How GARCH Drift Affects Expected Price Returns in Log Space
Summary
The document questions whether a standard GARCH model’s constant conditional mean for log returns is appropriate when volatility changes. For normally distributed log returns, it notes that the conditional expectation of the simple gross return depends on both the mean and variance. Its numerical illustration shows how changing volatility changes that expectation when the log-return mean is held fixed.
It proposes an alternative that adjusts the conditional mean as volatility changes so the expected gross return stays constant, and asks whether this is appropriate under physical probabilities. The text gives no estimation, forecasting, or empirical comparison; it presents an unresolved modeling question. Its premise that a stock’s conditional expected gross return changes little with volatility is an assumption to examine, not a result established by the document. It also distinguishes the issue from risk-neutral modeling, where expected returns are tied to the pricing measure.
Key ideas
- For normally distributed log returns, expected gross returns depend on both the conditional mean and variance.
- Holding the log-return mean fixed makes the expected gross return vary as conditional volatility changes.
- The proposed alternative adjusts the conditional mean to hold expected gross returns constant.
- The document raises the question in a physical-probability setting but does not provide empirical evidence or a definitive resolution.
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Full text
# Is GARCH assumption on constant drift wrong in log space?
# Is GARCH assumption on constant drift wrong in log space?
GARCH assumes constant drift $\mu$ - this imply $E[e^r]$ won't be constant and jump wildly. And it contradicts the reality, for stock prices $E[S_{t}/S_{t-1}]=E[e^r]$ doesn't jump with each time step.
Lets consider simple GARCH
$$r_t = \mu + \sigma_t N(0,1)$$ $$\sigma_t^2 = \omega + \alpha \sigma_{t-1}^2+\beta r_{t-1}^2$$
Let's consider variables at two times $t=1,2$ assuming $\mu=0$ and calculate expected value in linear space:
$t=1$, $\sigma_1=1$ likelihood $r \sim N(0, 1)$ implying $E[e^r] = e^{0 + 1^2/2 = 0.5} = 1.7$
$t=2$, $\sigma_1=2$ likelihood $r \sim N(0, 2)$ implying $E[e^r] = e^{0 + 2^2/2 = 2} = 7.4$
So GARCH implies that $E[e^r]$ jumps wildly with every time step. Which seems to be wrong for stock prices, where stocks with different volatilities have more or less close expected value for returns $E[S_1/S_2] = E[e^r]$.
Question 1:
How this thing even works? It seems the realistic way to vary both drift and scale and model likelihood as $r \sim N(\mu_t,\sigma_t)$. But what GARCH does seems to be totally wrong, it uses wrong location for distribution. How it converges and what exactly it predicts then?
Question 2:
Why not assume $E[e^r]$ constant? And rewrite GARCH as:
$$r_t = \mu_t + \sigma_t N(0,1)$$ $$\sigma_t^2 = \omega + \alpha \sigma_{t-1}^2+\beta r_{t-1}^2$$ $$\mu_t = log E[e^r] - \sigma_t^2/2$$
Is there something wrong with this approach? Is it used?
P.S.
Specific case - fitting GARCH on historical daily prices, in physical probabilities (not risk neutral measure).
UPDATE:
The assumption accepted as truth - for stock the $E[S_t/S_{t-1}]$ is constant and doesn't change with volatility, or changes very slightly so we can assume it's a constant.
GARCH violates that, if you plot $E[S_t/S_{t-1}]$ for each time step it would be zigzag, not a line. Or rephrasing it - I think $E[S_t/S_{t-1}]$ vary much less than $\mu_t$ - so it's closer to reality to treat $E[S_t/S_{t-1}]$ as a constant.
The root of the problem is this equality $E[S_t/S_{t-1}] = E[e^{r_i}] = e^{\mu_i + \sigma_i^2/2}$ - fixing the drift $\mu_i = const$ means that $E[S_t/S_{t-1}]$ won't be const and would vary.
UPDATE2:
It seems the GARCH under the risk neutral measure address exactly that issue.
My point is - I think GARCH should work same way under physical measure too. While $E[S_t/S_{t-1}]$ in reality is not a constant and vary slightly, it doesn't vary wildly, and it makes much more sense to fix it instead of fixing $\mu$ and not let it vary wildly with volatility.
By $E[S_t/S_{t-1}]$ - I meant theoretical conditioned expectation at each time t.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.