Creating Math is Harder Than Doing Math (and what this has to do with PT algos)
There is perhaps no age of life that features more villains than the middle school years: bullies who control the seating charts of the cafeteria with the spiteful precision of a Texan gerrymander, PE classes that seem designed to make you clumsily stumble from a volleyball hit to your head in front of your crush, and chaperoned dances more awkward than a cocktail mixer at a conference for sufferers of social anxiety. But the greatest villain of all is the math word problem.
If Janet has three math teachers who each present content in a negative way at rates of 0.1, 0.3, and 0.15 percentage of comprehensibility respectively, how long until Janet hates math?
The answer is: less time than it takes to finish reading the problem.
My personal hatred of math word problems may seem strange. Afterall, I love math! And I love words! Aren’t math word problems just the fusion of these two things that I love? No. They are a mockery of both, leaving me doubly offended.
The trouble is: math is already a language. It is intended to express and convey a reality of the world — whether a physical reality, or an abstract reality of ideas that shapes itself inside our minds. Starting with a rote, meaningless math problem, translating it into how many apples Alice has or how many bicycles Bob sells, and then trying to captivate children with it is like trying to unsquish a dead bug. The life is already gone. Few people can love math for its own sake.
But math that starts as an embodiment of meaning — rather than math that is meaninglessly embodied — is a truly compelling thing. Unfortunately it is rarely taught. We teach children how to solve math problems. Not how to discover them.
The best part of being a mathematician is getting to create math out of new situations. It is so fun, in fact, that we sometimes abuse the privilege and start using math to shuffle around problems that are, at their core, not mathematical. [*cough*… like blockchains … *cough*. You’ll have to excuse me, I must be getting a cold.] But in many situations, the math is readily there to be created. And there can be as many fruitful versions of it as there can be human perspectives on the same topic.
In the context of equities trading, there are no shortage of ways to turn the trading of a single order into a math problem. You can pick a metric: slippage vs. arrival, volume curve tracking error, some combination of short-term markouts, fill rate, etc. Then you can build a model for how your choices influence this metric. Then voila! — you have a math problem: finding the set of choices that optimizes your metric under your model.
The meaning of this math problem rests on the quality of your model and the wisdom of your metric choice, which makes the selection of problem as important (if not more important) to trading success than the solving of the problem.
But when we want to go beyond considering each order individually and instead want to think holistically about how to trade a basket of orders with collective constraints (like cash neutrality, for example), expressing our problem as a nicely contained and eminently solvable piece of mathematics becomes more fraught. For one thing, the set of reasonable choices of how to trade expands exponentially in terms of the number of orders in the basket. This means that solving for the best basket schedule according to some global metric could be very, very hard (if not impossible) to do in a feasible amount of time.
We might be tempted to think — well, when something is so complicated that it’s hard to put into math that can be solved quickly by computers, perhaps it’s best to just have humans do it. There are examples like this — where a computer’s need to “overthink” things by having them fully and exhaustively specified poses a deep challenge to automation. This is why things like folding laundry are pretty hard to get robots to do — a human is making a lot of little tiny adjustments to their motions based on the variations in the material and positioning without even thinking about them all specifically. Trying to automate this is much harder than, say, designing a dishwasher. [Aside: the fact that the robots in the Stepford Wives can presumably cook and fold laundry is perhaps the most unbelievable thing about the premise of that story. The misogyny, of course, I find believable.]
But in the case of portfolio trading, a human trader overseeing a basket is not really doing any subconscious actions that are hard to fully describe. Instead, they are probably doing rather well-understood things, like adjusting the parameters of orders that are getting too far ahead or too far behind. So we find ourselves in a mathematician’s favorite situation: a case where defining an appropriately meaningful and solvable math problem is delicate, but also should be doable.
Our initial approach to turning PT algo trading into a solvable math problem is detailed in our new whitepaper, available here. The main idea is this: let’s start by imagining a very specific execution of our basket that would satisfy all of our constraints. Maybe our imagined schedule achieves cash neutrality, for example, by closely following volume curves that aggregate to roughly the same balance for our buy and sell orders. This specific starting point does not have to be something we believe is “optimal” trading behavior. Rather, it serves two simpler purposes: 1) establishing that our desired constraints are well-formed and at least hypothetically satisfiable, and 2) encapsulating our human intuition and/or client preferences into a starting point for mathematical optimization, so that we are starting from a “reasonable” point in the vast universe of basket scheduling options, not searching arbitrarily for a needle in an exponential haystack.
Next, we want to define a total budget of acceptable deviation (or slack) from this initial basket schedule. How much imbalance are we willing to tolerate in search of better performance? If we find some good liquidity that would put us ahead of schedule in some order, can we afford to take it? Conversely, how far behind are we willing to get while waiting for passive fills, for example, before we switch to being more aggressive to catch up to our target schedule? This total budget of slack should reflect the “wiggle room” that we give ourselves around constraints like cash neutrality, intended to help us achieve lower impact than strictly and blindly sticking to a pre-formed schedule.
This concept of a slack budget then naturally splits our problem into two problems: 1) how much slack should each individual order get to use, and 2) how should an individual order be scheduled to achieve minimal expected impact while staying within its’ allocated slack? Both questions have a common mathematical basis: how do we model the value of the ability to deviate from the fixed schedule on a per order basis? Intuitively, such a model should account for characteristics of the order and the symbol. A small order in a highly liquid symbol, for example, should not benefit too much from having additional flexibility, while a larger order in a more illiquid symbol should be expected to have a greater benefit. Once we have a satisfying model of how slack benefits each order, we can use this to allocate slack to optimal global effect, and also to find the best schedules for the individual orders that stay within their slack allocations. For more details, see the whitepaper.
For those who are already lost in the word problem, however, it suffices to know a few key takeaways:
1) Holisitic optimization over large baskets of orders is a thorny mathematical problem, and the wrong formulation won’t be solvable in a realistic amount of time.
2) With careful design, however, versions of the problem are solvable and highly automatable. This is taking shape as a backbone of Proof’s upcoming portfolio trading algorithm.
Perhaps the most important take-away of all: who defines your math problems is probably ultimately more impactful than who solves them. But your kid is still going to have turn in those annoying middle school math worksheets anyway.
