Benoit Mandelbrot is one of the greatest mathematicians of our time. Most renowned for his work on fractals (including the famous Mandelbrot set), he applies a complexity science lens to financial markets. As with most transdisciplinary endeavors, the findings are truly astounding.
It was also fascinating to dive deeper into Modern Portfolio Theory, Capital Asset Pricing Model, and Black-Scholes. The 3 most important foundations in modern finance.
However, the book demands some knowledge of fractal geometry and finance. Some sections get technical and can be more difficult to follow. And so, these book notes have been written to distill the key insights in a more accessible form.
Financial orthodoxy is built on shaky assumptions.
The Efficient Market Hypothesis still dominates the financial world. It postulates that asset prices fully reflect all relevant information. Therefore, you cannot beat the market. Unless you go against the consensus, and the consensus turns out to be wrong.
But the chasm between theory and reality is strikingly apparent. Markets are not always efficient, price changes are not continuous, volatility itself is volatile, and the list of fallacious assumptions goes on.
“Many a grand theory has died under the onslaught of real data.”
Before exploring the flaws in the orthodoxy of financial markets, it first helps to understand its three most important elements: MPT, CAPM, and Black-Scholes.
These notes are thus composed of two parts.
In part 1, we cover what the financial orthodoxy is and how they were developed.
In part 2, we explore what makes them flawed. As well as sample a plate full of exotic concepts.
A few chapters in the book are dedicated to explaining fractal geometry, but these notes won’t go into these. Will save that for another time. (Meanwhile, check out Scale by Geoffrey West if you’d like to learn more about this).
Part 1. Financial orthodoxy and its development
Modern Portfolio Theory (MPT) is a method for building a portfolio that optimizes risk-adjusted return.
Developed by Harry Markowitz in the 1950s, it’s quickly gained popularity for its simplicity.
Theory says that everything about the prospects for a stock depends only on two factors: risk and return.
Return depends on the price you think the stock will sell for.
Risk can be thought of as how wrong you could be about the price when it’s time to sell. Therefore risk, Markowitz thought, depends on price volatility. And the most common way of measuring volatility is variance and standard deviation. (Note: standard deviation is just square root of variance)
But most portfolios are composed of more than just one stock. And most stocks are highly correlated with each other. So if you pick a combination of stocks that correlate negatively, you can lower the risk of your portfolio without too much compromise on potential returns.
Using this method, the portfolio to choose should be those closest to the Efficient Frontier. And along this Efficient Frontier, select a portfolio based on your risk appetite.
While simple in theory, MPT requires a good forecast on earnings and volatility for thousands of stocks. The outputs of a formula is only as good as the accuracy of the inputs. Garbage in, garbage out.
Then there’s the computational challenge. The covariance of each stock needs to be laboriously calculated against every other stock. A 30 stock portfolio demands 495 different calculations of mean, variance, and covariance. Plus, this calculation needs to be updated continuously.
Enter William Sharpe and his early 1960s contribution: CAPM.
Capital Asset Pricing Model (CAPM) is a method for valuing (i.e. assessing the return of) a security. It’s commonly used to estimate the cost of capital, including making investment decisions. For instance, whether it makes sense to open a factory in a certain location.
At its heart, CAPM says that the expected investment return (Ri) of a stock is equal to the risk-free return plus the beta of the stock multiplied by the market risk premium.
The risk-free rate (Rf) is the returns you expect to make with effectively zero risk: the treasury bill rate.
The market rate (Rm) is the returns you expect to make by investing in the market index. Market index here can be whatever it is you’re looking at investing e.g. S&P500.
The market risk premium (Rm-Rf) is the additional returns you expect by investing in the market index, compared to investing in treasury bills.
Beta (βi) is the factor by which a particular stock correlates with the market index. A stock with a beta of 1 moves exactly with the index. A beta of, say 0.1, is insensitive to market moves. A stock with a negative beta moves up when the market moves down etc.
This notion of a beta is what alleviated the computational pain of MPT. Once you forecast the overall market returns (Rm), you can then just estimate the beta for each stock. This way, 495 calculations in MPT for a 30 stock portfolio gets reduced to just 31 with CAPM.
Fast forward another decade, the next big step in modern financial theory came in the mid 1970s.
The Black-Scholes formula is a method for estimating the market value of options.
Brief detour on options for those less familiar, as it’s hard to follow along without understanding this terminology.
Options are contracts that gives you the right (but not an obligation) to buy or sell stock at a certain price (strike price) within a certain time (time to expiration).
The price you pay to purchase an option contract is the option premium.
An option is said to be in-the-money (ITM) if the current market stock price exceeds the strike price. Such options have intrinsic value as it can be used to purchase the underlying stock at the lower strike price, before profiting from immediately selling it at the higher market price.
An out-of-the-money (OTM) option is one where the market price has yet to reach the strike price.
Unlike trading a stock, there is a capped downside with options. The maximum loss is the premium. However, if you buy a call option (long call) and the price of the stock at the time of expiration is lower than the strike price, the option is rendered worthless.
The premium at which options are priced should reflect the favourable asymmetry they provide.
Herein lies the question: how do we figure out how much that premium should be?
