Thread by @muskk

(Thread) Stochastic processes

Thought of posting a primer on stochastic processes that’ll be useful for any future posts on whether deriving Black’s formula for pricing calls/puts (my next post and should be a quick one) or discussing interpolation of vol surfaces (SVI) etc.

This should also help understanding my VIX derivation post better.

Any let's get started.

Any underlying variable, be it a Nifty/BankNifty, USDINR, crude etc, can be represented as a stochastic process with a drift and a diffusion(random) component.
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Think of a stochastic process as a random variable evolving with time OR a collection of random variables that have been gathered at different times (Usually all stochastic processes, expectations are always defined under some probability measure but I’m not touching on
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measures right now as it’ll complicate the topic under discussion. Will explain it in some future post).

The Weiner process or a standard Brownian motion is a fundamental unit to “drive” the diffusion component of an underlying variable.
So, what is a Weiner process? See below
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A couple of more properties/applications of a Wiener processes in below pic.

Now that we defined a Wiener process let’s propose a typical stochastic differential equation (SDE) for a stock Index.
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Let’s work on the future/forward (these aren’t the same but let’s treat them for now without much harm) instead of spot to ignore the drift component. The SDE is in eq (1) below:

Ito’s formula:
If we look at any function V of F_t i.e., V(F_t) and try writing a Taylor series expansion of it, and ignoring orders greater than two, we get the eq.2 below.

Here we assume V(F_t) is atleast twice-differentiable (i.e., second derivative is defined).
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Difference here between standard calculus, where F is deterministic, and in this case where F is stochastic comes with how [dF_t ]^2 is treated. In standard calculus this term can be very small and often ignored. But with F_t stochastic let’s evaluate this term
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.., also called "quadratic variation" of F_t also represented as 〖<F>〗_t:

From (1), see pic below.

That is the Ito’s formula for the index assumed to follow the SDE in (1)

In the next post I’ll derive the Black’s formula

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