Uncorrelated but Dependent Variables in Financial Examples

# Correlation vs. dependence in finance


# Correlation vs. dependence in finance












I found an example that shows how two uncorrelated random variables can be dependent: a normally distributed variable $X$ is not correlated with its square $Y=X^2$. What can be $X$ and what can be $Y$ (in finance terms) so that they represent a shape close to a parabola when plotted in $(x,y)$ plane (both branches present)? This would give 0 correlation, but not independence. Is there such an example?

## Answer by Martin Vesely (score 2)

https://quant.stackexchange.com/a/59068

A correlation and a dependence cannot be interchanged. The dependence is more general term that two radnom variables are somehow linked. The correlation concerns linear dependence only. So, in your example variables $X$ and $Y$ are dependent because $Y=X^2$. As you pointed out, this is a quadratic dependency, not linear, hence there is no correlation.

A general measure for normally distributed random variables measuring how much are two variables linked is called covariance and it is defined as $$ \text{cov}(X,Y)=\text{E}\{[X-\text{E}(X)][Y-\text{E}(Y)]\}, $$ where $\text{E}(.)$ means expected value.

Here are some other measures of dependence.

## Answer by dm63 (score 2)

https://quant.stackexchange.com/a/59073

The simplest example might be Y= realized variance of a stock and X= return on the stock. Clearly these are dependent since they are both calculated from daily stock prices. X can be positive or negative , but Y is always positive. If large moves in the stock occur (up or down) , we would expect to measure high realized volatility. This might give a close to zero correlation for X and Y.

## Answer by Enrico Schumann (score 0)

https://quant.stackexchange.com/a/59069

I guess there are examples in options trading, whenever things depend on Gamma (which is essentially a squared term). For instance, delta hedging: the strategy is, in the textbook version, long the option and short delta times the underlier. If you follow the changes in profit/loss over time and plot them against changes in the underlier, you can often see a u-shaped curve.

An example (R-code):

```
library("NMOF")
steps <- 100

## simulate a path of the underlier
S <- gbm(npaths = 1, timesteps = steps,
         S0 = 100, v = 0.3^2, tau = 1, r = 0)

## compute option value + delta
option <- vanillaOptionEuropean(S = S,
                                X = 100,
                                tau = seq(1, 0.1, length.out = steps + 1),
                                r = 0,
                             v = 0.3^2)
plot(diff(S), -diff(S) * option$delta[-length(option$delta)] +
              diff(option$value),
     xlab = "Change in S", ylab = "PL of delta-hedged position")
```