Relating Intraday Log Returns to Daily Return Expectations
Summary
The document asks whether the expected return at an intraday sampling interval equals the final cumulative return, and whether returns measured from a fixed starting price should be added or multiplied. One response reframes the issue as whether price formation is diffusive across sampling scales. It notes that intraday returns may show multifractal behavior and points to Hawkes-process and cascade or stochastic-volatility approaches, which can differ from geometric Brownian motion at short horizons.
A second response distinguishes cumulative log returns from interval log returns: the latter are differences between successive log prices, and their sum telescopes to the total log price change. Under a stable average interval return, the total is related to the per-interval mean through the number of intervals; this is not a claim that the expected interval return equals the realized endpoint return. A third answer gives the drift-times-time result under a geometric Brownian motion assumption. The explanations rely on different assumptions and do not establish one universal intraday-to-daily model.
Key ideas
- Successive log returns are changes between adjacent log prices, and they add to the cumulative log return.
- The mean interval return and total return are related through the sampling horizon under suitable stability assumptions.
- Intraday price formation may depart from geometric Brownian motion and can exhibit multifractal or self-exciting behavior.
- A drift-times-time expectation follows under a geometric Brownian motion model, so the model assumption matters.
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Full text
# How to model the daily return using intraday data?
# How to model the daily return using intraday data?
Say, I have hourly returns $r_1,r_2,...,r_T$, where $r_t$ = $ln(p_t)$ - $ln(p_0)$ for $t = 1...T$. So what is the value of $E[r_t]$? Would $r_T$ be the $\prod{(r_t)}$?
Basically $r_t$s are the returns w.r.t to a fixed point $t_0$. My question then is how can I mathematically prove that the $E[r_t]$ is $r_T$ or it isn't?
## Answer by lehalle (score 4)
https://quant.stackexchange.com/a/3537
It seems that your real question is: is the PFP (Price Formation Process) diffusive from intraday to weekly sampling rate?
It is a very good question since on intraday, some academics found some multifractal features into intraday returns, meaning that the PFP is not a Geometric Brownian Motion at small scales (even considering stochastic volatility).
You have for instance successful modellings of the PFP using pint processes, and especially Hawkes ones (that are not diffusive and even not Markovian): Modeling microstructure noise with mutually exciting point processes by: E. Bacry, S. Delattre, M. Hoffmann, J. F. Muzy (forthcoming in Quant. Finance). They obtained some formula to express characteristics of the diffusive limit of such processes with respect to ones of the underlying Hawkes process, like the large scale diffusive volatility: $$\sigma=\frac{2\mu}{1-||\phi||_1}\,\frac{1}{(1+||\phi||_1)^2}$$ (with $\phi$ the kernel of the Hawks process linking its stochastic intensity with its realizations and $\mu$ is the deterministic part of its intensity).
But also more "classical" multifractal approaches: Modelling fluctuations of financial time series: from cascade process to stochastic volatility model by: J. F. Muzy, J. Delour, E. Bacry in Euro. Phys. Journal B, Vol. 17 (2000), pp. 537-548. In such cases, a classical "Hurst exponent" allows zooming in or out.
## Answer by rtybase (score 3)
https://quant.stackexchange.com/a/3052
Isn't this a simple mathematical rule?
$$\Delta r_{t}=r_{t} - r_{t-1} = ln(p_{t}) - ln(p_{0}) - ln(p_{t-1}) + ln(p_{0})=ln(\frac{p_{t}}{p_{t-1}})$$ i.e. logarithmic or continuously compounded return. As a result: $$E(\Delta r_{t})=\frac{1}{T}\sum_{t=1}^{T}\Delta r_{t} = \frac{1}{T}\sum_{t=1}^{T}ln(\frac{p_{t}}{p_{t-1}})=\frac{1}{T}ln(\prod_{t=1}^{T}\frac{p_{t}}{p_{t-1}})=\frac{1}{T}ln(\frac{p_{T}}{p_{0}})=\frac{1}{T}r_{T}$$
is hourly expectation.
Or $$\frac{p_{T}}{p_{0}}\frac{p_{T-1}}{p_{0}}...\frac{p_{2}}{p_{0}}\frac{p_{1}}{p_{0}}=(\frac{p_{T}}{p_{0}})(\frac{p_{T-1}}{p_{0}}...\frac{p_{2}}{p_{0}}\frac{p_{1}}{p_{0}})$$ and $$ln[\frac{p_{T}}{p_{0}}\frac{p_{T-1}}{p_{0}}...\frac{p_{2}}{p_{0}}\frac{p_{1}}{p_{0}}]= ln(\frac{p_{T}}{p_{0}}) + ln[\frac{p_{T-1}}{p_{0}}...\frac{p_{2}}{p_{0}}\frac{p_{1}}{p_{0}}]$$ or $$ln[\frac{p_{T}}{p_{0}}\frac{p_{T-1}}{p_{0}}...\frac{p_{2}}{p_{0}}\frac{p_{1}}{p_{0}}] - ln[\frac{p_{T-1}}{p_{0}}...\frac{p_{2}}{p_{0}}\frac{p_{1}}{p_{0}}]= ln(\frac{p_{T}}{p_{0}})$$ and if we assume convergence $$T\cdot E(r_{t}) - (T-1)\cdot E(r_{t})\approx r_{T}$$
## Answer by Suminda Sirinath S. Dharmasena (score -2)
https://quant.stackexchange.com/a/2958
If you assume GBM, then
Expected Change in price = Stock * Drift * change in time
The Weiner term disappears.
This implies, E[r(T)] = drift * T
Use hourly return to estimate drift and plug it in the formula.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.