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The Trading Mesh

High-Frequency Trading Inspires a Formula

Fri, 13 Jan 2012 02:27:14 GMT           

In a new working paper, Godfrey Cadogan, of Toronto’s Ryerson University, offers a stock-price formula designed to capture the “empirical regularities of high frequency trading.”

As is often the case, though, the discussion can leave those of us outside the quant world confused: does the rendering of facts as a formula make them clearer, or does it just create a potentially misleading patina of precision?

Given Cadogan’s ambitious-sounding program, linking HFT, bubbles, and crashes all into one formula, one remarkable feature of the result is his formula’s extreme simplicity or, as Cadogan puts it, its “parsimony.” I was reminded of the warnings in Emanuel Derman’s recent book, Models.Behaving.Badly, that the “simple models” of finance economists have failed “to reflect the complex reality of the world around them,”

The paper, “Trading Rules Over Fundamentals: A Stock Price Formula for High Frequency Trading, Bubbles and Crashes,” was inspired by recent studies indicating that HFT accounts for 77 percent of the volume of trading in the UK and 70 percent in the US, and that it is concentrated in the most popular and liquid stocks, commodities, and currencies. Cadogan takes it as evident that “high frequency traders quest for alpha, and [that] their concomitant trade strategies are the driving forces behind short term stock price dynamics.”

His formula, then, is as follows:

Translation into English…

Cadogan is claiming that if trading in the underlying stocks is dominated by HFT, then the movements of a stock price index (S) can be predicted by the exposure to () and the volatility () of E-mini contracts, which serve as a proxy for the strategies of traders seeking to control exposure.

One of the consequences of the formula, Cadogan writes, is that it shows how if volatility is above a historical average the price of S can enter “an exponentially downward spiral if volatility continues to increase,” because the HFT strategies become “phase locked,” each feeds off of the others’ efforts to reduce exposure to that downward move and compounds it.

Cadogan makes explicit the following assumptions behind his model, and his formula: first, that asset markets are competitive and frictionless with continuous trading of a finite number of assets; second, that asset prices and adapted to a filtration of background driving Brownian motion; third, that prices are ex-dividend.

Cadogan cites a 2010 paper by Frank Zhang, of the Yale School of Management, “High-Frequency Trading, Stock Volatility, and Price Discovery,” which found in the empirical data a strong positive correlation between volatility and HFT, “after controlling” as Zhang noted, “for firm fundamental volatility and other exogenous determinants of volatility.”

Cadogan accepts that result, and invokes Zhang’s definition of key terms. Thus, to both of these scholars, HFT is a subcategory of algorithmic trading, distinguished by holding periods and trading purposes. But what HFT has in common with the rest of the category of algorithmic traders is the use of an algorithm for the timing, price, and quantity of orders.


Cadogan also cites Ananth Madhavan, head of trading research at BlackRock, who has written of late on “Exchange-Traded Funds, Market Structure and the Flash Crash.” Madhavan reaches many of the same points as does Cadogan without the thick clustering of Greek letters. “Market structure reforms enacted since the Flash Crash,” Madhavan wrote, “should help mitigate future such market disruptions, but have not eliminated the possibility that another Flash Crash would occur, albeit with a different catalyst and perhaps in a different asset class.”

As Cadogan notes, Madhavan’s paper “distinguishes between rules based algorithmic trading” on the one hand, and “comparatively opaque high frequency trading” on the other. In the latter context, traders can place and immediately cancel their orders, in the usual expression they can engage in “quote stuffing” in order to jam the signals received by other high-frequency traders. Cadogan contends that his formula captures precisely such predictable tactics by such traders and their spiraling consequences.

One policy inference that Cadogan draws is that much of the attention paid to HFT has been misdirected. Other writers have contended that the dysfunctional character they attribute to HFT is the result of its speed advantages over non-HFT players. In Cadogan’s view, though, the problem is the interaction of HFT with HFT, not with non-HFT players. Further, in his view the above formula suggests that the policy solution is not to be found in mandating more deliberative trading. Rather, it “suggests that [a] limit on short sales would mitigate the problem while still permitting the [technological] status quo.”

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Re: High frequency trading inspires stock price formula


Renown physicist Stephen Hawkins, writing in A Brief History of Time stated "A theory is a good theory if it satisfies two requirements: It must accurately describe a large class of observations on the basis of a model that contains only a few arbitrary elements, and it must make definite predictions about the results of future observations." By that standard, the recent stock price formula for high frequency trading I introduced is a resounding success. In plain English, the formula says: 

log (end of period stock price [index])=log(begin period stock price [index]) + sum over period (volatility of stock [index] futures  x exposure to stock [index] futures x news about stock [index] futures)

where 'x" means "times". What some analysts call “news about stock [index] futures” is really "mispricing of stock [index] futures". A positive value means the asset is underpriced, whereas a negative value means its overpriced. When combined with textbook cost of carry models, that formula shows that HFT capital gains depends almost exclusively on the difference between short term interest rates and dividend yield on stock [index], and volatility of the stock futures [index] and stock price volatility. For a popular hedge factor like the E-mini, and its volatility, the formula not only predicts high frequency stock price behavior, it plainly shows how traders profit from and why they generate stock market volatility. Your article is quite correct to warn about the limitations of financial models by invoking Emanuel Derman’s recent book Models Behaving Badly. Ironically, Derman praises Black-Scholes option pricing formula, based on some of the underlying assumptions in my formula, as an example of a successful formula. 

A little history about the formula. It is a spin off of another paper on alpha strategy representation recently published by the American Statistical Association. In fact, the formula is a by product of a paper written to rebut Prof. Robert Jarrow (Cornell) paper which alleges that positive alpha is illusionary. So one can argue that Jarrow’s paper was a catalyst for the formula. I have been thinking about such a formula since my days in graduate school when renown Prof. Jan Kmenta, admonished me for paying attention to data analysis results, produced by Prof. Robert Engle and his students, without understanding the theory driving those results. 

In passing, the subject formula is different from Easley, De Prado and O’Hara formula which deals with market microstructure issues arising from toxicity of order flow. 

Again, thanks for taking the time to write about the formula. If you have any questions, please do not hesitate to contact me via email:, (preferred) or call 786-329-5489.


godfrey cadogan 2293 days ago,(2012/01/13)

All models are wrong, but some are useful.

Asia 2287 days ago,(2012/01/18)