MODERN QUANTITATIVE FINANCE
Introduction to quantitative finance Part 27: Lattice Theory is structurally a subset of Probability Theory
We consider applications in Finance
Section I: We start with the old battle
From the article Frequentism in Robotics and Quantitative Finance, we have seen that Frequentist Statistics have the sampling distribution, while Bayesian Statistics have the likelihood function. In this article, we go for further division; we have Frequentist Probability as the result of the limit of the relative frequency of an event happening. On the other hand, plausibility (aka Bayesian Probability) from the ‘plausible reasoning’ school is the quantification of how plausibile a proposition is — given our state of knowledge or ignorance (where 0 corresponds to impossibility and 1 corresponds to certainty). Rigorous reader, since the Bayesian school underpins most human thinking, we have the following two verses of MC’s track My All cementing how the plausibility is updated under changing knowledge states.
[Verse 1 — Heart updates MC’s love-sickness 😊]
I am thinking of you…
In my sleepless so-li-tude tonight.
If it’s wrong to love you…
Then my heart just won’t let me be right.
’Cause I’ve drowned in you,
And I won’t pull through,
Without you by my s-i-i-i-i-i-de.[Verse 2 — Distance updates MC’s sanity 😊]
Baby, can you feel me?
Imagining I’m looking in your eyes.
I can see you clearly…
Vividly emblazoned in my mind.
And yet you’re so far,
Like a distant star,
I’m wishing on tonight.
Section II: Let’s say we hypothetically consider modelling prices, instead of how we typically do the modelling of returns
Probability theory is an extension of logic; in fact, it contains classical logic as a special case.
- Classical logic is founded on the idea of truth tables.
- The reduction to atoms in the process of finding ‘if a statement is true’ (in classical logic) creates a lattice (with ordered elements).
- A fair coin game can be represented using a lattice. So, in Derivatives Finance, the Binomial Option Pricing Model (from the Binomial lattice) is essentially built on the same mathematical structure as repeated coin tossing.
- Remember, a fair coin game is uncertainty without structural advantage, whereas a casino game is uncertainty with structural advantage.
Historically, the Probability theory that is used nowadays in Actuarial Science, Data Science, and Quantitative Finance takes its roots in 17th century France, when a French aristocrat asked a friend for gambling advice. Specifically, the aristocrat was Antoine Gombaud, who wrote about proper behaviors and lifestyles under the pen name Chevalier de Méré, and the friend was Blaise Pascal, who by age thirty had already proved the existence of a vacuum and invented the Pascaline (one of the first mechanical calculators). De Méré asked Pascal to solve a problem that had been debated for centuries: If two people are playing a ‘fair game’ in which the winner is the first to reach a certain number of points, but the game is interrupted before either of them has won, what’s the fairest way to divide up the prize money? In the solving of the problem, Pascal (with some help from Pierre de Fermat) contributed in laying the foundations of Modern Probability theory — one that affects all our lives today, even people who don’t gamble!
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Frequentists say that prob of a fair coin landing on either side approaches 0.5 as trials approach infinity, while Bayesian say it is so because you must assign equal probabilities to what you don’t know. Now, being ‘fooled by randomness’ is a danger. So let’s now look at the following example to understand the connection between gambling in a fair game and stock trading.
In the following R code, the sample function will be used to obtain the outcome of tossing a coin. If we do not use replace, we won’t have any numbers after two coin tosses! Therefore, the replace parameter must be indicated as TRUE. The cumsum function calculates the cumulative sum after each sequence.
tossingcoin <- sample(c(-1,1), 1000, replace = TRUE)
cumsum(tossingcoin)
plot(cumsum(tossingcoin), type = 's')We obtain the following chart.
According to the results provided by R, we won $6 [this is the last element’s numeral and can just be seen from just the cumsum(tossingcoin) values instead of the plot] after tossing the coin 1000 times. Note that the figures will change if the code is re-run.
💣In Python, we have the following equivalent code.
import numpy as np
import matplotlib.pyplot as plt
# Generate 1000 coin tosses: -1 for tails, +1 for heads
tossingcoin = np.random.choice([-1, 1], size=1000, replace=True)
# Compute cumulative sum
walk = np.cumsum(tossingcoin)
# Plot as a step function (equivalent to type='s' in R)
plt.step(range(len(walk)), walk, where='post')
plt.xlabel("Number of Tosses")
plt.ylabel("Cumulative Sum")
plt.title("Random Walk from Coin Tosses")
plt.grid(True)
plt.show()The following is the resulting chart.
The graphs indicate random behaviour of the cumulative sum as we cannot control the results provided by ‘the tossing a coin’ event. Now, stock price evolution also exhibits a random behaviour.
Then, we note that the rudimentary example should be regarded with high precaution as stock prices do not exhibit a constant rate of evolution. Modelling stock price movements is a complex process as their evolutions are influenced by many factors — as seen in the article Introduction to quantitative finance Part 11: What makes asset prices move?
Sources:
YouTube.
Worldquant.
Internet articles.
