Thread by @muskk

(Thread) A basic math primer for people with non-math background (this will also help in understanding my post tomorrow on India VIX).

I’ll be simplifying a lot of math details here.

I’ll mostly talk on "expected value" and a bit on integration.
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Expected value is one of the most important terms in financial markets. When we want to find fair value or “price” of any financial derivative we mathematically try to find its “expected value”. We will define what it is later on. But first let’s talk about random variables.
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Random variables: When we talk of random variables we talk of what values/outcomes a variable can take and what is the probability of each of these outcomes. So, two important terms here: outcomes & their probabilities.

Nifty, BNF (and their vols etc) are all random variables
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Also,
Discrete = countable (you can possibly count all outcomes. Ex: outcomes when you roll a dice)

Continuous = uncountable (infinite/large number of outcomes i.e. cannot count all of them. Ex: real number 'x' lying between 2 & 3)

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Expected value: Coming back, expected value of a random variable is simply the "average" of all outcomes that a random variable takes during an "experiment" when we measure a very large number of them. Important thing to note here is average of ALL and NOT possible outcomes!
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Let’s take the PnL of an intra-trading strategy. If you repeat the same strategy a very large number of times and record the PnLs each time, the average of them is your Expected value.
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If the outcomes are countable, meaning you can actually count the number of outcomes then such random variables are called “discrete”. In this case each outcome has a probability associated with it.
And Expected value is in eq(1) below:

If the outcomes of a random variable are potentially infinite/very large in number then we call it a “continuous” random variable. In this case we talk of outcomes being in a particular “interval” rather than look at their actual individual values.
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And in this case a probability density function (pdf) is defined which measures the probability of this random variable being inside a given range. Say BN staying between 35k & 35.5k next expiry has a probability ‘p’ for example.
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An equation for the probability of a random variable X with pdf φ(x) and lying between a & b is given in eq(2) below

And the Expected value of the continuous random variable with pdf φ(x) is defined in eq(3)

For all practical purposes all indices are assumed to be continuous!
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Integration: Finally, when we say integration(∫) of a function all we mean, in crude terms, is area under the function when it's plotted vs x. We can approximate that with rectangles. So equation (3) can be approximated as eq(4) below. A visual demo is also given in the pic

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