Suppose we are analyzing two stocks, Stock A and Stock B, for potential pairs trading opportunities. The first step involves examining the marginal distribution of each stock.

Through a detailed analysis, we find that Stock A follows a normal distribution with a mean return of 5% and a standard deviation of 2%. This marginal distribution provides insights into the
expected returns and volatility of Stock A in isolation.

Stock B, on the other hand, has a marginal distribution that is skewed to the right, indicating that it has occasionally provided high returns, although its mean return is also around
5%.

Now, we’re interested in understanding the behavior of Stock A given the performance of Stock B and vice versa.

Let’s say on days when Stock B’s returns exceed its mean return of 5%, Stock A has a tendency to underperform, yielding returns below its mean. We calculate the conditional distribution of Stock
A’s returns given Stock B’s performance exceeding its average returns.

When Stock B’s returns are expected to be above 5%, the trader might consider shorting Stock A, expecting it to underperform due to the observed conditional relationship.

The marginal distributions give the trader insights into the individual behavior of each stock, while the conditional distribution illuminates the interdependencies between the two
stocks.

For instance, if Stock B’s performance is exceeding expectations, the conditional distribution informs the trader about the likely underperformance of Stock A. Consequently, a pairs trading
strategy could involve shorting stock A and going long on stock B during such scenario

A judicious integration of both distributions empowers traders to devise nuanced, data-driven pairs trading strategies, optimizing profitability while mitigating risks.

Copulas connect marginal distributions to joint distribution and reveal an advanced analysis landscape.

In the realm of copulas, conditional probability functions play a pivotal role. Defined by the differentiation of the copula with respect to its parameters, these functions unveil nuanced
insights into asset dependencies.

For instance, consider the conditional probabilities P(U > u|V = v) and P(V > v|U = u).

These are calculated using the formulas:

- P(U > u|V = v) = ∂C(u, v)/∂v

- P(V > v|U = u) = ∂C(u, v)/∂u

C(u, v) is the copula function that captures the dependency between the two stocks, and the partial derivatives with respect to u and v give the conditional probabilities.

Where:

- U and V are the transformed uniform marginal distributions of the returns of Stock A and Stock B, respectively.

- "u" and "v" are specific values within these distributions.

- For example, u = 0.7 would represent the return of Stock A at the 70th percentile of its uniform distribution, and v = 0.8 would represent the return of Stock B at the 80th percentile of its
uniform distribution.

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