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?

• Not sure I understand. Why would you give a reference to a textbook on portfolio theory? – Qwerty Nov 1 '20 at 3:07
• You're right, that book was too hard. Here is a suppy and demand curve example from a high school economics course. oocities.org/vuumanj/BusinessAlgebra/Quadratic.html – Dimitri Vulis Nov 1 '20 at 3:24
• What about an asset‘s return and it’s squared return? – Kermittfrog Nov 1 '20 at 6:04

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.

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.

• Covariance is not a general measure for how much random variables are linked. Basically, covariance is just the scaled (Pearson) correlation coefficient. A standard example (you mention it yourself): $\mathrm{cov}(X,X^2)=0$ if $X\sim\mathrm{Uni}(-1,1)$, see here. Zero covariance implies $X$ and $Y$ are uncorrelated but not necessarily independent (unless they are e.g. normal). – Kevin Nov 1 '20 at 7:44
• @Kevin: Thank for pointing that out. I edited my answer. – Martin Vesely Nov 2 '20 at 10:54

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")


• Your plot does not seem correct to me. I would expect that if you're making money on the large moves, you should be losing money on small moves (in your case, <~3 change in S) from the theta of the option. – will Nov 1 '20 at 13:26
• But it is losing money on the small moves: the points "in the middle" are all below zero. Note that the PL is computed along a path, so delta/gamma changes as a result of the simulated spot price. – Enrico Schumann Nov 1 '20 at 14:48
• haha. My bad. I didn't even read the scale and just assumed that zero was the bottom of the plot / that there'd be an axis on the plot. Where it crosses through zero lines up with expectation too, appologies. – will Nov 1 '20 at 16:49