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Why Symmetric Log Returns Produce Skewed Price Forecasts

Article Quant Q&A · Author: Ludo
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Summary

The document explains an apparent asymmetry in simulated currency prices generated from a GARCH model. The model is fitted to log returns, which may have a symmetric innovation distribution, but the simulation is transformed back into price levels by exponentiating accumulated returns. That nonlinear transformation creates a positively skewed price distribution even when the return distribution itself is symmetric.

The response therefore attributes the unequal upper and lower price quantiles to the conversion from log returns to prices, rather than to a directional prediction from the volatility model. The example provides simulated quantiles as context, but offers no additional diagnostic or empirical test. The explanation assumes the model’s log-return distribution is symmetric; a skewed innovation specification, such as the one described in the question, can introduce further asymmetry that this short answer does not analyze.

Key ideas

  • Symmetry in log returns does not imply symmetry in price levels.
  • Exponentiating accumulated log returns produces a positively skewed price distribution.
  • Unequal simulated price quantiles can result from the return-to-price transformation.
  • A skewed innovation distribution may add asymmetry beyond the effect of exponentiation.

Tags

ForexVolatilityStatisticsMachine learning
Full text
# Log returns and GARCH models


# Log returns and GARCH models












I try to model currency rates volatility using GARCH models through the RUGARCH package in R.

Starting from the observed currency rate series, I compute the log-return through:

```
data <- diff(log(series)) # Log-return
```

Then (after some statistical analysis) I decide to use a GARCH(1,1) model with a skew-student distribution, hence I use

```
spec_final <- ugarchspec(mean.model=list(armaOrder=c(0,0),include.mean=FALSE),variance.model=list(model="sGARCH",garchOrder=c(1,1)),distribution.model="sstd")
fit_final <- ugarchfit(spec_final,data=data)
```

I then try to simulate future outcomes of this series with an horizon of 260 days with the code

```
horizon <- 260
exp(diffinv(ugarchsim(fit_final,n.sim=horizon)@simulation$seriesSim))[horizon+1]
```

If I perform this a great number of times (200,000) I can compute quantiles. More especially I see that the quantile at 0.5% is equal to 0.605 and the quantile at 99.5% is equal to 1.623. The distribution has a mean very close to 1 but is not symmetric.

I would like to understand why there is a lack of symmetry in the simulated distribution, even if the GARCH model is known to be symmetric. It does not happen only for one currency but for all those I tried to model. This is really an issue to me as I do not have any particular reason to explain why the model predicts larger upward shocks than downward shocks.

Thanks.

## Answer by Fortranner (score 2)

https://quant.stackexchange.com/a/16183

If log returns have a symmetric distribution, prices will have a positively skewed distribution, since exponentiating induces positive skew.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.