by the interplay between structural and reduced form credit models, we propose to model the firm value process as a time-changed Brownian motion that may include jumps and stochastic volatility effects, and to study the first passage problem for such processes. We are lead to consider modifying the standard f…

The paper approximates CARMA models for option pricing.

The aim of this paper is to evaluate geometric Asian option by a mixed fractional subdiffusive Black-Scholes model. We derive a pricing formula for geometric Asian option when the underlying stock follows a time changed mixed fractional Brownian motion. We then apply the results to price Asian power options on the stoc…

New findings show independent subordination is not relevant for accurate option pricing.

TCNF models SDEs using time deformation of Brownian motion.

Time-subordinated Brownian motion models improve financial market stochastic distribution.

Modeling price dynamics in AMMs with fees using geometric Brownian motion.

We consider structural credit modeling in the important special case where the log-leverage ratio of the firm is a time-changed Brownian motion (TCBM) with the time-change taken to be an independent increasing process. Following the approach of Black and Cox, one defines the time of default to be the first passage time…

The paper prices weather contracts using a complex temperature model.

A new framework for pricing the European currency option is developed in the case where the spot exchange rate fellows a time-changed fractional Brownian motion. An analytic formula for pricing European foreign currency option is proposed by a mean self-financing delta-hedging argument in a discrete time setting. The m…

This paper establishes a non-stochastic analogue of the celebrated result by Dubins and Schwarz about reduction of continuous martingales to Brownian motion via time change. We consider an idealized financial security with continuous price path, without making any stochastic assumptions. It is shown that typical price …

We compute the value of a variance swap when the underlying is modeled as a Markov process time changed by a Lévy subordinator. In this framework, the underlying may exhibit jumps with a state-dependent Lévy measure, local stochastic volatility and have a local stochastic default intensity. Moreover, the Lévy subordina…

We investigate methods for pricing American options under the variance gamma model. The variance gamma process is a pure jump process which is constructed by replacing the calendar time by the gamma time in a Brownian motion with drift, which makes it a time-changed Brownian motion. In general, the finite difference me…

This paper extends subordinated models to include stochastic time changes, improving financial modeling.

We present a new numerical method to price vanilla options quickly in time-changed Brownian motion models. The method is based on rational function approximations of the Black-Scholes formula. Detailed numerical results are given for a number of widely used models. In particular, we use the variance-gamma model, the CG…

Study on determinants of unitary Brownian motion and their asymptotic laws.

Researchers created a continuous Markov martingale that mimics Brownian motion but lacks the strong Markov property.

Study refracted skew Brownian motion, find densities and asymptotics.

Paper defines multi-dimensional fractional Brownian motion under volatility uncertainty.

New model uses generalized fractional Brownian motion for stock price prediction.

Geodesic walks converge to Brownian motion on Finsler manifolds.

Subordination is an often used stochastic process in modeling asset prices. Subordinated Levy price processes and local volatility price processes are now the main tools in modern dynamic asset pricing theory. In this paper, we introduce the theory of multiple internally embedded financial time-clocks motivated by beha…

New SDEs use $G$-Brownian motion, extending mean-field models.

We extend the now classic structural credit modeling approach of Black and Cox to a class of "two-factor" models that unify equity securities such as options written on the stock price, and credit products like bonds and credit default swaps. In our approach, the two sides of the stylized balance sheet of a firm, namel…

This paper derives the non-analytic solution to the Fokker-Planck equation of fractional Brownian motion using the method of Laplace transform. Sequentially, by considering the fundamental solution of the non-analytic solution, this paper obtains the transition probability density function of the random variable that i…

Replacing Black-Scholes' driving process, Brownian motion, with fractional Brownian motion allows for incorporation of a past dependency of stock prices but faces a few major downfalls, including the occurrence of arbitrage when implemented in the financial market. We present the development, testing, and implementatio…

The discrete sum of geometric Brownian motions plays an important role in modeling stochastic annuities in insurance. It also plays a pivotal role in the pricing of Asian options in mathematical finance. In this paper, we study the probability distributions of the infinite sum of geometric Brownian motions, the sum of …

The paper studies the question of whether the classical mirror and synchronous couplings of two Brownian motions minimise and maximise, respectively, the coupling time of the corresponding geometric Brownian motions. We establish a characterisation of the optimality of the two couplings over any finite time horizon and…

The book explores stochastic areas and heat kernels on manifolds.

Modeling financial markets with memory using fractional calculus and Brownian motion.

Estimates spectral gap for Brownian motion on sticky-reflecting domains.

Two insurance companies collaborate to maximize the probability of none going bankrupt.

The paper extends Merton model to price equity warrants under subdiffusive fractional Brownian motion of the short rate.

Solves optimal liquidation problem for stock price following geometric Brownian motion.

Study Brownian motions and heat kernel bounds on Kähler and quaternion Kähler manifolds.

Universal approximation for stochastic processes using Brownian motion.

Quaternionic Brownian motion on flag manifold linked to sphere diffusion.

Upper bounds on constants for Brownian motion with sticky boundary.

Proves CLT for Brownian paths on pinched negative curvature manifolds.

The sub-fractional Brownian motion (sfBm) is a stochastic process, characterized by non-stationarity in their increments and long-range dependency, considered as an intermediate step between the standard Brownian motion (Bm) and the fractional Brownian motion (fBm). The mixed process, a linear combination between a Bm …

Rough volatility models are becoming increasingly popular in quantitative finance. In this framework, one considers that the behavior of the log-volatility process of a financial asset is close to that of a fractional Brownian motion with Hurst parameter around 0.1. Motivated by this, we wish to define a natural and re…

Study bounds for Brownian motion on manifolds with sticky boundary conditions.

An innovative extension of Geometric Brownian Motion model is developed by incorporating a weighting factor and a stochastic function modelled as a mixture of power and trigonometric functions. Simulations based on this Modified Brownian Motion Model with optimal weighting factors selected by goodness of fit tests, sub…

A new option pricing model uses a time-varying Hurst exponent for more accurate financial predictions.

We provide an explicit formula giving the optimal number of paths needed to simulate two correlated Brownian motions.

In this paper we revisit the integral functional of geometric Brownian motion $I_t= \int_0^t e^{-(μs +σW_s)}ds$, where $μ\in\mathbb{R}$, $σ> 0$, and $(W_s )_s>0$ is a standard Brownian motion. Specifically, we calculate the Laplace transform in $t$ of the cumulative distribution function and of the probability density …

This paper constructs Brownian motion on complex flag manifolds and finds joint distribution of stochastic areas.

In this paper, we develop a theory of common decomposition for two correlated Brownian motions, in which, by using change of time method, the correlated Brownian motions are represented by a triplet of processes, $(X,Y,T)$, where $X$ and $Y$ are independent Brownian motions. We show the equivalent conditions for the tr…