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Aug 29, 2021 12 tweets 4 min read Read on X
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
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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
Replacing φ(x) and splitting the integral into two with the x-axis split at forward “F”. We use (OTM) puts for the integral below F and (OTM) calls for integral above F. We proceed as follows (you can ignore the below if not interested in details): ImageImage
Hence, we have our most important result in the pic below.

This is called "static replication" of the valuation of the european payoff f(x). Image
Based on this any twice-differentiable European payoff can be priced using OTM calls and puts with weights (f^'' (k)) determined by the form of the payoff.

This result holds not only for stocks but also fixed-income instruments and those from other asset classes.
This result holds not only for stocks but also fixed-income instruments and those from other asset classes. Let’s look at a few examples quickly.

YOU CAN SKIP THIS SECTION IF FOCUS IS ONLY ON VIX
+ ImageImage
Now let’s look at the “Log” contract: f(x)=Log(x/F)
In this case,

+ Image
Now let’s look, very briefly at the (stochastic) process for the underlying (I’ll cover stochastic processes, without being boring, in a separate thread but for now just focus on the bottom line!)
+ Image
Now that we have already deduced the first equation pic below we proceed from that point,

+ Image
Hence we have our VIX below! Finally! (This has been painful… writing all the above in Word Equation editor!)

+ Image
Remember above derivation is model independent i.e. no assumptions made on distribution of underlying.

Next, we will try to use the above formula to get some actual values of VIX and look at it’s connection with expected fwd realized variance i.e. Variance swaps!

(END)

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More from @muskk

Oct 22, 2022
(Thread) Event Volatility & Implied move

Objective: an event coming up next day and we want to calculate the expected move in the index/stock implied by option prices

Bro @l_thorizon mentioned on a call about implied move formula the other day (not for the first time)
+
..so thought it's time to derive it.
(it’s been ages since I posted a thread so excuse any mistakes pls)

Events can be FED/RBI policy meetings, CPI data release, stock earnings. You can compute these for (say) Reliance, ICICIbank this weekend or for Nifty(or S&P500 if you’re
+
into it) before 2nd Nov Fed meeting and see if they’re of any use.

Event Volatility:

We need ATM options with two different expiries T1 and T2 say T2>T1
Event vol, ev = sqrt (365) * sqrt [ IV(T1)^2 * T1/365 - FV(T1,T2)^2 * (T1-1)/365]
Where,
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Read 7 tweets
Apr 18, 2022
(quick thread)
Option strategies from probability density functions (pdfs)

Ref: papers.ssrn.com/sol3/papers.cf…

I'll summarize the paper...

The author presents an easy way to transform one's expectation of how returns distribution will pan out in the future("user-expected"
+
distribution) given current market implied distribution into appropriate trading strategies.

Let's say b(x) is what you expect index/stock returns "x" to be distributed in the future and m(x) is current market implied pdf based on prevailing option prices (see thread

+
below to get market implied density function from option prices. To get a continuum of option prices fit an SVI IV surface on option price grid)

Then the strategy that needed to be entered is simply given by:

f(x) = b(x) / m(x)

+
Read 5 tweets
Dec 30, 2021
#OptionExpiry
Quick note on Nifty/BankNifty options

Asian/Averaging options

Given option prices are settled based on average of last half hour prices (correct me if interval is wrong) perhaps in 5min intervals (trust me I couldn't find an NSE doc on this piece🤦‍♂️) our options
+
are actually Asian options.
So,

Asian call payoff : max(avg(S(ti) - strike,0) where S(ti) last 30mins, 5 min intervals prices (say)
and not european i.e. max(S(T) - strike,0)

The impact of averaging feature will be much less when we are away from expiry but gain significance
+
As we are closer and specially in the second half of the expiry.

Now I won't bore you with the math of pricing them (which isn't trivial and done by matching 1st, 2nd moments of an assumed log-normal distribution of the averaged price and finding parameters i.e. mean/variance
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Read 4 tweets
Dec 27, 2021
I usually don't do the "role model" thingy but if there's anyone I look up to and want to be like (in a couple of years time!) that's Benn Eifert @bennpeifert

@mnopro pointed me to his handle a couple of months ago and I recently watched his mixing with models YT video & gone
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through some of his posts and given my experience interacting with HF friends or interviewing with HFs during my 10years working in London I can honestly say he's right up there with the best. Don't get me wrong there's absolute garbage as well on the HF side (and I'll share
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my thoughts on that some other time), but the CIO folks are altogether different class and Benn seems no different. The clarity with which he speaks about anything Vol related is mind-boggling(!) & something I'd like to aspire to.

(Hello Benn - this is not fake praise
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Read 4 tweets
Dec 20, 2021
(Thread) One of my more important posts from a trading point of view (hardly any quanty stuff in this one). Might be relevant now hence posting.

Just so it's clear for anyone wanting to trade call ratio spreads in the future in a bear market:

(this is my approach anyway..)

Best time to enter zero-cost call ratios or a ladder (shorts at multiple strikes) is when IVs are shooting up combined with mkt going down. Today was a classic example.

If we see the kind of fall we saw today in Nifty keep long leg 100pt OTM and
+
short leg atleast 300pts further away and you can increase the ratio higher than 1:3 (you'll have to, to keep it zero-cost). You can make it credit depending on your risk-appetitie and your skill in adjusting these.

So when you enter this you will be net short vega.
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Read 10 tweets
Dec 14, 2021
#Strangles A little note on getting into strangles in a bear market scenario

If you're an intra-day trader selling delta-neutral strangles on non-expiry days in the morning and holding them till day end then be careful of the following when market is expected to be bearish.
+
Given short strangles are short vol trades (-ve vega) the view is not just on benefitting from theta but more importantly on vols going down during the day. So in a bearish scenario when vols are expected to go up with market going down your strangle position
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will get affected in the following way:

- Loss due to vols going up with this being a -ve vega position. There are other vol factors that you'll be short on such as skew (below), convexity of the vol surface and these increase as well affecting the position negatively.

+
Read 7 tweets

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