Conditional GBM Moments for Paths That Stay Above a Barrier
Summary
The document considers geometric Brownian motion paths that remain above a lower price barrier throughout a fixed horizon. It starts from a supplied survival probability and frames the requested conditional mean and variance of the terminal price in terms of truncated moments: the expected terminal price, and its second moment, restricted to paths whose running minimum stays above the barrier. Dividing these quantities by the survival probability gives the conditional moments, from which variance follows.
The response identifies the numerator for the mean with a zero-strike down-and-out call quantity and suggests obtaining the second moment through replication with down-and-out calls across strikes. Another route is to derive the joint distribution of terminal price and running minimum, then integrate over surviving paths. It cautions that discounting must be handled correctly. The question about expected time spent below the barrier is left unresolved and may involve local time, so the note does not provide a complete solution or explicit formulas for all requested quantities.
Key ideas
- Condition on survival by restricting terminal-price moments to paths whose running minimum exceeds the barrier.
- Compute the conditional mean as a survival-weighted terminal-price moment divided by survival probability.
- The conditional second moment can be used with the conditional mean to obtain variance.
- A zero-strike down-and-out call is related to the numerator for the conditional mean.
- Expected time below the barrier remains unspecified and may require a local-time treatment.
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Full text
# Trying to derive some properties of geometric brownian motion
# Trying to derive some properties of geometric brownian motion
I am trying to derive some properties of geometric brownian motion: $dS_t = \mu S_t dt + \sigma S_t dW_t$
I am interested in analyzing paths that 'survive' a lower boundary $X$ i.e. always stay above $X$ during time $T$.
I have found that the probability of survival can be calculated by: $P\left( S_t > X, \, \forall t \in [0, T] \right) = \Phi\left( \frac{\ln\left(\frac{S_0}{X}\right) + \left(\mu - \frac{1}{2} \sigma^2\right) T}{\sigma \sqrt{T}} \right) - \left( \frac{X}{S_0} \right)^{\frac{2\mu}{\sigma^2}-1} \Phi\left( \frac{-\ln\left(\frac{S_0}{X}\right) + \left(\mu - \frac{1}{2} \sigma^2\right) T}{\sigma \sqrt{T}} \right)$
I am interested in three things:
- What is the expected value of $S_T$ of all the paths that 'survived' and never went below $X$ $\mathbb{E}[S_T \mid S_t > X, \forall t \in [0, T]]$
- What is the variance of $S_T$ of the survived paths?
- For the paths that touched or went below $X$: How can I calculate the expected time the price went below?
Ideally I'm looking for a formula and a simple proof, but if someone could give me a link to a paper or search terms that would be a great help too. Thank you!
## Answer by Andrea (score 2)
https://quant.stackexchange.com/a/81304
You could start writing out exactly what you need
- Call $M_T = \min_{t<T} S_t$
- for the mean, I guess you want $\mathbb{E}[S_T \mid M_T > X] = \frac{\mathbb{E}[S_T \mathbb{1}_{M_T > X}]}{P(M_t>X)}$
- 2nd moment, similar: $\mathbb{E}[S_T^2 \mid M_T > X] = \frac{\mathbb{E}[S_T^2 \mathbb{1}_{M_T > X}]}{P(M_t>X)}$
And from this the variance. Do you agree so far? (For your point 3, it would be better to spell it out, it smells of local time).
You already have a formula for $P$. The numerator for the mean is the same as a 0-strike down-and-out call (which there are formulas for).
The only way I know how to get $\mathbb{E}[S_T^2 \mid M_T > X]$ is via replication with a portfolio of down-and-out calls (with multiple strikes).
If you do the above, double check you are handling the discount factor correctly.
Alternatively, you could study the derivation of the $P$ you copied above and see if you can get the joint distribution of $S_T$ and $M_T$.
You can find something very similar in Karatzas & Shreve in proposition 8.1, but it is for a Brownian Motion without a drift (which you do have). This question asks the same thing: https://math.stackexchange.com/questions/3333028/joint-density-function-of-brownian-motion-with-drift-and-its-running-maximum
Once you get it, you just integrate it.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.