Choose Correlation Inputs to Match the Asset Simulation Model
Summary
The document asks whether correlations should be calculated from simple returns, log returns, or log prices, and whether the choice is merely conventional. The response ties the answer to the intended use: when correlations will drive a simulation, estimate them from variables that correspond to the model’s random shocks rather than selecting a transform without regard to the model.
For a lognormal price process, the response shows how to invert the price evolution equation to recover standardized shocks from consecutive prices, accounting for drift, volatility, and the time step. Correlations among those implied shocks can then be used to generate correlated shocks in a new simulation. The central lesson is model dependence: this procedure should be adapted to the chosen dynamics, since correlations of returns, prices, or inferred shocks need not be interchangeable. The exchange does not compare alternative models or discuss estimation error, changing correlations, or empirical diagnostics, so it gives a modeling principle rather than a universal prescription.
Key ideas
- The appropriate correlation inputs depend on how the estimated correlation will be used.
- For simulation, estimate dependence between the model’s underlying random shocks when those shocks drive the price process.
- Under lognormal dynamics, consecutive prices can be transformed into standardized implied shocks using model parameters.
- Correlations estimated for one price model should not automatically be reused with a different model.
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Full text
# What is the appropriate transform before calculating the correlation between two assets?
# What is the appropriate transform before calculating the correlation between two assets?
When calculating the correlation between two assets, what is the appropriate transform before taking the correlation?
- PChg = (P2/P1) - 1.0
- LChg = Log(P2) - Log(P1)
- x = Log(P2)
where P2 is the newest price and P1 is the previous price of the asset in question.
Also, is the answer just a matter of convention or is there a statistical/mathematical/computational reason we would prefer one over the other two?
As I understand it, we need some kind of transform first, because prices are non-stationary.
I think 1. makes sense, but I've read/heard that 3. is sufficient for correlation calculations.
Is this right? Even if it is, can someone rigorously explain why 3. is appropriate/sufficient?
## Answer by will (score 1)
https://quant.stackexchange.com/a/58444
It depends how you plan on using the correlation you calculate.
If you plan on simulating the assets using some model, and you want to sample the returns to estimate the correlations, then you need to invert your model and calculate the random numbers underlying the samped data.
Take a pair of underlyings which you decide are lognormal. In this case you would simulate them using the following:
$$ S_{t_i} = f(S_{t_{i-1}}, r, \sigma, \Delta t, \tilde{X}_{t_i}) = S_{t_{i-1}} e^{(r-\frac{1}{2}\sigma^2)\Delta t + \sigma\sqrt{\Delta t}\tilde{X}_{t_i}} $$
So, if we have the time series of $S_t$, then we can invert the above:
$$ \tilde{X}_{t_i} = f^{-1}(S_{t_i}, S_{t_{i-1}}, r, \sigma, \Delta t) = \frac{\ln \left( \frac{S_{t_i}}{S_{t_{i-1}}} \right) - (r-\frac{1}{2}\sigma^2)\Delta t}{\sigma\sqrt{\Delta t}}$$
to give you a time series of random variables, as implied by a. the series of prices, and b. the model you've chosen. You can do the same for any model you choose to use.
You can then measure the correlation between these implied random variables, and then use the same correlation to generate more to simulate your new time series.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.