# Tag Info

9

Why not using the so simple Monte-Carlo estimator $\hat{p}_N =\frac{ \sum_{i=1}^N 1_{|A_{i+1}-A_i|>0 \cap |B_{i+1}-B_i|>0}} {\sum_{i=1}^N 1_{|A_{i+1}-A_i|>0 }}$ where $1_{|A_{i+1}-A_i|>0}$ is $1$ if stock $A$ has moved at time $i+1$

8

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 ...

7

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, ...

5

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 ...

5

...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.

3

You can use copulas. The probability that B rises given A rises is $P(- R_B < 0 | - R_A < 0) = \frac{P(-R_B < 0, - R_A < 0)}{P(-R_A < 0)} = \frac{C(F_{-B}(0),F_{-A}(0))}{F_{-A}(0)}$. You can specify the marginals as a GARCH process and use either non parametric or parametric copulas to get your final conditional probability.

2

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 ...

2

if you agree that the marginal probability $P(u\le Y\le v)=F_Y(v)-F_Y(u)$, then your formula follows immediately, because next you simply plug the marginals into the copula. your 3rd equation for the joint probabilities is incorrect for $P(Z\le z,u\le Y\le v)$, I'm not sure where you got it from

1

A few initial observations but the quick answer is your Option 2. (1) Assuming both prepayment and default can occur only at discrete time-points is not strictly correct since a borrower has the right to payoff the loan in full at any time between payment dates. (2) Default is a nebulous concept - are you referring to the act of missing a payment itself, or ...

1

Suppose you would like to compute \begin{align} Q_1(x_1,x_2;B) &= \Bbb{E}[X_1\max(B-X_2,0)]\\ Q_2(x_1,x_2;B) &= \Bbb{E}[X_2\max(X_1-B,0)] \end{align} where you know the marginal probability density functions $p_{X_1}(u)$ and $p_{X_2}(v)$. Let's start by focusing on $Q_1$. By definition, the expectation equivalently writes:  Q_1(x_1,x_2;B) = \...

1

I have written R code for some time-varying bivariate fat-tailed copula functions (ripped off Patton's Matlab code) and played around with various optimizers. You can then use Rsolnp, nloptr, alabama or DEoptim packages to find an optimisation solution. Here is some R code where I play around with different optimisation algorithms. Note that the data2.csv ...

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