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.
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):
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).
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
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Now let’s look at the “Log” contract: f(x)=Log(x/F)
In this case,
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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!)
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Now that we have already deduced the first equation pic below we proceed from that point,
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Hence we have our VIX below! Finally! (This has been painful… writing all the above in Word Equation editor!)
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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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#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)..
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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.
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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.
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:
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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.
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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.
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(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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(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)