Discrete Barrier Simulation and Brownian Bridge Corrections
Summary
The document compares two estimates of the probability that a drifted Brownian process stays below an upper barrier over a trading year. A discrete Monte Carlo simulation checks the process at each daily step, while a Brownian bridge calculation uses only the start and end values to account for possible barrier crossings between them. The reported estimates are about 63% and 61%, respectively.
The difference arises because the bridge formula assumes continuous monitoring, whereas the simulation observes the process only at discrete times. The document points to a continuity correction for discrete barrier options, which adjusts the barrier using a term based on volatility and the monitoring interval. It does not provide a derivation, implementation details, or a wider accuracy study, so the reported gap should be read as an example rather than a general error bound.
Key ideas
- Discrete monitoring can miss barrier crossings that occur between observation times.
- A Brownian bridge uses endpoint values to estimate crossing probability under continuous paths.
- The example reports different survival probabilities for daily simulation and continuous bridge estimation.
- A continuity correction adjusts the barrier to account for discrete monitoring.
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Full text
# Barrier Reaching Probability using Monte Carlo and Brownian Bridge
# Barrier Reaching Probability using Monte Carlo and Brownian Bridge
Considering a normal distributed stochastic process $X_t$, such that \begin{equation} X_t = X_0 + \mu t + \sigma\sqrt{t}Z_t, \qquad Z_t\sim\mathcal{N}(0,1) \end{equation} Assume that a year has $T = 250$ trading days, my goal is to find probability of "all 250 trading days" are below the upper barrier $U$, $X_0<U$.
Firstly, I generated 10 Million x 250 random paths using annualized $\sigma = 0.003$, $U = X_0 + 20$bp and find out the probability is about 63%.
Secondly, I selected the 1st and 250th column of random path only, then calculated probability using Brownian Bridge such that
\begin{equation} P_i(X_0^i,X_T^i,T,U,\sigma) = \left\{\begin{matrix}1, \text{if $X_0$ or $X_T$ is above $U$}\\ \exp\left(-\frac{2\left(U-X_0^i\right)\left(U-X_T^i\right)}{T\sigma^{2}}\right), \text{otherwise}\end{matrix}\right.\qquad \text{for}~i=1,\cdots,10000000 \end{equation}
By taking the average of $P_i(X_0^i,X_T^i,T,U,\sigma)$, I got approximately $61\%$. Why there is a $2\%$ differences between two methods? Is it because $T$ is too large? Or $\sigma$ is too large comparing with the gap between $U$ and $X_0$? How can I improve the estimation accuracy?
Many thanks!
The closed-form formula (the second one) is developped with the assumption that the brownian motion is continuous while in the MC method, the brownian motion is discretized. The Glasserman's paper "A Continuity Correction for Discrete Barrier Options" offers a solution by adjusting the upper barrier by $\zeta(\frac{1}{2})/\sqrt{2\pi} \sigma /\sqrt{252}$, where $\zeta(\cdot)$ is the zeta function.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.