$\begingroup$

The CAPM, of course, doesn’t have a clean process. It is a single period model. That implies that there could exist a $\mu_t$ and a $\sigma^2_t$. Additionally, there is an assumption in mean-variance models that the parameters are known with certainty, making chance the only source of uncertainty. Risk exists, but uncertainty does not exist.

A few additional implicit or explicit assumptions are present. Firms do not go bankrupt, and mergers cannot happen because, in equilibrium, no firm should be materially undervalued and so there would be no economic gain from engaging in such a transaction. Liquidity costs do not exist. Either there is infinite liquidity, or there do not exist market makers.

Implicitly, the wealth equations of motion are $p_{t+1,i}=\mu{p}_{t,i}+\epsilon_{t+1,i}$. This is arithmetic Brownian motion with drift. However, you can show through Donsker’s scale invariance theorem that this maps to geometric Brownian motion with drift, which is one way to derive the Black-Scholes Option Pricing Model. If this is a Brownian motion, then continuous time and discrete time models are interchangeable.

The problems with these models are that there can be no empirical solution for them. Models of the form $$x_{t+1}=\mu{x}_t+\varepsilon_{t+1}$$ have no solution in this case. To understand why, the maximum likelihood estimator for $\mu$ is the least squares estimator for all $\varepsilon$ drawn from any distribution with a finite positive variance that is centered on zero. It is also the MVUE since it minimizes squared loss. $\hat{\mu}$ then, as an estimator, can be thought of as the estimator of the mean slope. So far we have not departed from any standard element of economic theory.

Notice though that if $\mu\le{1}$ then no investment should happen due to rationality. That is important because White in 1958 showed that the sampling distribution of $\hat{\mu}-\mu$ is the Cauchy distribution. The Cauchy distribution has no mean, so the least squares estimator has zero power and is perfectly inefficient. There is no difference between a sample of size one and a sample of size one million.

To understand the implication, there does not exist a Frequentist or Likelihoodist estimator that is also consistent with the underlying theory. Models like the CAPM or Black-Scholes cause vacuous proofs, even if they are true.

To get a better understanding as to why this is happening, begin with two observations. First, if stocks are in equilibrium, but for a random shock, then they can be thought of as $p^*_t+\epsilon_t$. If we ignore mergers and bankruptcies as the CAPM does and pretend liquidity costs do not exist, then a return could be defined as $$\frac{p_{t+1}^*+\epsilon_{t+1}}{p_t^*+\epsilon_t}.$$

Since stocks are traded in a double auction, the winner’s curse does not obtain and the rational behavior is to bid your expectation. There cannot be a “market order” because there can be no market maker. As there are many buyers and sellers, the static distribution of orders must converge to the normal distribution by the central limit theorem. So the shock terms are intrinsically normal under the Markowitzian assumptions.

Since the buying price and selling prices are bivariate normal and centered around $(p_t^*,p_{t+1}*)$ and returns are the ratios of the prices, it follows that returns are a ratio distribution. If you translate the problem from price space to error space, then it could be thought of as happening around $(0,0)$. This is rather fortunate as this distribution is well known and understood. The ratio of two normals around $(0,0)$ can only be the Cauchy distribution.

Then, a fortiori, the stochastic process can only be a Levy flight. Unfortunately, it is shown that there is no Frequentist or computable Likelihoodist solution that is unbiased and admissible in the case where there is a limitation of liability.

Truncation eliminates the ability to create an admissible estimator. There is a Bayesian estimator, however. This takes you into Bayesian predictive distributions and out of the Frequentist ones. This is a different math.

I would recommend you start in Bayesian decision theory.

Harris, D.E. (2017) The Distribution of Returns. Journal of Mathematical Finance, 7, 769-804 White, J.S. (1958) The Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case. The Annals of Mathematical Statistics, 29, 1188-1197.

EDIT

If your concern is math and not finance or economics, where all parameters are known, then you should read "Brownian Motion" by Peter Morters and Yuval Peres ISBN 978-0521760188. I say this because there are components in Brownian motion that you may not be aware of.