Conceptually, how an option’s worth today simply depends on the price of the underlying stock at expiration – that is, how far “in the money” the option would end up being. But this is obviously an impossible endeavor.
Fisher Black and Myron Scholes’s key insight was this:
“When valuing an option, you do not need to know how the game will end – that is, what the stock price will finally be when the option expires. Instead, all you need to know is what the traders themselves know, the terms of the option (the strike price and time to expiry) and how volatile the stock is.”
This was captured by a complicated differential equation, which yields the Black-Scholes model:
C is call option price; N is the cumulative distribution function of the normal distribution; St is the spot price of asset; K is strike price of asset; r is risk-free interest rate; σ is volatility of asset.
Black-Scholes provided several ground-breaking conveniences.
One, it provided a relatively simple method to value options based on well-known parameters plus the volatility. If a stock has low volatility, it’s unlikely that the stock price will exceed the strike price before expiry. Such options should be worthless. However, if a stock is highly volatile, options can pay off handsomely.
Two, the formula allowed easy, frequent re-calculations over time, as the option matures.
Three, the model revealed the implied stock price volatility based on options prices. Options were first traded since the opening of the Chicago Board Options Exchange (CBOE) in 1973). And Black-Scholes permitted the rise of an entirely new type of speculation – betting on volatility itself: VIX was listed as a product in 1993 (bet on how volatile S&P500 will be in 30 days).
Effectively, Black-Scholes mechanically places a price on risk.
However, like all financial alchemy, the mechanistic nature of the formula instills false confidence.
Warren Buffet and Charlie Munger speak to this in this video.
“Black-Scholes is an attempt to measure the market value of options and it cranks in certain variables. But the most important variable it cranks in… is past volatility… But past volatility is not the best judge of value… For longer term options in particular Black Scholes can give some silly results… it’s a mechanical system and any mechanical system is going to misprice things from time to time.” – Warren Buffet
“Black-Scholes is a know-nothing formula… If you don’t know anything at all about value compared with price… then it’s a pretty good guess… on a 90 day option on some stock… The minute you get into longer term options, it’s crazy to use Black-Scholes… At Costco for instance… we issued stock options at 30, and also issued stock options at 60, and Black-Scholes valued the options we issued at 60 as a strike price way higher than the options we issued at 30. Well this is insane.” – Charlie Munger
Part 2. Problems with the financial orthodoxy
There’s a simple way to explain the stark discrepancies between theory and reality. Look at the assumptions underpinning the theory.
(i) Assumption: People are rational and aim to only get rich. Reality: People are not utility-maximizing rational agents.
The predictably irrational behaviour of humans has been well known for decades. Misinterpreting information, miscalculating probabilities, emotionally distorted decision making inc. loss aversion etc. You’ve heard this all before.
(ii) Assumption: All investors are alike with homogeneous expectations. Reality: Investors are far from alike.
Consider two types of investors. Value investors look for under-priced value based on fundamentals. Growth investors jump on and off momentum. Computer simulations have shown that the two groups interacting in a market gives rise to unexpected emergent behaviour.
“Price bubbles and crashes arise spontaneously. The market switches from a well-behaved linear system… to a chaotic non-linear system… And that is with just two classes of investors. How much more complicated and volatile is the real market, with almost as many classes as individuals?”
(iii) Assumption: Price changes are continuous. Reality: Price changes are discontinuous and can jump substantially.
Even in highly liquid FX markets, 80% of quotes end in 0 or 5, skipping the intermediate digits.
(iv) Assumption: The magnitude of price changes are normally distributed. Reality: Fat tails and power laws abound.
The Gaussian assumption implies that the magnitude of most price changes are small, and large changes occur in predictable and rapidly declining frequency.
However, a closer look at historical data reveals that big prices changes occurred far more frequently than what the standard normal distribution model predicted. This insight holds true for a range of asset classes: commodities, stocks, and currencies.
“Large changes, of more than five standard deviations from the average, happened two thousand times more often than expected. Under Gaussian rules, you should have encountered such drama only once every seven thousand years; in fact, the data showed, it happened once every three or four years.”
Kurtosis is a number used to measure how closely real data fits with the ideal Gaussian. “Think of it as how much spice is in the statistical broth. A perfect, unseasoned bell curve has a kurtosis of 3. A spicy, hot-tailed curve would have a higher number.”
The S&P500 between 1970 and 2001 had a kurtosis of 43.36. Even if the spiciest data point, the Black Monday Crash of October 1987, the kurtosis is still a high 7.17. Far above the expected 3.
(v) Assumption: Each change in price is independent of the last. Reality: Price changes actually exhibit some short-term dependence.
Financial orthodoxy assumes that price changes obey a Brownian independence. What happened yesterday has no effect on what will happen today. Even if a coin flips heads 10 times in a row, the chance of a another heads remains equal to a tails.
But studies reveal that prices do exhibit some short-term dependence. Momentum effects come to play. If an index falls in a given month, it has slightly higher odds of falling again in the next month. And vice versa.
“The data overwhelmingly show that the magnitude of price changes depends on those of the past… markets can exhibit dependence without correlation. The key to this paradox lies in the distinction between the size and the direction of price changes.”
