Bivariate Normal Method for a Joint GBM Threshold Event

Summary

The document considers the probability that one of two correlated geometric Brownian motion prices finishes above the other, while the lower price also exceeds a threshold. It rewrites the event using shifted Brownian variables as two simultaneous inequalities: a condition on the difference between the Brownian terms and a condition on the second term. The transformed pair is jointly normal, so the probability can be expressed using a bivariate normal cumulative distribution function.

This approach turns a region involving two terminal prices into a standard bivariate normal probability problem. Applying it requires mapping log prices under geometric Brownian motion into normal variables and retaining the induced means, variances, and correlation. The document gives the event transformation but does not write out the final standardized CDF formula or show numerical examples. Its presentation also compresses the GBM-to-normal transformation into scalar shifts, so those parameters must be derived carefully for the specific initial prices, drifts, volatilities, and horizon.

Key ideas

• A joint terminal-price threshold event can be rewritten as two inequalities in transformed normal variables.
• The difference of correlated Brownian terms and one original term form a bivariate normal pair.
• The event probability can therefore be evaluated with a bivariate normal cumulative distribution function.
• The shifts and correlation must reflect the GBM parameters and horizon, though the document does not give the full standardized formula.

Tags

# Closed form expression for $\Bbb E(\mathbb{I}_{\{S_{1,T}>S_{2,T}>K \}})$


# Closed form expression for $\Bbb E(\mathbb{I}_{\{S_{1,T}>S_{2,T}>K \}})$












Is it possible to calculate analytically $\Bbb E(\mathbb{I}_{\{S_{1,T}>S_{2,T}>K \}})$, using the 2-dimensional normal probability function $\Phi_2$, where $S_{1,T}$ and $S_{2,T}$ follow geometric Brownian motion with corrrelation $\rho$.

In fact, calculate $\Bbb E(\mathbb{I}_{\{S_{1,T}>S_{2,T}>K \}})$ is equal to calculate $$\Bbb P(W_{1,T} + a>W_{2,T} +b > K)$$ where $a,b, K$ are scalar, $K>0$ and $W_{1,T},W_{2,T}$ are two Brownian motions of $S_{1,T}$ and $S_{2,T}$.

I believe it is possible but have not found yet the formula!

Finally, I found the closed form expression for this. It suffices to re-write the integration region as $\{W_{1,T}-W_{2,T}>b-a, W_{2,T}>K-b \}$. As the bivariate $(W_{1,T}-W_{2,T},W_{2,T} )$ follows a bivariate normal distribution, we can obtain easily the probability via the bivariate normal probability function $\Phi_2$.

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.