To provide an explicit example, two dimensional Brownian motion has properties that are strictly unique to planar, or two-dimensional Brownian motion and these properties could materially impact you. You should also get a copy of Krzysztof Burdzy's doctoral dissertation. I don't have the bibliographic references though.

I bring up planar motion because even if you model a Brownian motion in one thousand dimensions, if it can be instead understood as a stock and a market, you are on a plane.

There are several properties unique to planar Brownian motion. The first is that the area the motion covers will always be zero. The CDF always sums to zero. In all other dimensions it sums to one. In addition, any point reached once will be reached infinitely many times. However, due to this fact, some places will never be reached so you end up with "holes" as $t\to\infty$.

To think about that for a minute, consider that this implies that some configurations of market and security will be seen repeatedly and that some configurations will never happen. Those configurations won't be known ex ante.

Additionally, planar Brownian motion projects onto $R^1$ as the Cauchy distribution. If you think of returns as being a two dimensional number, such that $r_t=f(p_t,p_{t+1})$ in a similar since that a complex number is two dimensional, so that it is a number like $a+bi$ expressed as $(a,b)$, then you will begin to see the connections between prices, returns and dimensions.

If you are simulating the math, definitely pick up Morters and Peres's book.

If you are approaching this as an economist or finance professional, then you need to treat this as a Levy flight. It is a Brownian motion, arithmetic or geometric, if and only if, the parameters are known with probability one. If there is any source of uncertainty, then it is a Levy flight.

In that case, you need to approach this problem as a Bayesian process and not a Frequentist one. This is due to the fact that the Cauchy distribution lacks a sufficient statistic. Because of this, only Bayesian solutions are admissible. If you have no practical experience with Bayesian methods, I would recommend the undergraduate level textbook by Bolstad as a good starting book.

In the Bayesian analysis you need to construct the Bayesian predictive distribution and if you need one point, then you need to minimize a cost function over the density.

Bayesian methods contain three distributions. The prior distribution contains all information you have about the true location of the parameter, prior to seeing the data or from outside the data. I will denote it $$\pi(\mu;\sigma)$$. Please note that this is the distribution of your beliefs about the location. For a doctoral level treatment, which also won Leonard Jimmie Savage the Nobel in Economics, see his book "The Foundations of Statistics," at ISBN 978-0486623498.

The second distribution is the posterior distribution, which is the distribution of your beliefs about the location of $(\mu,\sigma)$ after seeing the data. I will denote it $$\pi(\mu;\sigma|x),$$ where $x$ is the data set.

Because the Bayesian likelihood function is guaranteed to be minimally sufficient, you cannot lose information by using the entire distribution. You can by using just one point on the distribution. This is why you use the predictive density.

The predictive density is $$\pi(\tilde{x}|x).$$ It is the distribution of the next value of $x$ given all prior information, both data and outside knowledge. It is important because, you will notice, there are no parameters in it. It does not depend upon the parameters, but instead on the data. Its relationship to the posterior is $$\pi(\tilde{x}|x)=\int_{\mu\in\Theta}\int_{\sigma\in\Theta}\mathcal{L}(\tilde{x}|\mu;\sigma)\pi(\mu;\sigma|x)\mathrm{d}\sigma\mathrm{d}\mu.$$ You do not know the value of the parameters, but you do not need to as their impact is integrated out of the equation using the likelihood function $\mathcal{L}$.

If you have not worked with Bayesian methods before and fell queasy about using "beliefs" as a basis for statistics, then maybe you should spend some time thinking about what beliefs are. It is a purely subjective method. It takes more work. It is the only choice though if you want admissibility. I am working on a paper that looks at the difference between the standard Frequentist treatment and the Bayesian treatment of an ordinary problem and there is a 16:1 improvement in the efficiency of the Bayesian to Frequentist estimator in the simulation. A 16:1 improvement in efficiency is a huge reduction in the search area for parameters. Effectively, that is a 4:1 improvement in the interval estimates.