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In this scenario, the "joint dynamics" are trivially computed since the option value is a known deterministic function of stock price. For example, the mean of the option value for time $\tau$ is $$ \mu_O = \int_0^\infty BSM( S_\tau ) p(S_\tau) dS_\tau $$ which is best computed using quadrature as available in standard numerical libraries like scipy. The ...

So you want to calculate $\mathbb{P}[B_1 > B_0 + \varepsilon \;|\; A_1 > A_0 + \varepsilon]$? If you truly have the joint distribution of $A_1$ and $B_1$ and the current prices $A_0$ and $B_0$, this just becomes a simple exercise in integration, by the definition of probability density. Are you asking how to find a conditional probability in general, ...

...do you know of any standard libraries that can calculate the joint probability of stocks A and B, given a time series of historical data? Using R and the LSPM package with the code posted here might be what you are looking for.

I think an extremely interesting strand of research on this topic is represented by extensions of vine copulas with time-varying parameters. For vine copulas in general have a look at this site from the Technische Universität München: Vine Copula Models One of their research projects, which is the most relevant in this context, is:Time varying vine copula ...

Once we start building time-varying copulas like Lopes suggests in that paper, I think we are better off venturing into the world of state space models. When viewed in a bayesian context, the similarities between the approaches are striking to me. The advantage of the copula, as I understand it, is that it is a quick and dirty way to understand the ...