# Convergence of Milstein Brownian bridge Monte Carlo methods and stable Greeks calculation

## TL;DR

This paper introduces a new stable Monte Carlo method combining Brownian bridge and one-step survival techniques for accurate and efficient sensitivity analysis of barrier options, achieving stable Greeks and variance reduction.

## Contribution

It develops a novel one-step survival Brownian bridge approach that ensures stable second-order Greeks and improves computational efficiency for barrier option pricing.

## Key findings

- Proves weak convergence and variance bounds for the new method.
- Demonstrates stable second-order Greeks with unbiased estimates.
- Develops a multilevel Monte Carlo algorithm for cost reduction.

## Abstract

We consider the pricing and the sensitivity calculation of continuously monitored barrier options. Standard Monte Carlo algorithms work well for pricing these options. Therefore they do not behave stable with respect to numerical differentiation. One would generally resort to regularized differentiation schemes or derive an algorithm for precise differentiation. For barrier options the Brownian bridge approach leads to a precise, but non-Lipschitz-continuous, first derivative. In this work, we will show a weak convergence of almost order one and a variance bound for the Brownian bridge approach. Then, we generalize the idea of one-step survival, first introduced by Glasserman and Staum, to general scalar stochastic differential equations and combine it with the Brownian bridge approach leading to a new one-step survival Brownian bridge approximation. We show that the new technique can be adapted in such a way that its results satisfies stable second order Greeks. Besides studying stability, we will prove unbiasedness, leading to an uniform convergence property and variance reduction. Furthermore, we derive the partial derivatives which allow to adapt a pathwise sensitivity algorithm. Moreover, we develop an one-step survival Brownian bridge Multilevel Monte Carlo algorithm to greatly reduce the computational cost in practice.