(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.
+
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
+
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
+
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.
+
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).
+
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
+
.., 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

(END)

• • •

Missing some Tweet in this thread? You can try to force a refresh
 

Keep Current with ExoticQuant

ExoticQuant Profile picture

Stay in touch and get notified when new unrolls are available from this author!

Read all threads

This Thread may be Removed Anytime!

PDF

Twitter may remove this content at anytime! Save it as PDF for later use!

Try unrolling a thread yourself!

how to unroll video
  1. Follow @ThreadReaderApp to mention us!

  2. From a Twitter thread mention us with a keyword "unroll"
@threadreaderapp unroll

Practice here first or read more on our help page!

More from @muskk

25 Oct
#Distribution Below is how an index return distribution can "potentially" evolve with time AS OBSERVED at starting time t=0.

As an example, one can view this as a potential #Nifty return distribution with PDFs given by Nov month end options (t=1), Dec month end options (t=2)..
+ Image
and so on (T=1 can be weekly also but I reckon weekly distributions won't look that smooth based on what I observed of option price/IV behavior).

Things to note:

At t=0 nothing is random, everything deterministic and pdf is a dirac-delta function.

+
And with time, probability of index moving away from its mean (colored with orange ticks) goes up and so pdf spreads wider and it's peak value keeps coming down in order to assign more weight to returns away from its mean.

+
Read 4 tweets
17 Oct
(Thread) Forecasting

Once upon a time (many many many years ago!) one of my friends was asked to implement a forecasting project as the last stage of getting an offer from a prop trading firm in Europe. Below is the problem statement:
+
Input data consists of (several hours of) trade and order book data for a listed product.
Order book data consists of time, bid/offer price and size resp. whereas the trade data
consists of time, price and volume.
Objective: Build a quantitative model to trade this instrument.
+
My friend was a uni. chicago booth grad but had a lazy arse so I helped him implement it with the help of a common HFT friend. We both knew shit and the project was all implemented through the HFT guy’s guidance. I’m just presenting the report below. Check out.
+
Read 5 tweets
29 Aug
Thread on deriving India VIX!

Get your pen and paper ready!

The NSE India VIX white paper (link below) only gives the formula and we will derive it in this thread. That'll be the only focus of this thread with more in future threads.
www1.nseindia.com/content/indice…
This is going to be mathematical and my post yesterday about expectation and integration should help. But I’ll try to reduce jargon and leave out unnecessary mathematical details. Some topics such as stochastic processes have been touched upon here. Will post more on that later
+
Let’s say f(x) is any function of a stock (or any other tradable underlying) ‘x’ and whose 1st & 2nd derivatives exist. Following on from the derivation last time of the PDF of any underlying,
‘x’ has a PDF, φ(k), given in fig below.
Image
Read 12 tweets
28 Aug
(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.
+
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.
+
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
+
Read 11 tweets
21 Jul
(Let’s do some math!) Thread on how to retrieve probability density function (PDF) of any underlying from its option prices. We will use this result later on in another thread I’ll post in the future to derive an equation for VIX.

Let's go...
Let’s first look at equation to price a call option at any time t, maturing at time T and with Strike K:

(refer equation1 pic below)

here F is forward, E[] is expectation & B(t,T) is discount factor. I’m excluding a few math details like measures & numeraires to keep it simple.
Let φ be the probability density function (PDF) of the underlying we are trying to recover. Let’s try to solve the expectation above (ignoring discount factor and other parameters C depends on to make equations look simpler)

(refer equation 2 pic below)
Read 7 tweets

Did Thread Reader help you today?

Support us! We are indie developers!


This site is made by just two indie developers on a laptop doing marketing, support and development! Read more about the story.

Become a Premium Member ($3/month or $30/year) and get exclusive features!

Become Premium

Too expensive? Make a small donation by buying us coffee ($5) or help with server cost ($10)

Donate via Paypal

Thank you for your support!

Follow Us on Twitter!

:(