# Towards the Exact Simulation Using Hyperbolic Brownian Motion

## TL;DR

This paper develops an expansion for the transition density of Hyperbolic Brownian motion with drift, aiding in option pricing and hedging under stochastic volatility models, with a simplified proof under certain drift conditions.

## Contribution

It introduces a new expansion for the transition density of Hyperbolic Brownian motion with drift, simplifying the analysis for financial applications.

## Key findings

- Provides a new transition density expansion for Hyperbolic Brownian motion with drift.
- Simplifies the proof under specific drift conditions.
- Potentially useful for pricing and hedging in stochastic volatility models.

## Abstract

In the present paper, an expansion of the transition density of Hyperbolic Brownian motion with drift is given, which is potentially useful for pricing and hedging of options under stochastic volatility models. We work on a condition on the drift which dramatically simplifies the proof.