Who needs $\sigma$ anyways?

Quantitative finance follows in the tradition set by Bachelier in his 1900 paper "Théorie de la spéculation" (Bachelier (1900)), in which he models asset prices as fundamentally random. In quant finance, we throw up our hands and leave the business of trying to forecast prices to the value investors and the day-traders. Once we acquiesce to a nondeterministic universe¹, we ask "ok, now what can we do?"

Bachelier essentially started with an arithmetic binomial model and considered the transition probability between timesteps to obtain a stochastic diffusion process:

Into the 60s and 70s, however, a preference emerged to model asset prices as stochastic processes which were geometric, both in drift and diffusion. This is certainly an improvement, but it leads to subtleties which are not present in Bachelier’s arithmetic model.

In this probabilistic setting, descriptive statistics can grant us an initial foothold in describing the jagged, chaotic time series of asset prices. Of course, efficient markets strip these statistics of any useful information, but their properties are instructive.

We begin by considering a discrete time series indexed by time.

We might consider the entire sequence $\{S_t\}$ to be a random variable, i.e. $\{S_t\} \equiv S(t, \omega)$, but it’s not clear how we would extend this definition or sample it.

The individual prices $S_t$ might be another choice, but each value seems to depend on the previous value. Bachelier considered the sequence of absolute differences $\{\Delta S_t\}$ to solve this problem, but we can do better.

To proceed, we make an empirical observation.

This suggests that the stock’s movement has something to do with exponential growth, and it’s why a geometric stochastic process model is a better choice.

This also means that, while studying the properties of the $\Delta S_t$ indeed leads to the same answers, the math gets a bit cluttered, as the distribution of $\Delta S_t$ depends on $S_t$. Thus, a natural choice is to reframe the model in terms of the star of this essay: returns.

Returns are noisy and somewhat Gaussian, and since annual return is simply compounded daily returns, it’s fair to think of the asset price path as "growth plus noise." The subtle interplay of this growth and noise is the subject of this essay.

A tale of two growth rates

Let’s start by formalizing our choice of random variable. Emperically, we approximated that

so a natural first step is to model a sequence of simple returns:

Here, I introduce the notation $R_t^{(\Delta t)}$ to indicate that this return is defined across a timestep $t_{i+1} - t_i = \Delta t$. The notation may seem cumbersome, but it will prove quite useful later on.

However, we can eliminate our dependence on $S_t$ another way, by considering log returns.

Using this view, the sequence becomes

Again, the implication of modeling the asset price as random is to claim that it could have ended differently. So what is the event space? Recall the emperical observation from earlier, this time in terms of returns:

We’ll label these proportionality constants $\mu$ and $\tilde{\mu}$ respectively, so that

Now that we have chosen $R^{(\Delta t)}$ or $R_{\log}^{(\Delta t)}$ as the engine of the price sequence, we can reason about the event space. Both approaches have the result of turning the geometric problem into a linear one. Furthermore, we can see that, under our assumptions, $\widebar{R^{(\Delta t)}}$ and $\widebar{R_{\log}^{(\Delta t)}}$ do not depend on time. Thus, if we have a sufficiently large number of returns, we can replace our sample averages with the expectations.

This is a good start, but we have a mystery on our hands. If we think of a stock price as "growth plus noise", which growth rate is the correct one? And what does the other one represent?

The growth rate associated with simple returns, $\mu$, feels more legitimate. After all, simple returns are all that really matters in finance – they are the actual amount of money gained or lost.

For kicks, let’s superimpose these growth rates atop the same SPY time series.

In some sense, this view makes $\tilde{\mu}$ seem more legitimate, since it preserves total return! If indeed our stock price is "growth plus noise," surely this growth curve is the ideal deterministic basis to compose with the noise.

We can rewrite these random variables equivalently as discrete random growth factors (i.e. price ratios).

Interestingly, these expectations are the arithmetic and geometric mean of price ratios, respectively. The AM-GM inequality tells use that $\mu$ will always be larger than $\tilde{\mu}$.

This view also explains why log returns recover the total return.

The mean log return simply answers the question "what growth rate, if compounded $n = t / \Delta t$ times, would result in the correct total growth?"

You might not think that this difference is a big deal; for this sampling of SPY prices, we have only a difference of about 3.6%.

But consider pathological cases where these two quantities have different signs. As it turns out, in markets, this happens all the time. I tested 252-day windows for 15 stocks, and for most stocks, a whopping 5% of their year-long windows exhibit this behavior – overall positive daily returns but with a negative total return! This should convince you that understanding the difference between these two growth rates is worthwhile.

Think about the implication:

• An investor who buys the asset each morning and sells the following morning, while buying the asset again at the exact same price makes money.

• An investor who buys and holds loses money.

This exercise also gives us a clue – more volatile stocks have a higher frequency of these pathological periods, and for any single stock, the periods occur during volatile periods. Whatever this thing is, it has something to do with volatility.

