### 3 Levels of Quant World

Last
month I was invited to give a talk at the Princeton-UChicago Quant Trading Conference in
Chicago. My topic was "Volatility Trading: Theory and Practice". Even
though Chicago is the land of volatility trading, I was the only one
there covering this topic, which generated quite a bit of interest. The
preparation of this talk got me to reflect on the general aspects of
quantitative trading, and how the volatility trading stands in the arena
of quant strategies.

In physics, there are 2 general approaches for applying math: axiomatic method and modeling method -- both have the same requirements: minimize the assumptions (axiomatic or heuristic), and push the deductive math as far as possible.

It is important to remember that economics and finance are about human activities, thus quantifying them like physics is a tall order. The deductive math chain can be short while assumptions are inserted liberally at every turn. It is better to realize such a limitation than to regard the math of economics and finance as similar to physics.

I put the quantitative "truth" in finance in 3 levels: axiomatic; functional and statistical. Let's look at some detail examples:

1. Axiomatic (or arbitrage-free): though it seems almost inconceivable that there are quantitative axioms in finance, actually there are quite a few:

#1. Equity = Asset - Liability

#2. Interest = Principal * exp(Rate*Time)

#3. EquityChange = -Dividend_Distribution

#4. Bid < Ask

#5. Price of Futures = Spot + Carry - Yield

#6. Call/Put Parity: Call - Put = Spot - Strike + Carry - Yield

These 6 axioms, though seemingly simple, are quite powerful, as in the case of "principles" in physics. The #1 formula is where the entire accounting business is going round. The #2, though it may be a result of the "definition" created by human, is still quite magical, as Einstein once said: “Compound interest is the eighth wonder of the world. He who understands it, earns it ... he who doesn't ... pays it.” #3 is extremely simple, but it is a difficult concept for many to comprehend. Try explaining it to financial reporters or even mutual analysts that dividends do not create any value... The inequality of #4 is what makes all the financial markets work, and it is a crucial though often being dismissed by quant researchers as noises or nuisance. But ignoring it, you had the Knight Capital $440 million debacle in 2012. #5 and #6 are the results of the so-called "arbitrage-free" deductive process. The call/put parity is particularly amazing, considering the fact that it first appeared in practice in medieval ages, and discussed in books 70 years before the Black-Scholes option pricing formulas was published.

2. Functional: In this level, the "truth" is result of some constraint conditions, while some model building processes are needed to put in the "meat" of the mathematics. Two examples:

#7: Bond price as function of interest rate, default probability and length of term;

#8: Option price as function of spot, volatility, dividend, interest rate and time to expire.

For #7, the constraints are the contractual conditions of a bond that dictates the exact settlement conditions of interest payment and principal repayment. Similarly, for options in #8, there is a rigorous condition for option settlement at the time of expiration. To get the detailed mathematical form, some models are assumed and an analytical form (may or may not be closed form) is derived. For example, in the Black-Scholes formula (closed-form), a normal distribution of stock returns is assumed. It is important to realize that - even though a statistical model is used for deriving the formulas - the relationship between option prices and spots (or other elements) are analytical, not statistical. Black-Scholes is just one of the many analytical forms.

Functional relationship is relatively more reliable. The risk is not in whether such a relationship exists or not (which intrinsically exists due to the settlement contract constraints). The risk is in picking a not-so-perfect formula. Nevertheless, it is very different from the following type of relationship -- the statistical relationship.

3. Statistical: In this level, quant relationship is derived by looking at statistical analysis of quantities of interests. Statistics are everywhere and can be readily applied – whether they are reliable is another matter. One can look vertically at history, or look horizontally by mixing different quantities and objects and do tallying, or one can calculate pairwise in all kinds and orders of correlations. The vertical, horizontal and pairwise statistics generate multitude of numbers. Unfortunately, their reliability depends on 2 fragile assumptions: 1) Historical may repeat itself; 2) Stable statistics emerges out of very large number. The fallacy of 1) is obvious; nevertheless it makes the foundation of back-testing. 2) has solid mathematical basis from the Law of Large Numbers. However, I doubt a few hundred stocks would form a "large" enough number to generate reliable statistics. Pairwise correlation is based on the "History repeats" assumption.

Unlike the second type of "functional" relationship, there are no rigorous constraints applicable for statistical relationship. We are in a free-for-all zone where conclusion can be plentiful (but dangerous), and if trading strategy is to be based upon such conclusions, one can either become very prolific in a strategy, or can get trapped in the "in-sample"/"out-of-sample" hide-and-seek cycle. Unfortunately, the most common "quant" strategies: portfolio optimization and StatArb, live in this level. Here, the risk is in whether any relationship ever exists or not.

The trustworthiness of the above 3-level quant world drops down pretty rapidly. As quant traders, we usually live in the non-axiomatic world (Level 2 and 3) since uncertainties and unknowns are where the opportunities lie. As volatility traders, we live mostly in the second level of the "functional" world, and should feel lucky that we have a relatively reliable (but not rigorous or axiomatic) to grind on. In the meantime, we should be extremely cautious when we venture out and use various statistical relationships to help our analysis and strategy. Statistics and correlations are the two most dangerous words in trading. For quant researchers and analysts, it is worthwhile to put the common quant strategies in the above 3-Level perspective.

Derek Wang - dwang at bellcurvecapital.com