Probability toolbox for quant finance
Background
Probability theory and related concepts are one of the key building blocks for a variety of concepts in quant finance. Most of us had the chance to learn probability concepts at different levels during our academic years and also on projects on the job. I am of the view that probability theory is one of the main pillars (other pillars include differential equations, integral calculus, linear algebra etc.) for a thorough understanding of quant finance fundamentals.
There are a lot of great reference books on probabilities, however, at times these references may become heavy from a math perspective; thus a reader who may not come from a math background may find a few concepts to be a bit tricky to relate to from a practical application standpoint. I will try to keep this article light on mathematical concepts and as far as possible avoid usage of too many dreaded notations that we commonly come across in math and probability references!
Also, all examples mentioned in this article will be specific to banking and finance. However, the concepts that we discuss may be used for applications in other industries too, for example just to name few — industrial engineering, operations research etc.
Why this fuss on probabilities in quant finance?
We begin with an idea of a random variable. From the name random we can guess that the changes in such a variable are random in nature.
We rely on probabilities because there is an uncertainty associated with the way this quantity will behave. With ideas in probabilities that we will be discussing below, we can understand better the way financial assets (i.e. also random variable) behave. Probability concepts also help us comprehend advanced concepts like stochastic process modelling, simulations, pricing, martingale etc. which are very important for industry applications.
Therefore it would be safe to say that having a clear understanding of probability concepts can take us a long way in understanding quantitative finance.
Relating a random variable with a distribution function
We begin with the standard notation of tuple:
(Ω, I, P)
Where, Ω is defined as a sample space;
I: is an information set (also called Filtration set in some texts)
P: Probability measure (we may view this loosely like a set)
Technically, a random variable X can be viewed as a function or a mapping on a certain information set I. Thus, given a certain event the random variable will take on a certain numerical value.
Let the probabilities for this variable be defined by a distribution function given by G(x)
We can say G(x) = P(X<=x)
where, X is the random variable, and x is a observed value.
When the function G(x) is smooth and differentiable, we can define a density function given by g(x) such that
g(x) = dG/dx
In easy words, a density function helps us relate the outcomes with associated probabilities. This becomes a useful concept in modelling exercises for simulating say asset paths.
Distributions and moments
Every distribution has a set of properties that are of relevance to finance. We use the concept called as “moments” to understand the properties like shape, skewness etc. of distributions in addition to the mean and standard deviation.
Most of the times we use mean and variance to explain a certain distribution eg: N ~ (μ, σ2). A quick review of the first four moments. Generally for financial applications, these suffice most of the times:
a. First moment: Mean. Also represented as E[X] — the expected value of the random variable
b. Second moment: Variance. Also represented as E[X — E[X]]^2- the deviation of the values about the mean.
c. Skewness: Represents the symmetry of a distribution. Also represented as E[X — E[X]]^3 For a Gaussian distribution we have a nice symmetric distribution. Gaussian/normal distribution also happens to be a popular distribution to model asset returns. Speaking of modelling returns, we often leverage the central limit theorem and say that returns follow a normal distribution.
There may be certain distributions which can show a kind of skewness, i.e. one of the tails could be longer than the other. So what does this imply for finance, for instance risk management: a right skewed distribution has higher weights applied to the right tail viz-a-viz left and vice-versa for a left skewed distribution.
d. Kurtosis: Represents the height of the distribution. Also represented as E[X — E[X]]^4 . Related terminologies include leptokurtic (taller than a Gaussian); platykurtic (shorter than a Gaussian). We know area under the curve has to be 1 (or 100%), therefore the height of the distribution plays a role in deciding the thickness of tails. Tail thickness is important because compared to a normal distribution, distributions with a higher kurtosis place more weights to extreme outcomes in the tails of the distribution, something which is important again for example for risk measurement applications.
We won’t be discussing any further moments in this article. There are however other higher order moments that one may find in literature for reference.
Conditional Probability
Conditional probability explains the probability of observing a certain event given the information about another event. Represented for example by P(Stock Rise | favorable analysts call). This explains what the probability of the stock price rise is given that the analysts call was favorable on the stock.
Also in risk management in particular credit risk, we use this concept often. There is a concept called as default intensity in credit risk modelling, wherein we attempt to model the probability of default in time interval, given that a particular entity has survived uptil the previous time instance. There are multiple such examples used in finance. We often apply the Bayes rule to understand these conditional measures.
A few key distributions used frequently
a. Normal distribution — We all love this isn’t it?! The most popular and well know distribution for modelling asset returns
b. Log-normal distribution — A popular distribution used for modelling equity or FX prices as it guarantees that there cannot be negative values generated
c. Poisson distribution — Popular for credit risk modelling. It lends well to exponential treatment which makes it elegant to use in models
d. Binomial distribution — A famous discrete distribution used for modelling assets binomial moves in assets. For example, the binomial options pricing model happens to be an important application
e. Uniform distribution — Arguably, the most widely used out of the lot. Most softwares (eg: Excel, Python etc.) allow us to easily draw uniformly distributed numbers. These can be converted to a normally distributed number by running an inverse transform operation and can be consumed in our models
The above list is not exhaustive, in the interest of brevity, I have included only a few prominent ones. There are many more distributions that one will find in literature. Its however important though, to know which distribution to be used for what kind of an application.
Markov Processes for finance
In quant finance, we are often interested in the way an asset evolves over a period of time i.e. the sequence of random numbers observed at subsequent time steps.
The Markov process is a memoryless process i.e. the probability statement about X(t+s) will depend only on the latest observation x(t) and nothing else. This property of markov process comes handy for financial applications.
For example, if we have spot interest rate r(t). The idea of markov process says that r(t+s) depends only on r(t) and nothing else. Assuming a discrete time setup for simplicity, we can say:
r(t+s) — r(t) = E[(r(t+s) — r(t) | I(t)] + σ(I(t),t) . ΔW(t)
where, ΔW(t) is the random variable with variance Δ.
If we assume r(t) is a markov process, and that I(t) contains only the current and past values of r(t), then the conditional mean and variance will be functions of r(t) only. So we can say:
E[(r(t+s) — r(t) | I(t)] = μ(r(t), t) Δ; and
σ(I(t),t) = σ(r(t),t).
However, if interest rates were not Markov, these steps could not be followed, since the conditional mean and variance of the spot rate r(t) may potentially depend on observations other than the immediate time step in the past. Thus the markov process becomes relevant for such interest rate products.
Conclusion
In this article, we discussed the building blocks for probability theory. We also linked it with a few applications in quant finance. Probabilities continue to be a main pillar for gaining full clarity of math finance concepts, and this is a topic that must be pursued by every quant finance enthusiast. It would not be incorrect if I said, that not just quant finance, but also other analytical fields including data analytics, machine learning models etc. also use ideas of probabilities pretty extensively.
I had earlier written an article on Martingales, which also finds its grounding in probability theory. Interested readers may refer to the earlier article here: Introduction to Martingales for quant finance
The idea of this article was to introduce a few key concepts from probability theory that we use frequently in the industry. Interested readers are suggested to pursue further concepts on this topic so as to gain the required comfort on probabilities.
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