Cross-Sectional and Pathwise Volatility in Simulated Returns

# Cross-sectional volatility vs temporal volatility


# Cross-sectional volatility vs temporal volatility












Volatility is usually defined as the standard deviation of returns, but sometimes it is calculated as the standard deviation of cross-sectional return divided by the square root of time, where other times it is simply the standard deviation of returns for each time period.

I guess for equities that follow Brownian motion, the two don't really matter, since its variance equals time. But this property does not usually hold. For example, if you simulate bond price paths using transition matrices, assuming a fixed spread for bonds with the same credit rating, then the resulting price paths are clearly not Markovian: A bond that has been downgraded will have even a higher chance of being downgraded, thus low price bonds have stronger tendency to drift downward, and the opposite is true since upgraded bonds have even a smaller chance of being downgraded.

This creates an unusual situation. Suppose we simulate many price paths, there seems to be two ways to calculate the return volatility:

- Find the standard deviation of the cumulative return at the end, then divide by the square root of time.

- For each path, find the standard deviation of returns for all the time periods, and the take the average for all paths.

I have ran some simulations and the two does not match very well. The cross-sectional volatility is usually 50% to 200% larger than the time-wise volatility.

What would be a good way to illustrate volatility using these price paths?

## Answer by Evan Wright (score 3, accepted)

https://quant.stackexchange.com/a/14623

The cumulative return over the entire path is the sum of the returns on the individual periods: $$X = X_1 + X_2 + \ldots + X_N.$$ Two potential definitions of the volatility of this process would be $Std(X) / \sqrt{N}$ (which is exactly your "cross-section" volatility) or $Std(X_i)$ (assuming each $X_i$ has the same unconditional distribution). If the $X_i$ are independent, then these two are equal, whereas if the $X_i$ are positively correlated, then $Std(X) / \sqrt{N} > Std(X_i)$.

You could also ask: what is the standard deviation of $X_i$ given the returns I've already seen for $X_1, X_2, \ldots X_{i - 1}$? This will generally depend on $i$, unless you have some special conditions, like a Markov process. Your "time-wise" volatility calculation is taking all of these standard deviations for different $i$'s and taking some kind of average, which doesn't seem to be that meaningful, since they have different distributions.

## Answer by NeverMind (score 1)

https://quant.stackexchange.com/a/81721

Given a sample of multiple return paths $\{(R_{1,1}, \ldots, R_{n,1}), \ldots , (R_{1,m}, \ldots, R_{n,m})\}^T$, where $R_{i,j}$ is the return at time $t_i$ of the $j$th path, we can define two types of volatilies or standard deviations:

- Given a fixed point in time $t_i$:

We measure the volatility of the full sample of return paths at a fixed point in time $t_i$:

$$ \mathrm{Var}_i = \frac{1}{m} \sum_{k=1}^m \left(R_{i,k} - \frac{1}{m}\sum_{l=1}^m R_{i,l}\right)^2 $$

We can call the square root the cross-sectional or ensemble volatility.

- Given a fixed return path $j$:

We measure the volatility of the returns of the individual path $j$:

$$ \mathrm{Var}_j = \frac{1}{n} \sum_{k=1}^n \left(R_{k,j} - \frac{1}{n}\sum_{l=1}^n R_{l,j}\right)^2 $$

We can call the square root the pathwise or temporal volatility.

Please note that these two volatilities are fundamentally different in their conception, as the first refers to a sample of multiple return paths, while the second targets a single return path in the form of a sample of pathwise returns over time.

The mathematical definition of sample variance does not know about time, but can use returns measured at different points in time as input.

## Answer by Analyst (score 0)

https://quant.stackexchange.com/a/14279

Do you mean by cross sectional volatility that you take results from the returns of several assets?

Of course then volatility is different since you are averaging across returns.

For one asset, it is more useful to calculate volatility over time.