Additionally, big changes often come together in rapid succession.
But only in the short-term. And short here can hours, days, or even years, depending on what asset you’re looking at.
“If you zoom in on an individual cluster of big changes, you find it is made up of smaller clusters. Zoom again, and you find even finer clusters. It is a fractal structure.”
(More on multifractal nature of trading time below.)
Having said that, corrective mean reversion does appear in the medium term.
This phenomenon shares parallels with yearly flood level variations on the Nile. Engineers assumed that flood levels in one year were statistically independent of the previous. But turns out that the historical sequence of flood levels influences levels today.
A consequence of the momentum-continuing-tendency of price changes is a rigged game. In Reddit vernacular: “Stonks always go up.”
Mandelbrot explores this tendency towards ‘cheating’ further by introducing the Hurst number, H. (min 0, max 1)
- H = 0.5 is perfect, random, Brownian motion. Zero short-term dependence.
- H > 0.5 is short-term price momentum persistence.
- H < 0.5 is anti-persistence to short-term momentum. A price move in one direction is shortly followed by a reversal in the other.
Word of caution though:
“Forecasting volatility is like forecasting the weather. You can measure the intensity and path of a hurricane, and you can calculate the odds of its landing; but… you cannot predict with confidence exactly where it will land and how much damage it will do. Nevertheless, work on such meteorological ideas has begun in finance.”
(vi) Assumption: Price changes exhibit statistical stationarity. Reality: The mechanisms driving the price changes, whatever they may be, continues to change over time.
Coin tosses have statistical stationarity as the coin does not get switched or weighted in the middle of the game. CAPM presumes statistical stationarity via beta.
But in practice, it’s evident that many other factors moves price.
So academic economists continue to devise new tools like Arbitrage Pricing Theory.
Similarly, when it became clear that volatility itself is volatile – that volatility tends to cluster in time – new tools such as GARCH (Generalized AutoRegressive Conditional Heteroskedasticity) were developed.
“Most established financial models say little with much. They input endless data, require many parameters, take long calculation. When they fail, by losing money, they are seldom thrown away as a bad start. Rather, they are “fixed.” They are amended, qualified, particularized, expanded, and complicated. Bit by bit, from a bad seed a big but sickly tree is built, with glue, nails, screws, and scaffolding.“
“They work around, rather than build from an explain, the contradictory evidence.”
(vii) Assumption: Trading time flows at the same rate as clock time. Reality: Price changes behave as if time itself does not flow at a regular rate.
On most days, the price action on some asset might be boring. On other days, say when news breaks out, the price moves erratically. Time seems to compress on these eventful days, and stretch out over the dull ones.
“Price changes in a financial market cluster into zones of high drama and slow evolution.”
Mandelbrot says that this is exactly how one should analyze a financial market.
There’s normal clock time which we’re all used to. And then there’s a deformed trading time.
Price is a function of both trading time and clock time. The result is a fractal market cube.
Time itself is multifractal. Volatility clusters. Volatility is itself volatile.
This is the fundamental problem with Black-Scholes. It assumes constant volatility.
The simplest fractals are self-similar. These are typically the same recurring visual patterns at different scales. Multifractals, in contrast, scale in different ways at different points.
“The genius of fractal analysis is that the same risk factors, the same formulae apply to a day as to a year, an hour as to a month. Only the magnitude differs, not the proportions… Statistically speaking, the risks of a day are much like those of a week, month, or a year. But the price variations scale with time.”
(vii) Assumption: Average returns and average risk are meaningful indicators for investors. Reality: The ergodic assumption is fallacious.
In a previous post, I illustrated that the average returns for a group of individuals is a misleading indicator of the average returns for an individual. As the saying goes, “averages lie”. This is because of a fallacious ergodicity assumption.
Consider two ways of making purple.
Method 1: Alternate red and blue squares. With enough squares, purple emerges.
Method 2: Alternate red and blue at various time intervals. With small enough intervals, purple emerges.
Method 1 made changes in the space (ensemble) dimension.
Method 2 made changes in the time (temporal) dimension.
Both converged to give the same result (purple). This is ergodicity.
A process is said to be ergodic if the ensemble-average is equal to the time-average.
(Continue reading more on ergodicity here.)
Modern financial orthodoxy is riddled with such ergodic assumptions.
In short, the odds of financial ruin are a lot higher than what the models assure us of.
The intent of this post was to share selective notes on the book, rather than attempting to be a comprehensive summary. If you found it interesting, I’d highly recommend reading the actual book: Amazon US; Amazon Australia.
Some other ideas I found interesting:
“…the movement of security prices, the motion of molecules, and the diffusion of heat could all be of the same mathematical species.”
Consider 3 types of randomness: mild, slow, and wild.
- Mild is like the solid phase of matter: low energies, stable structures, well-defined volume.
- Wild randomness is like the gaseous phase of matter: high energies, no structure, no volume.
- Slow randomness, like liquid, is in between mild and wild.
“The prime mover in a financial market is not value or price, but price differences; not averaging, but arbitrating. People arbitrage between places or times.”
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