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I am an applied math postdoc and I have been presented with the option of leaving academia to work in high frequency trading. I wanted to get a feel for the field and the theory underlying it so I scanned through several books in the library and it seems there are almost no books on the mathematical theory of this field. All the books I have looked at contain lots of explanations of the various aspects of trading such as 'market participants', 'limit order books', 'market microstructure', etc..which of course are very important to know, and some relatively basic math on things like 'statistical arbitrage strategies'. But where is the rigorous mathematical underpinning?

I would have expected to find books containing the same type of theory as in books on mathematical finance, i.e. a deep treatment of measure theory and probability theory, mathematical statistics, stochastic processes etc..

Why are these topics not covered in HFT books? Is advanced math not needed? If this is the case, what are the main skills needed for a high frequency trader?

Hah! There is no such thing as the “rigorous mathematical underpinning” of high frequency trading - because HFT, like all trading, is not primarily a mathematical endeavour.

It’s true that many people who work in HFT have a mathematical background, but that’s because the tools of applied math and statistics are useful when analysing the large amounts of data that are generated by HFT activity. So the math that is useful to know is linear algebra, statistics, time series and optimisation (to some extent it’s useful to be familiar with machine learning, which encompasses all of the above).

Don’t go into HFT thinking that you will primarily be doing advanced math. If you are lucky, you will mostly be doing data analysis. More likely, you will spend a lot of time cleaning data, writing code, and monitoring trading systems.

I would argue, taking a note from John von Neumman, that quantitative finance lacks rigorous underpinnings. Von Neumann warned in 1953 that many things that look like proofs in economics and finance depended on problems that were yet to be solved in mathematics, and where economists were assuming solutions into existence. As the problems were solved in math, economists did not go back and check to see if their solutions matched.

Let me give you an example of why it is a problem. Quantitative finance assumes, in the general case, though the actual practice in the wild varies, that the distributions of returns are either normal or log-normal.

Let us assume that wealth at a given point in time is defined as $w=ptimesq$, where $p$ is the price, and $q$ is the quantity of shares. If we assume that $q_t=q_t+1$, then return at time $t$ is $$r_t=fracp_t+1p_t-1.$$ That makes returns a ratio distribution.

If we make the assumption that is standard in mean-variance models of many buyers and sellers and that a double auction is happening, then the rational behavior of the actors at each point in time $t$ is to bid their expectation.

The limit book converges to normality as the number of actors becomes very large. I would note that this requirement is not necessary, far weaker assumptions could be used, but we would be here for thirty to forty pages.

So prices are normally distributed, and returns are a function of prices, which implies that the distribution of returns is the distribution of a statistic, whose distribution should be derived from the distribution of data.

If we assume that prices converge around an equilibrium and treat the equilibrium as $(0,0)$ in error space, then we can integrate around that point.

The ratio of two centered normal distributions is the Cauchy distribution, which has no first moment. Mean-variance finance is impossible. Indeed, right now, I am trying to put rigor around quantitative finance, but it is very difficult.

To see it a bit more directly, if you transform the data into polar coordinates you will note that the relationship between angles and returns is $tan(theta_t)=r_t.$ It follows that $theta_t=arctan(r_t)$. The arctangent is the kernel of the cumulative density function of the Cauchy distribution. You can quickly arrive at obvious disproofs of the underlying basis for the economic proofs. Do note that I vastly oversimplified the real world as disproof by counter-example doesn't really require the detailed case if one small subset is sufficient and the rest wouldn't remove the cause.

Quantitative finance violates the laws of general summation, in the general case. As a mathematician, dig deep. I have several papers out right now trying to add rigor, but it is hard to see how that will work out. I am proposing a new calculus for options pricing.

High-speed trading is a statistical concept and a key element of statistical theory that most people fail to notice is the absence of uniqueness theorems. There are a few non-existence proofs available, but generating THE solution isn't usually going to happen.

If I were wanting to ground high-frequency trading in sound math, I would avoid Kolmogorov (pace). I actually happen to have a copy of Kolmogorov's original work on probability about three meters from me at the moment, but I believe it will make your work more difficult. I would instead turn to Bruno de Finetti's coherence principle. You can derive Komogorov's axioms from de Finetti's coherence principle. Coherence is important because it is possible to wipe out a market maker who fails to use coherent measures. Generally speaking, Frequentist methods give rise to incoherent probabilities and incoherent prices. I have also worked out the conditions where a neural network will generate incoherent trading instructions(too long for this post).

If you are in want of greater rigor, then start with Leonard Jimmie Savage's "Foundations of Statistics." Again, the threat is incoherence if you do not. Another interesting grounding is Cox's 1961 book "The Algebra of Probable Inference."

The main skill is related to data mining. It may not actually be required that you are either good at it, or use sound methods because it may be the case that the people judging your work do not know calculus or statistics beyond t-tests. That is not a criticism, so much as a deep concern for soundness. Having spent a good chunk of my life inside financial institutions, I have more than a passing concern for the black-box system that is in place.

On the assumption that you want to do a very good job, then what I would do is work out the determinants of supply and the determinants of demand. I would factor the changes and risks to dividends, mergers, and bankruptcy. I would have to include liquidity costs. It would make it more like a very boring supply and demand model. It would likely not be very fancy and it would almost certainly lack pizzazz.

Boring is awesome if it makes you money.

EDIT
I need to give a thanks to @Accumulation because I have been looking at this problem too long. Let me be a bit more rigorous.

Let observed return $r$ be defined as $$r=r^*+gamma,$$ where $gamma$ is a random variable and $r^*$ is the equilibrium return and the center of location.

Also, let observed return be defined as $$r=fracp_t+1p_t.$$ Let equilibrium return be defines as $$r^*=fracp_t+1^*p_t^*.$$

Let us defined prices with respect to equilibrium prices using Wold's decomposition theorem as $$p_t=p^*_t+epsilon_t,$$ and $$p_t+1=p_t+1^*+epsilon_t+1.$$

So, $$fracp_t+1^*+epsilon_t+1p^*_t+epsilon_t=fracp_t+1^*p_t^*+gamma.$$ It follows that $$gamma=fracp_t+1^*+epsilon_t+1p^*_t+epsilon_t-fracp_t+1^*p_t^*.$$

$$gammaapproxfracepsilon_t+1epsilon_t.$$

The author acknowledges that in the general case, the ratio of two normal random variates shifted by a price are not a Cauchy distribution but rather a Cauchy distribution scaled by $(1+eta)$ where $eta$ is a finite variance distribution. In this case, $eta$ would become vanishingly small in effect. Out of equilibrium, that would not be true.

Note that $epsilon$ is normal as described above centered on zero. Also, note that in the general case, $sigma_t+1>sigma_t$ or there would be a violation of rationality. It implies, in the general case, price heteroskedasticity.

Optimal stochastic control. Hamilton jacobi bellman