Scaling Relations in Gaussian Random Walks

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Preliminary Discussion

For a time series $x i$ the deltas are assumed to be normally distributed, i.e.,

$\Delta x_i := x_{i+1} - x_i \sim \mathcal{N} ( 0, \sigma^2), \qquad (1)$

with mean zero and variance $σ 2$ . Then for a sample consisting of $N + 1$ ticks

$\sum_{i=1}^N | \Delta x_i |^2 \approx N \sigma^2. \qquad (2)$

Also note that

$\sum_{i=1}^{N/k} | \Delta x_i^k |^2 := \sum_{i=1}^{N/k} | x_{1+ik} - x_{1+(i-1)k} |^2 \approx N \sigma^2, \qquad (3)$

assuming that $N$ is divisible by $k$ and there are $N + 1$ ticks. Introducing time stamps for every tick, i.e., $x (t i): = x i$ , the deltas become a function of fixed time intervals $Δ t$ :

$\Delta x_i (\Delta t) := x(t_i+ \Delta t) - x(t_i ) \sim \mathcal{N} ( 0, \sigma^2). \qquad (4)$

Note that the number of observations $n$ depends on the sample size and the implicit $Δ t$ . E.g., if there are $N$ observations for a specified $\Delta \hat t$ , then for $Δ t$ there are

$n = \frac{\Delta \hat t}{\Delta t} N, \qquad (5)$

observations and $k = N / n$ . The variable $\Delta \hat t$ can be set to one second without loss of generality because the series $Δ x i$ can always be interpreted as $Δ x i (Δ t = 1)$ . Hence Eq. (3) becomes

$\sum_{i=1}^n | \Delta x (\Delta t) |^2 \approx n \Delta t \sigma^2.\qquad (6)$

Furthermore

$\langle | \Delta x(\Delta t) | \rangle_2 = \sqrt{\frac{\sum_{i=1}^n | \Delta x_i (\Delta t) |^2}{n}} \approx \sigma \sqrt{\Delta t}. \qquad (7)$

It is apparent that the only source of variability in Eq. (7) is the number of observations $n \propto \Delta t^{-1}$ .

If one takes the logarithm of $x i$ , the numerical result is

$\sum_{i=1}^n | \ln(x_{i+1}) - \ln(x_{i})|^2 \approx n |\ln(1+\frac{\sigma}{\bar x})|^2 =: n \hat \sigma^2,\qquad (8)$

where $\bar x$ is the median value of $x i$ . Note that the series of delta log (mid) prices, $x_{i+1}^{\ln} - x^{\ln}_i$ with $x_i^{\ln} := \ln((a_i+b_i)/2)$ , is distribute equivalently to a series of $\hat x_i^{\ln} := (\ln a_i+ \ln b_i)/2$ for constant spread, so the right-hand side of Eq. (8) is applicable, where $a i$ and $b i$ are the ask and bid price at time $t i$ , respectively.

It is interesting to note that compared to Eq. (2), Eq. (8) still has an explicit reference to the price series, although switching to log space eliminates all absolute price references.

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Scaling Laws - Part I

For the scaling law mentioned in U. Müller et. al, "Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis", Journal of Banking & Finance, Elsevier, vol. 14(6), 1990, relating the average $Δ x (Δ t)$ , or the volatility, to the time interval of sampling $Δ t$ ,

$\langle |\Delta x (\Delta t)| \rangle_p = \left( \frac{ \Delta t }{ C (p)} \right)^{E (p)},\qquad (9)$

the above mentioned calculations can be employed. Setting $p = 2$ , Eqs. (9) and (7) yield

$\sigma \sqrt{\Delta t} = \left( \frac{\Delta t}{C} \right)^E,\qquad (10)$

$C= \frac{1}{\sigma^2} , \qquad E=0.5.\qquad (11)$

In terms of logarithmic deltas, Eq. (8) yields

$C_{\ln} = \frac{1}{\hat \sigma^2} , \qquad E_{\ln}=0.5.\qquad (12)$

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Scaling Law Relations - Part II

Recall from Eqs. (2), (6) and (8) that fixed time intervals are assumed, i.e., $Δ t$ is constant. In analogy to Eq. (1), a new dimension of randomness can be added to the process by letting the time intervals vary as well:

$\Delta t_i = t_{i+1} - t_i \sim \mathcal{N} ( \mu_t, \sigma_t^2). \qquad (13)$

So this new time series, $Δ x i (Δ t i)$ , has two independent sources of variability: one for the price and another for the time increments. Note however that for the calculations, like the left-hand side of Eq. (6), there is still a fixed sampling period assumed, $Δ t$ . This requires an interpolation scheme, if there is no price tick for the sampling time $x (t i + Δ t)$ . For the current analysis, previous-tick interpolation is employed.

It follows from Eq. (13) that the scaling properties will be impacted, as the number of observations in Eq. (6) depends on $Δ t$ and the scaling law of Eq. (9) now reads

$\sigma \sqrt{\frac{\Delta t}{\mu_t + \sigma_t}} = \left( \frac{\Delta t}{C_x(2)} \right)^{E_x(2)}, \qquad (14)$