Indeed, this phenomenon is called volatility drag. To understand it better, we need to make a stop at the casino.

Great average returns do not a winner make

An American roulette table has 38 squares. 2 squares are green, and 18 are red, and 18 are black. The game starts by betting money on red or black. A ball is then spun around the wheel, randomly landing on a color. A correct guess doubles your money, and an incorrect guess loses the entire bet.

The expected winnings $\Delta S$ on a $1 bet is trivial to calculate:

Indeed, after $k$ games, the expected P&L is $\mathbb{E}[R] = \mathbb{E}\left[\sum R_i\right] = \sum \mathbb{E}[R_i] = k * -$$$0.053$.

So far so good. This is why no mathematicians play routlette. We can make roulette bets analogous to stock investing by simulating a single-period "buy and hold" strategy for some finite number of plays – we let our bet ride for $n$ rounds. Unsurprisingly, the expected final payout is abysmal:

But something quite interesting happens when we change the game slightly. Let’s imagine a roulette table which has inverted the odds in the gambler’s favor. At this table, the green squares are feebies - they always count as a win.

Now the expected return for a single play is positive.

Not only that, but the expected growth in wealth after $n$ plays is also positive, guaranteeing that the casino loses money in the long run.

Should we play this game?

We’ve just calculated the expected payout, but that number would only be realized after a sufficiently large number of $n$-round plays. We might also wonder about the expected outcome of a single play. Because individual rounds multiply the starting wealth, the order doesn’t matter. Thus, the final wealth is fully determined by the number of wins. Each win occurs with probability $p$, so the expected number of wins in an $n$-round play is simply

Then the final return in this case is

So it’s most likely that you will walk away with nothing after each play! Sure, you could walk out the door with $2^{10} =$$$1024$ after 10 rounds, but that happens only with a $\left(\frac{20}{38}\right)^{10} = .16\%$ chance!

On the other hand, if you play a 10-round game 400 times, it’s more likely than not that you’ll have won $2^{10} =$$$1024$ at some point, only sacrificing $399 for the privilege.

Whether or not you would take this bet given finite money (i.e. finite number of plays) depends on your level of risk tolerance.

This is a variation on the St. Petersburg Paradox, which highlights the difference between the mathematical value of a game and a rational individual’s willingness to play it. It was also explored in a setting similar to this one by Samuelson (1971) in 1971.

Now, take a look at the quantity $2^p * 0^{1 - p}$. Taking the log of both sides of the above equation,

This is the log return!

To summarize,

• Simple returns tell us about the dynamics of the expected value of the price across many possible realizations of the path.

• Log returns tell us about the dynamics of the expected path that the asset price takes.

This tracks with our discussion about the SPY time series. By accepting that time series data as our sample and declaring that it perfectly models the population statistics, we are not say that "this sample has the mean growth rate;" rather, we are saying that "this sample is the most likely path."

So, we’ve tracked down the difference in these two growth rates, but we’ll get a better feel for the machinery by building a bridge between the stock price and the casino. So far we’ve only been concerned with the first moment of our random returns, but in order to develop our model, we need to consider the second moment as well.

The binomial model

The binomial model provides the perfect setting for a random walk. It simplifies the math while providing some key ingredients:

• a natural notion of exponential growth

• sufficient degrees of freedom to tune distributions to our liking

A geometric version of the binomial model was introduced by Cox, Ross, and Rubinstein (1979) in 1979 to price options. I’ll refer to it as the Geometric Binomial Random Walk (GBRW). They also demonstrated that this option pricing method converges on the Black-Scholes-Merton solution. We won’t be pricing options here, but we’ll use a similar convergence technique to study and extend our stock price model.

In the GBRW model, the asset price $S_t$ rises to $u \cdot S_t$ with probability $p$ or falls to $v \cdot S_t$ in a single time step².

One time step is not enough for our needs – we need to develop a procedure for extending the model to timescales larger and smaller than $\Delta t$ so we can examine the statistics.

We begin by examining a simpler case: the Arithmetic Binomial Random Walk (ABRW).

Arithmetic binomial random walk

The arithmetic binomial random walk is characterized by step sizes that do not change with position or time. Here, I’ve kept the model general by allowing for assymetric step sizes and probabilities.

There are many ways to frame and study this setting mathematically, but as a computer scientist by trade, I like the framing that facilitates simulation (i.e. one that allows repeated sampling). As it turns out, this approach also builds a nice bridge to stochastic calculus.

We will analyze the binomial model via the properties of a random walk along its tree-like structure. The engine of the random walk is the random variable $\Delta y^{(\Delta t)}$.

This is already enough setup to do some interesting math, but the real meat comes from extending the definition of our model.

I like to think of it this way: by defining the sample space and probability measure at a particular time horizon, what we’re really doing is setting the parameters of a distribution at that time horizon. Then, we analyze the properties of the model at discrete interval lengths $k \, \Delta t$ to find a continuous map $M$ from time to the distribution.

