# Intraday versus daily volatility in slippage estimation

On page 21 of http://www.cims.nyu.edu/~almgren/papers/costestim.pdf Almgren has the formula

$\displaystyle{\text{Slippage} = \frac{1}{2}\gamma\sigma\frac{X}{V}\left(\frac{\Theta}{V}\right)^{\frac{1}{4}} + \text{sign}(X)\eta\sigma\left|\frac{X}{VT}\right|^{\frac{3}{5}}}$

Where:

$\gamma = 0.314$

$\sigma \text{ is volatility}$

$X \text{ is the trade size}$

$V \text{ is the 10-day average volume}$

$\Theta \text{ is the shares outstanding}$

$\eta = 0.142$

$T = \text{fraction of day in which trade is placed}$

In the paper [page 11], Almgren says $\sigma$ is an "an intraday estimator that makes use of every transaction in the day."

Does anyone have an idea of how this estimator works? I'm using a 40 day moving standard deviation of daily returns in my simulations, since I don't have intraday data. Do you think this provides an overestimate or underestimate of the appropriate volatility in the slippage formula?

There are two questions packed in here. I will attempt to answer one at a time.

Does anyone have an idea of how this estimator works?

A much more concise practical guide to this estimator is found here: http://corp.bankofamerica.com/publicpdf/equities/Equity_Mkt_impact.pdf

But I will try to break it down anyway.

This estimator appears to be a specialized form of a general slippage model in which market impact is proportionate to the b/a spread, trade size as fraction of the total liquidity, shares outstanding, and price variation:

$I \propto s + \alpha \cdot \sigma_n \cdot (rate_\text{participation})^\beta$

where $I$ is the impact function, $s$ is the b/a spread as percentage, $\sigma_n$ is the daily price volatility, and $\alpha$ and $\beta$ are constants. This is of the form of a generalized power function regression.

Typically, market impact models are expressed as square root functions; see: Typical coefficients uses in square-root model for market impact. Almgren added some refinements to fit empirical impact and to express the intuitions that selling provides liquidity and that impact is proportionate to volume intensity with respect to time. Notably, he removed the term for the b/a spread since it didn't fit the data.

The first term of the expression, $\gamma\sigma\frac{X}{V}\left(\frac{\Theta}{V}\right)^{\frac{1}{4}}$, is intended to model the permanent price change due to a trade. It is also of a generalized power function, but adds a term $\frac{\theta}{V}$ to express the expected long-term impact of removing shares from the market. $\frac{X}{V}$ is the standard expression for participation rate.

The second term, ${ \text{sign}(X)\eta\sigma\left|\frac{X}{VT}\right|^{\frac{3}{5}}}$, is intended to express the temporary impact. The term reflects two intutions:

1. a long-trade (X is positive) removes liquidity and therefore result in additional impact; vice versa, a short trade (X is negative) provides liquidity; and

2. Trades spread out over longer periods of time result in less impact, thus: $I \propto \frac{1}{T}$.

Combining these intuitions together yields the following statement regarding trading costs: expected total cost from trading are $\frac{1}{2}$ of permanent market impact (assuming you VWAP into the position) plus the temporary impact.

Interestingly, it appears feasible under this model to have negative slippage. There is also an peculiar outcome in which selling over very short intervals of time results in very negative slippage. It seems intuitive to me that you could avoid most cases of asymptotic or non-sensical values by assuming both terms following square root laws, but this obviously didn't fit the data as well. I guess models are just models.

I'm using a 40 day moving standard deviation of daily returns in my simulations, since I don't have intraday data. Do you think this provides an overestimate or underestimate of the appropriate volatility in the slippage formula?

Whether or not standard deviation of daily returns under or over-estimates actual variance is idiosyncratic; it depends on how closely the underlying stochastic process follows the "root time" rule. For example, daily sampling of mean-reverting processes will tend to understate the intraday volatility; i.e., the close-to-close sample variance understates the intraday variation for choppy price action (vis-a-vis "penny stocks" in which the tick size is large compared to price).

On the other hand, the opposite tends to be true for trending processes.

I recommend that those interested in making the most out of OHLC data check out intraday volatility estimators. I've had much success with the Yhang Zhang estimator found here: Understanding Yang-Zhang Volatility Estimator.