Conditional GBM Probabilities from an Observed Price

Summary

The document explains how to find the distribution of a geometric Brownian motion price at a future time when the price at an intermediate time is known. Under constant drift and volatility, the future price is the observed intermediate price multiplied by a lognormal factor driven by a standard normal random variable. This gives a direct way to calculate the chance that the future price exceeds a threshold.

For an interval of possible prices, the answer describes the conditional probability as the difference between the conditional cumulative probabilities at its upper and lower endpoints. The result relies on the GBM model and its independent future increments; the stated formula conditions on the information available at the intermediate time. The original threshold event compares the later price with the earlier price, which is treated as known. The interval example assumes a compact interval and does not elaborate on more general measurable sets or boundary conventions.

Key ideas

• With constant drift and volatility, GBM evolves from a known intermediate price through a lognormal multiplier.
• The future price conditional on information at the intermediate time depends on the remaining time interval.
• A threshold probability can be evaluated using the conditional future-price distribution.
• The probability of an interval can be found by subtracting cumulative probabilities at its endpoints.
• The derivation assumes the GBM model and its independent increments.

Tags

# $P(S_T > S_u \mid S_v = s_*)$


# $P(S_T > S_u \mid S_v = s_*)$












Let $u < v < T$ and assume $S_t$ follows a lognormal $((\mu - \sigma^2/2)t, \sigma^2 t)$ process. I'm interested in computing the conditional probability $$ P(S_T > S_u \mid S_v = s_*) $$ where $s_* \in (0, \infty)$ is some known value (as is $S_u$). The idea is that we've already generated the value $s_*$ and, based on this, I'd like to compute the probability the stock is above some value at time $u < v$. My thought is that I can simply write $$ P(S_T > S_u \mid S_v = s_*) = P(s_* e^{(\mu - \sigma^2/2)(T-v) + \sigma\sqrt{T - v}Z}> S_u) $$ but I'm not confident about the derivation.

Even better, I'd like to work out $$ P(S_T \in B \mid \mathcal{F}_v) $$ for general $B \in \mathcal{B}$, but I thought I'd start with a concrete probability first.

## Answer by Quantuple (score 2)

https://quant.stackexchange.com/a/30154

If you assume that $(S_t)_{t\geq0}$ is a GBM with constant drift $\mu$ and volatility $\sigma$, then conditionally on $\mathcal{F}_v$ you can indeed write that: $$ S_T = s_* \exp\left({\left(\mu - \frac{\sigma^2}{2}\right)(T-v) + \sigma\sqrt{T - v}Z} \right)$$ for any $T \geq v \geq 0$ provided $Z \sim N(0,1)$ and $S_v = s_*$ a.s.

This can be shown using Itô's lemma. Thus what you wrote is perfectly fine.

Now assuming that $\mathcal{B}$ is a compact set $\mathcal{B}:=[a,b] \in \Bbb{R}$, then $$ P(S_T \in \mathcal{B} \,\vert\, \mathcal{F}_v) = P(S_T < b \,\vert\, \mathcal{F}_v) - P(S_T < a \,\vert\, \mathcal{F}_v) $$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.