Estimating Barrier-Touch Probabilities for Correlated Baskets

Summary

The document asks how to calculate the probability that a weighted basket of assets touches a barrier during a time interval when each asset follows geometric Brownian motion with constant volatility and correlations. The endpoint prices are known, so the target is a conditional probability. It also asks how the problem changes for best-of and worst-of baskets.

No solution or evidence is provided; the text describes the modeling question and its assumptions. It highlights that the single-asset barrier-touch calculation does not directly settle the basket case, where dependence between components matters. Any practical calculation would need to account for the joint paths of the assets, not only their endpoint values or individual touch probabilities.

The discussion is limited to framing the question: it gives no proposed method, derivation, numerical results, or references. Best-of and worst-of payoffs are mentioned as extensions, but are not analyzed.

Key ideas

• The question concerns the conditional probability that a weighted basket touches a barrier within a time interval.
• The assumed asset dynamics are geometric Brownian motion with constant volatility and correlation.
• Known endpoint prices make the desired probability conditional on the assets' ending values.
• Best-of and worst-of baskets are raised as related cases, without a solution.

Tags

# Taking into account the correlation in Barrier options on a Basket


# Taking into account the correlation in Barrier options on a Basket












In a Barrier option (where the contract cancels when the underlying hits the barrier) I succesfully found the way to compute the probability of a single underlying touching the barrier (with constant volatility). The problem comes when I have an average basket of underlyings $B_t$ $$ B_t = \sum_{i=1}^n \omega_iS_t,\qquad \sum_{i=1}^n\omega_i=1\ . $$ I don't know how to properly calculate the probability of the basket $B_t$ touching the barrier. I think I need to take into account the correlations $\langle dW_t^{i},dW_{t}^{j}\rangle=\rho_{ij}$ between assets but i just don't know how to do it.

I'm considering constant volatility and constant correlation thoughout an arbitrary time step.

Extra: Same question if the basket is Best Of/Worst Of type.

I'm not expecting a complete developement of the answer but I would find very helpfull any indications or references. Thanks a lot.

Edit: Yes, i'm considering that Stock prices follow geometric Brownian motions. I also forgot to say that the prices at the end of the time interval are known, so the probabilities are actually conditioned probabilitites.

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