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Equity returns have persistent negative skewness and excess kurtosis[1] over longer periods. So yes you're right: a majority of the daily returns is positive and small and a minority of the returns is negative and larger. This can be quite extreme, for example Black Monday. I don't have the data right now but you can get returns on major indices freely. [1]...


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I found this power point and this paper to be an excellent source on this topic. Here is a quote from the paper: A square-root singularity for small traded volumes is highly non-trivial, and certainly not accounted for in Kyle’s classical model of impact [11], which predicts a linear impact ∆ ∝ Q. A concave impact function is often thought of as a ...


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My understanding (devoid of any mathematical grounding) is as follows. v = Turnover PER UNIT TIME n = Shares you need to execute therefore n/v = Number of units of time required to execute your size at the normal turnover rate Realized vol follows a SQRT(T) heuristic. Given that we can now rewrite the transaction cost formula purely in terms of vol ...


4

In practice all impact models are sub-linear. Despite this is fact (seen in many academic publications, commercial and proprietary models), there is an interesting argument for using a linear impact models (other than being careful and pessimistic). Would anyone try to build a model, this approach would be also more parsimonious with less parameters to fit. ...


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The reference papers for market impact decay are: Waelbroeck, H., and Gomes, C. (2013). Is market impact a measure of the information value of trades? market response to liquidity vs. informed trades. Social Science Research Network Working Paper Series. URL http://ssrn.com/abstract=2291720 Brokmann, X., Serie, E., Kockelkoren, J., and Bouchaud, J. P. (2014)...


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There is no official definition of market depth (this is only a qualitative concept), only the cost of a roundtrip for a given number of shares of contracts. Take $V$ shares, on average, knowing the shape of the book at time $t$, what is the cost of buying and selling them immediately? You obtain a cost $C(V,t)$. Then you need to average or to choose an ...


3

Bootvis accurately describes the math - Skew plus Kurtosis. What's interesting is that many of the efficient market theorists (example: Eugene Fama) observed this phenomenon too. I consider two intuitive reasons for this: 1) Behavioral - According to prospect theory, the mental benefit of gaining a dollar is lower than the fear of losing a dollar. This ...


3

The blog post "A slice of S&P 500 skewness history" http://www.portfolioprobe.com/2012/01/16/a-slice-of-sp-500-skewness-history/ has a bit of data on this question. It appears that log returns might have some negative skew, but symmetry is a possibility.


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The square root law is a quite simple and popular model for price impact estimation: $$\Delta p = Y\sigma\sqrt{\frac{Q}{V}}$$ where: $\Delta p$ is the price impact, $Y$ is a constant (needs to be calibrated). $\sigma$ annualized daily volatility of the returns $Q$ daily trading volume. There are a lot of papers around this model (e.g Gomes and Walbroeck ...


2

Please note that my answer is primarily opinion/experience based. If it is not appropriate I will take it down or edit accordingly. How should I begin to think about optimal execution given a choice of execution methods? What simplifying assumptions or heuristic frameworks could be useful in identifying quasi-optimal execution strategies? I think optimal ...


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IMHO there is a general shift toward algorithmic execution for institutionz over the last 5 to 10 years, and depending on your method of execution the price impact can vary, so I am not certain whether you will get meaningful results using long histories. Some older papers look at tickets data and aggregate impact of large tickets as proxy for ...


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See Kandel and Pearson (1995) and Kim and Verrecchia (1991, 1994, 1997).


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This is actually a deceptively good question because, as we all know, estimates of variance are extremely sensitive to sampling frequency, sampling intervals, and lags. This is because not all stock prices perfectly adhere to Brownian Motion (i.e., the variance doesn't adhere strongly to the root time rule). It is also not entirely clear from the paper how ...


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There is an ongoing trend in GARCH modeling that seeks explain the why of this phenomena. Some economic explanations iv'e read revolve around the effects of leveraged positions and flights to liquidity. The GARCH model derivatives used are usually classified as asymmetric power models I believe.


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We will know the answer to this question when Navinder Singh Sarao is put on trial in the United States on charges of doing precisely this.


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It's like a giant game of tug-of-war except there are many ropes and thousands of participants. The SPY tugs at the ES, which tugs at the SPY, which tugs at other ETFs. I suppose in theory you could manipulate the ES in a low volume setting by selling a huge block in the thinly-traded period before 12AM PST, selling enough to make the price fall 1/2 -1% ...


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The easiest way ,i suppose, would be to analyze the market depth. If there is a 20 cent gap between each 100 shares on the bid then to sell 1000 shares instantly would have an impact of $2. Your average price is the midpoint. There are more complicated formulations, but this seems to be how it works on simple examples such as bitcoin exchanges.


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