Think of $\Delta t$ as the resolution of the grid.

Now, consider the process

The logarithm turns the geometric binomial grid above into an arithmetic one.

At this point, the grid is only defined for times $t + k \, \Delta t; k \in \{0, 1, 2, \dots\}$. So we look at how the expectation and variance evolve for $k > 1$. It’s relatively easy to work this out using the binomial distribution, or you can simply recognize that we are accumulating independent Bernoulli trials.

Because the quantities $\frac{\mathbb{E}\left[\Delta Z^{(\Delta t)}\right]}{\Delta t}$ and $\frac{\mathbb{V}\left[\Delta Z^{(\Delta t)}\right]}{\Delta t}$ are invariant for any $t = k \, \Delta t$, this gives us a natural procedure to extend the binomial grid definition to times between $t$ and $t + \Delta t$; i.e. to increase the resolution: iteratively partition the interval $\Delta t$ while preserving these two invariants³.

With this framing in mind, we’ll indicate the hard-coded parameters for the $\Delta t$-resolution grid by defining constants.

We did a similar substitution at the beginning of this essay for the SPY time series. But here, this is a direct result of how we’ve defined the binomial grid, not an emperical observation. Writing $\tilde{\mu} \, \Delta t$ is useful because later on we will manipulate $\Delta t$ like a variable. It is justified because, as shown above, $\mathbb{E}\left[\Delta Z^{(\cdot)}\right]$ is linear in time, insofar as you consider $\Delta Z^{(\cdot)}$ a discrete function of time:

This essentially recognizes $\tilde{\mu}$ and $\tilde{\sigma}^2$ as growth rates per unit time of expectation and variance of $Z^{(k \, \Delta t)}$ respectively.

If the goal is to simulate a random walk on the binomial grid (e.g. with a computer), all we really need is a formula for the binomial random variable from which to sample to construct a random walk with timesteps of arbitrary length. We simply extend $k$ to $\mathbb{R}^+$, and define $n = 1/k$. Let

Then, we can simply increase $n$ until we have sufficient precision.

These two constraints in two unknowns gives a unique solution (up to a reflection about the first moment):

What’s more, we can let $n$ become large, and with a healthy bit of applied math hand-waving, let

Voilà! We’ve discovered Brownian motion and stochastic increments!

Here we’ve approached $dZ_t$ through a limiting process of smaller and more frequent Bernoulli trials, so $dZ_t$ takes the form of a Bernoulli random variable. The more common (functionally equivalent) form is to write $dZ_t$ as a Gaussian random variable.

with $\phi \sim \mathcal{N}(0, 1)$.

All that’s left to do is integrate and exponentiate to obtain our simulated path.

Geometric binomial random walk

Now let’s find out what happens when we apply the same procedure for the geometric case.

Start by choosing a random variable which drives the random walker across the grid.

We would rather not have our random variable depend on $k$ (via $S_{k \, \Delta t}$, such that its distribution changes after each trial), so instead, we express the problem geometrically.

The expectation and variance across $\Delta t$ are

Because these values are just constants, we’ll write them as such.

Again, we write them this way to create quantities $\mu$ and $\sigma^2$ which are expressed per unit time in anticipation of scaling the time interval of our trials later on. But at this point, $\Delta t$ is just a number, not a variable!

The next step is to find the first and second moments of the derived random variable, $R^{(k \, \Delta t)}$. Recall from the linear example that the goal is to establish a relationship between $R^{(\Delta t)}$ and $R^{(k \, \Delta t)}$ which can be extended to all values of $t$.

Across $k$ timesteps of length $\Delta t$, the expectation is binomial.

This confirms that the expected total growth and thus the expected value of $S_t$ grows exponentially, which is unsurprising.

The variance isn’t quite as clean:

Nonetheless, we have the necessary formulae to determine our binomial random variables for any grid resolution. Again, extend $k$ to $\mathbb{R}^+$, with $n = 1/k$.

Since $\delta t$ becomes small, we can simplify.

Now we notice something very interesting. Just like in the linear case, we have the scaling of the random variables proportional to $\delta t$. But instead of the $\mu$ and $\sigma$ which describe the final distribution, we have new quantities.

What’s more, we can see that these values represent an expected value and variance of some other distribution on the $\Delta t$ timescale.

Now, let’s write our random variable using the new quantities.

What we’ve done here is no small feat – we’ve created a "ghost" distribution $\mathcal{N}\left(\mu_\dagger \, \delta t, \sigma_\dagger^2 \, \delta t\right)$ from which to sample simple returns, which nonetheless results in the correct behavior for $S_t$! Again, we can convert this to an equivalent stochastic differential equation.

The presence of $\Delta t$ in the distribution at any grid resolution means that simple returns result in growth rates which are tethered to the original sampling resolution. In other words, defining the problem starting at any particular timescale results in a different return growth rate, which is compensated for in $\mu_\dagger$ and $\sigma_\dagger$.

Indeed, this technique works:

References