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Random Variable or Stochastic Variable

A variable whose value is determined by the outcome of a random experiment is called a random variable. It is also known as the chance variable or Stochastic variable.

In stock analysis, we often look to the past to predict the future. We take great pains in understanding the historical prices, their formation (trends) and other information such as trading volumes, company's performance and corporate actions to make a prediction about future prices. To face future uncertainity, we take comfort in the past. There is nothing wrong in this approach, except that the past may not always guide us to the future. It is also difficult to judge whether this approach works or not because to do that we need to again depend on historical data.

For example, based on the scores of the last 100 ODI matches, we can derive that the average (mean) score of Virat Kohli is 58 runs per match. This does not mean that when he plays the 101st match, he would score 58 runs. We can expect (predict) that he would score 58 runs but we do not know this for certain. Let's assume that he scores exactly 58 runs. Does this mean that our technique of prediction is good? Certaintly, no. To validate that our prediction technique is good, we may conduct this experiment or prediction many more times and see how we fare. Let's say that we predict the score of the 102nd match to be 58 again, but he scores 60 runs. There is a deviation of the actual score when compared to our predicted score - by 2 runs. We can use statistical techniques such as Standard Deviation to define and make sense of such deviations. So, instead of just looking at the averages, we now look at both averages and standard deviation to predict the future. But, still, we are relying on historical data to predict the future.

An alternative approach is the "Efficient Market Hypothesis", which states that the current stock prices reflect all information and future prices are not predictable and any efforts to predict them would be futile. Under this hypothesis, we cannot predict the future with any certainity, irrespective of whether we have historical data or not, or the strong empirical evidence of its capacity to predict consistently. We assume (irrespective of whether we believe or not) that the variables that affect the movement of stocks are random in nature (that means they move in an unpredictable manner) and that their random movements produce an outcome which is random in nature (the future stock price).

One can argue that if we cannot predict future then what is the use of such hypothesis. We are looking for help to predict future and not something to tell us that we cannot do it.

Well! the answer to our problems lie in its assumptions and the way it compels us to look at stock analysis from a different perspective.

All along we have been trying to predict future stock prices (or returns) based on historical prices. This approach fails us more often than not. The Efficient Market Hypothesis (EMH) assumes that stock prices and the variables that affect it follow a random path. So, it is telling us that our focus on historical prices is faulty to start with. The alternative (which EMH did not propose) is to focus on risk instead of returns. While prices (or returns) cannot be predicted, we can predict risk using statistical methods such as probability and normal distribution and mathematical tools such as calculus and logarithms.

In otherwords, we may not be able to directly predict what Virat Kohli would score in the 102nd match, but we would be able to predict a range of scores along with their respective probabilities.

One might again argue that what possible use is a range of scores and their probabilities. How is anyone ever going to make decisions based on such information? Well! it may not be possible to use such information for cricket betting but when it comes to financial markets, the various probabilities when clubbed with assumptions under 'risk-neutral world' pave the way for developing models which can predict future prices to a certain degree of accuracy and thereby help in risk management.

The concept of "risk neutral world" assumes that all individuals are indifferent to risk. In such a world, investors require no compensation for risk, and the expected return from all securities is the risk-free interest rate. This assumption may seem a bit odd, but it makes perfect sense if you look through the window of Efficient Market Hypothesis (EMH). Under the EMF, markets are highly competitive and investors can hedge away all risks, thereby creating a portfolio of assets whose returns should be equal to risk-free rate. Once this assumption is in place, we can price any financial instrument because all such instruments should yield us the risk-free return. We know the risk-free return at any point of time as it is available to us from market - such as yields on Treasury Bills, Notes and Bonds or Repo instruments. The future price of a stock at time (t1), under the risk-neutral world, should be equal to the current price plus the risk-free interest rate for time period (t1 - t0).

Both the Efficient Market Hypothesis and Risk Neutral World concepts are arbitrary because the real markets are neither efficient nor risk-neutral. However, just like in mathematics (particularly Algebra) we can use arbitrary concepts to solve real world problems. Without these assumptions, it is impossible to price or value financial instruments and understand the risk involved. We use these arbitrary concepts to get the expected results and then we can tweek them based on the real-world scenarios. As you can see, under this concept, we are not predicting the stock price, we are estimating it under the risk-neutral world. In other words, we created a world of our own using various assumptions and in this world we are able to estimate the future prices.

The obvious question now is: are these estimates under the arbitrary world any good in real world? The answer is yes, to a certain extent (what extent?.. don't have the answer for this as of now. I am still doing research on this.) There are quite a lot of critics of these concepts and models, including the ones who teach and practice them for a living. One particular comment I recall is from a lecture session of Mr. Paul Wilmott, an Oxford qualified lecturer in Quantitative Finance. In one of his recent lectures on Quantitative Finance, he commented that on a scale of 1 to 10 (10 being high), he would give interest rate models a score of 4, FX models a score of 3 and credit models a score of 1, with an additional comment that he was being generous. This is coming from a well-acknowledged expert and practitioner on quantitative finance.

Anyway, the estimates may not be good right now but surely they would evolve in a better way over time. What is important to note is that the concept of random variables and random outcomes is a fundamental argument that cannot be denied. Randomness is vague, but it is a fact of life. The concept of dealing with randomness is not unique to stock analysis or finance. Many of these concepts were developed for other branches of science and humanities. We are just trying to apply these concepts to stock analysis.

The original observation of randomness was carried out by Robert Brown, a botanist, while observing the movement of pollen in water, and the concept was named after him as "Browian Motion". The related concept that is applicable in finance is called as "Stochastic process". Any variable whose value changes over time in an uncertain way is said to follow a stochastic process. The stochastic processes can be discrete time or continuous time, and discrete variable or continuous variable. In essence, we have four varieties of stochastic processes - discrete time and discrete variable, discrete time and continuous variable, continuous time and discrete variable and continuous time and continous variable. In general, the continous time and continous variable stochastic process is widely used. Based on this process, there are many models that have been created over the years to describe the randomness and estimate stock prices. The following are some of the models.

1. Binomial Trees - calculates the price of options under the assumptions of risk-neutral world


2. The Markov Property or Model - It is one where only the present value of the variable is relevant for predicting the future. The past history of the variable and the way in which the present has emerged from the past is irrelevant.


3. Wiener Processes - It is a process which describes the evolution of a normally distributed variable.


4. Ito's Lemma - It is an extension of the Wiener Process that describes the drift and variance of stock prices.


5. Monte-Carlo Simulation - Used to simulate the behaviour of the variable by dividing the time interval into small time steps and randomly sampling possible paths for the variable.


6. Black-Scholes-Merton model of option pricing - used for pricing option and other contingent contracts with the assumption that the stock prices follow a geometric brownian motion.

All of these models assume that the stock prices are random and follow a geometric brownian motion and has a lognormal distribution. An explanation of these terms is beyond the scope of this article. Maybe I will cover them in a separate article. For now, let's note that each model (listed above) is not a complete model in itself. They are essentially concepts that try to incorporate the randomness and describe it in some meaningful way. For example, a Wiener process follows the Markov property, Ito's Lemma is an extension of the Wiener property, which uses partial-differential calculus to describe what happens to a random variable at an infinitesimal small change in time, the Monte-Carlo simulation is a Markov process or property, and the Black-Scholes-Merton model relies on Ito's Lemma.

These models (and others which are being developed) are various ways to deal with random variables and random outcomes. After all, accepting the fact that we cannot predicting anything for certain is a good thing because it forces us to think differently and find out workable solutions.

END OF MY NOTES